1. Introduction

Suppose \(\mathbb{S}^{n-1}\) for \(n\geq 2\) is the unit sphere in \(\mathbb{R}^{n}\) equipped with the normalized Lebesgue measure \(\text{d}\sigma\). Further suppose that \(\Omega\) is a homogeneous function of degree zero on \(\mathbb{R}^{n}\) satisfying \(\Omega\in L^{1}(\mathbb{S}^{n-1})\) and

For \(0< \rho< n\), the parametrized Marcinkiewicz integrals is defined as; $$\mu_{\Omega}^{\rho}(h)(x)=\left(\int^{\infty}_{0}|F^{\rho}_{\Omega,t}(h)(x)|^{2} \frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2},$$ where \(F^{\rho}_{\Omega,t}(h)(x)=\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h(y)\text{d}y,t>0.\)

For \(m\in\mathbb{N},b\in \mbox{BMO}(\mathbb{R}^{n}),\) the higher-order commutator of parametrized Marcinkiewicz integral is defined as;

It is easy to see that when \(\rho=1,\) and \(\mu^{\rho}(h)=\mu^{1}(h)\), then (2) is the classical Marcinkiewicz integral \(\mu(h)\) introduced by Stein in [1]. It has been proved in [1] that if \(\Omega\in Lip_{\gamma}(\mathbb{S}^{n-1})(0< \gamma\leq1)\) and \(\Omega\) is continuous, then the operator \(\mu(h)\) is of the type \((q,q)\mbox{for}1< q\leq2\) and of the weak type \((1,1)\). Benedek et al., [2] proved that if \(\Omega\in C^{1}(\mathbb{S}^{n-1})\), then \(\mu(h)\) it is of type \((q,q)\) for any \(1< q\leq \infty\). The \(L^{p}\) boundedness of the \(\mu(h)\) has been studied in [1, 3, 4, 5].

In 1960, Hörmander [4] introduced the parametrized Marcinkiewicz integral operators proved that if \(\Omega\in Lip_{\gamma}(\mathbb{S}^{n-1}),0< \gamma\leq1,\) then it is of strong type \((q,q)\) for \(1< q\leq2\). Sakamoto and Yabuta [6] proved the boundedness of the operator \(\mu^{\rho}(h)\) on \(L^{q}(\mathbb{R}^{n})\). Shi and Jiang [7] considered the weighted \(L^{q}-\)boundedness of parametrized Marcinkiewicz integral operator and its higher order commutator. Note that the Littlewood-paley \(g\)-function played very important roles in harmonic analysis and the parameterized Marcinkiewick integral is a special case of the Littlewood-paley \(g\)-function. Many authors studied properties of \(\mu^{\rho}(h)\) on different function spaces, for examples [8, 9, 10, 11, 12, 13, 14].

In the last three decade, the generalized Orlicz-Lebesgue spaces and the corresponding generalized Orlicz-Sobolev spaces have been extensively studied by many researchers. The variable Lebesgue spaces are special cases of generalized orliz spaces which introduced by Nakano in [15] and developed in [16, 17]. In addition, for properties of \(L^{p(\cdot)}\) spaces we refer to [18, 19, 20], and the fundamental paper of Kováčik and Rákosník [21] appeared in 1990. By virtue of this works many function spaces appeared [22, 23, 24, 25]. Recently, in 2015, Lijuan and Tao established the Herz spaces with two variable exponents \(p(\cdot),q(\cdot)\) in the paper [26].

The main purpose of this work is to discuss the boundedness of parameterized Marcinkiewicz integral and it's higher order commutators with rough kernels on Herz spaces with two variable exponents. The boundedness of higher order commutator generated by BOM function and parameterized Marcinkiewicz integral is also obtained.

Let \(\Upsilon\) be a measurable set in \(\mathbb{R}^{n}\) with \(|\Upsilon|> 0 \).

Definition 1. Let \(p(\cdot): \Upsilon \rightarrow {[1,\infty)}\) be a measurable function. The Lebesgue space with variable exponent \(L^{p(\cdot)}(\Upsilon)\) is defined by $$L^{p(\cdot)}(\Upsilon)= \left\{{ h \mbox{is measurable} : \int_{\Omega}\left(\frac{|h(x)|}{\eta}\right)^{p(x)} dx < \infty} \mbox{for some constant } \eta > 0\right\}$$

The space \(L _{loc}^{p(\cdot)} {(\Upsilon)}\) is defined by $$L_{loc}^{p(\cdot)} {(\Upsilon)}= \{ \mbox {h is measurable} : h\in {L^{p(\cdot)} {(K)}}\mbox{for all compact}K\subset{\Upsilon}\}$$ The Lebesgue spaces \(L^{p(\cdot)} {(\Upsilon)}\) is a Banach spaces with the norm defined by We denote $$p_{-}= \text{essinf} \{p(x): x \in \Upsilon\},p_{+}=\text{ess}\sup \{p(x): x \in \Upsilon\},$$ then \(\mathcal{P}(\Upsilon)\) consists of all \(p(\cdot)\) satisfying \(p_{-} > 1\) and \(p_{+} < \infty\). Let \(M\) be the Hardy-Littlewood maximal operator. We denote \(\mathcal{B}(\Upsilon)\) to be the set of all function \(p(\cdot)\in \mathcal{P}(\Upsilon)\) such that \(M\) is bounded on \(L^{p(\cdot)}(\Upsilon)\). Now, let us recall the definition of Herz spaces with variable exponents.

Definition 2.[26] Let \(\alpha \in\mathbb{R}^{n} ,q (\cdot),p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\). The homogeneous Herz space with variable exponent \(\dot{K}_{p(\cdot)}^{\alpha,q(\cdot)}(\mathbb{R}^{n})\) is defined by $$ \dot{K}_{p(\cdot)}^{\alpha, q(\cdot)}(\mathbb{R}^{n})= \{h\in {L_{loc}^{p(\cdot)}}(\mathbb{R}^{n}\backslash\{0\}) : \|h\|_{\dot{K}_{p(\cdot)}^{\alpha, q(\cdot)}(\mathbb{R}^{n})}< \infty \},$$ where \begin{eqnarray*} \|h\|_{\dot{K}_{p_{(\cdot)}}^{\alpha,q(\cdot)}(\mathbb{R}^{n})}&=&\left\| \{ 2^{k \alpha}|h\chi_{k}|\}_{k=0}^{\infty}\right\|_{l^{q(\cdot)}(L^{p(\cdot)})}=\inf\left\{ \eta> 0 : \sum\limits_{k=-\infty}^{\infty} \left\| \left( \frac{2^{k\alpha}|h\chi_{k}|}{\eta}\right)^{q(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q(\cdot)}}}\leq 1\right\}. \end{eqnarray*}

Remark 1.Let \(v\in \mathbb{N},a_{v}\geq 0,1\leq p_{v} < \infty\), then $$\sum\limits_{v=0}^{\infty} a_{v}\leq \left(\sum\limits_{v=0}^{\infty} a_{v} \right)^{p_{\ast}},$$ where $$ p_{\ast}= \left\{\begin{array}{ll} \min\limits_{v\in \mathbb{N} }p_{v},\quad\quad\quad\sum\limits_{v=0}^{\infty} a_{v}\leq 1,\\ \max\limits_{v\in \mathbb{N} }p_{v},\quad\quad\quad\sum\limits_{v=0}^{\infty} a_{v}>1. \end{array}\right.$$

Remark 2.[26]

1. If \( q_{1} (\cdot),q_{2}(\cdot)\in \mathcal{P} (\mathbb{R}^{n})\) satisfying \( (q_{1})_{+}\leq (q_{2})_{+}\), then \({K}_{p(\cdot)}^{\alpha, q_{1}(\cdot)}(\mathbb{R}^{n}) \subset {K}_{p(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n}),
\dot{K}_{p(\cdot)}^{\alpha, q_{1}(\cdot)}(\mathbb{R}^{n}) \subset \dot{K}_{p(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n}).\)
2. If \( q_{1} (\cdot),q_{2}(\cdot)\in \mathcal{P} (\mathbb{R}^{n})\) and \( (q_{1})_{+}\leq (q_{2})_{-}\), then \(\frac{q_{2}(\cdot)}{q_{1}(\cdot)}\in \mathcal{P}(\mathbb{R}^{n})\) and \(\frac{q_{2}(\cdot)}{q_{1}(\cdot)}\geq 1 \).

By Remark 1, for any \( h\in \dot{K}_{p(\cdot)}^{\alpha, q(\cdot)}(\mathbb{R}^{n}) \), we have \begin{eqnarray*} \sum\limits_{k=-\infty}^{\infty} \left\| \left( \frac{2^{k\alpha}|h\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}} &&\leq \sum\limits_{k=-\infty}^{\infty} \left\| \left( \frac{2^{k\alpha}|h\chi_{k}|}{\eta}\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}^{p_{v}}\leq \left\{ \sum\limits_{k=-\infty}^{\infty} \left\| \left( \frac{2^{k\alpha}|h\chi_{k}|}{\eta}\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}^{p_{h}} \right\}^{p_{*}}\leq 1; \end{eqnarray*} where \begin{eqnarray*} &p_{v}&= \left\{\begin{array}{ll} (\frac{q_{2}(\cdot)}{q_{1}(\cdot)})_{-},\quad\quad\quad\frac{2^{k\alpha}|f\chi_{k}|}{\eta} \leq 1,\\ (\frac{q_{2}(\cdot)}{q_{1}(\cdot)})_{+},\quad\quad\quad\frac{2^{k\alpha}|f\chi_{k}|}{\eta}>1, \end{array}\right . \end{eqnarray*} and \begin{eqnarray*} &p_{*}&= \left\{\begin{array}{ll} \min\limits_{v\in \mathbb{N} }p_{v},\quad\quad\quad\sum\limits_{v=0}^{\infty} a_{v}\leq 1,\\ \max\limits_{v\in \mathbb{N} }p_{v},\quad\quad\quad\sum\limits_{v=0}^{\infty} a_{v}>1. \end{array}\right . \end{eqnarray*} This implies that \(\dot{K}_{p(\cdot)}^{\alpha, q_{1}(\cdot)}(\mathbb{R}^{n}) \subset \dot{K}_{p(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n})\).Similarly, we get \({K}_{p(\cdot)}^{\alpha, q_{1}(\cdot)}(\mathbb{R}^{n}) \subset {K}_{p(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n}).\)

Definition 3. For all \(0< \gamma \leq 1,\) the Lipschitz space \(\dot{\Lambda}_{\gamma}(\mathbb{R}^{n})\) is defined by $$\dot{\Lambda}_{\gamma}(\mathbb{R}^{n})=\left\{h:\|h\|_{\dot{\Lambda}_{\gamma}(\mathbb{R}^{n})}= \sup\limits_{x,y\in \mathbb{R}^{n};x\neq y}\frac{|h(x)-h(y)|}{|x-y|^{\gamma}}< \infty\right\}.$$

Definition 4. The BMO function and BMO norm are defined by \begin{align*} \mathrm{BMO}(\mathbb{R}^{n})&:=\left\{b\in L^{1}_{loc}(\mathbb{R}^{n}):\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}< 0\right\},\\ \|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}&:=\sup\limits_{Q:\text{cube}}|Q|^{-1}\int_{Q}|b(x)-b_{Q}|\text{d}x. \end{align*}

From here, we suppose that \(B_{k}=\{ x\in\mathbb{R}^{n} : |x|\leq 2^{k}\},\) and \( C_{k}= B_{k}\backslash B_{k-1} , \chi_{k}= \chi_{C_{k}} , \) \; \( k \in{\mathbb{Z}}.\)

2. Preliminary Lemmas

Proposition 1. [27] Let a function \(p(\cdot): \mathbb{R}^{n} \rightarrow [ 1 , \infty).\) If \(p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) satisfies

and then \(p(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\).

Lemma 1. [21] (Generalized Hölder Inequality) Let \(p(\cdot),p_{1}(\cdot),p_{2}(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\), then

1. for every \(h\in L^{p_{1}(\cdot)}(\mathbb{R}^{n})\text{and}g\in L^{p_{2}(\cdot)}(\mathbb{R}^{n})\), we have \(\int_{\mathbb{R}^{n}}|h(x) g(x)| dx \leq C\|h\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{p'(\cdot)}(\mathbb{R}^{n})},\) where \(C_{p}=1+\frac{1}{p_{-}}-\frac{1}{p_{+}}\);
2. for every \(h\in L^{p_{1}(\cdot)}(\mathbb{R}^{n}),g\in L^{p_{2}(\cdot)}(\mathbb{R}^{n})\),when \(\frac{1}{p(\cdot)}=\frac{1}{p_{2}(\cdot)}+\frac{1}{p_{1}(\cdot)}\),we have \(\|h(x)g(x)\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq C \|g(x)\|_{L^{p_{2}}(\mathbb{R}^{n})}\|h(x)\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})},\) where \(C_{p_{1},p_{2}}=[1+\frac{1}{p_{1-}}-\frac{1}{p_{1+}}]^{\frac{1}{p_{-}}}\).

Lemma 2. [18, [19] Let \(p(\cdot) \in \mathcal{B}(\mathbb{R}^{n})\). If there exists a positive constants \(C,\) \(\delta_{1},\) \(\delta_{2}\) such that \(\delta_{1},\delta_{2}< 1\), then, for all balls \(B\subset\mathbb{R}^{n}\) and all measurable subset \(R\subset B,\) we have $$\frac{\|\chi_{R}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\leq C \frac{|R|}{|B|}, \frac{\|\chi_{R}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p_{u}^{\prime}(\cdot)}(\mathbb{R}^{n})}}\leq C\left(\frac{|R|}{|B|}\right)^{\delta_{2}},\frac{\|\chi_{R}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}{ \|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\leq C\left(\frac{|R|}{|B|}\right)^{\delta_{1}}.$$

Lemma 3.[28] Let \(p(\cdot) \in \mathcal{B}(\mathbb{R}^{n}),\) then there exists a constant \(C > 0\) such that for any balls B in \(\mathbb{R}^{n}\), we have $$ \frac{1}{|B|}\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \|\chi_{B}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}\leq C. $$

Lemma 4. [29] Let \(p(\cdot) \in \mathcal{B}(\mathbb{R}^{n}),\) and \(b\in \mathrm{BMO}(\mathbb{R}^{n})\). If \(i,j\in\mathbb{Z}\) with \(i< j\), then we have

1. \(C^{-1}\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})} \leq\sup\limits_{B}\frac{1}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\|(b-b_{B})\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})};\)
2. \(\|(b-b_{B_{i}})\chi_{B_{j}}\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C( j-i)\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{q(\cdot)}(\mathbb{R}^{n})}.\)

Lemma 5.[26] Let \(p(\cdot),q(\cdot) \in\mathcal{P}(\mathbb{R}^{n}).\) If \(h\in L^{p(\cdot)q(\cdot)}\), then $$ \min ( \|h\|_{L^{p(\cdot)q(\cdot)}}^{q_{+}}, \|h\|_{L^{p(\cdot)q(\cdot)}}^{q_{-}} )\leq\||h|^{q(\cdot)}\|_{L^{p(\cdot)}}\leq\max ( \|h\|_{L^{p(\cdot)q(\cdot)}}^{q_{+}}, \|h\|_{L^{p(\cdot)q(\cdot)}}^{q_{-}}). $$

Lemma 6.[30] Let \(a>0,0< d \leq s,1\leq s\leq\infty\) and \(\frac{-sn+(n-1)d}{s}< v< \infty,\) then $$\left(\int_{|y|\leq a|x|}|y|^{v}|\Omega(x-y)|^{d}\mbox{d}y\right)^{1/d}\leq C |x|^{(v+n)/d}\|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}.$$

Lemma 7.[31] Let the variable exponent \(\tilde{q}(\cdot)\) is defined by \(\frac{1}{p(x)}=\frac{1}{\tilde{q}(x)}+\frac{1}{q}(x\in\mathbb{R}^{n})\), then we have $$\|hg\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq C \|g\|_{L^{q}(\mathbb{R}^{n})} \|h\|_{L^{\tilde{q}(\cdot)}(\mathbb{R}^{n})}.$$

Lemma 8. Let \(p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}),\Omega\in L^{s}(\mathbf{\mathbb{S}}^{n-1})\) and \(0<\rho0\) independent of \(h\), then \(\mu_{\Omega}^{\rho}\) is bounded from \(L^{p(\cdot)}\) to it self.

Lemma 9. Let \(b\in\mathrm{BMO}(\mathbb{R}^{n})\) and \(m\in\mathbb{N}\). Further let that \(p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}),\Omega\in L^{s}(\mathbf{\mathbb{S}}^{n-1})\) and \(0< \rho< n\). If there exists a constant \(C>0\) independent of \(h\), then \([b^{m},\mu_{\Omega}^{\rho}]\) is bounded from \(L^{p(\cdot)}\) to itself.

Lemma 10. Let \(b\in\dot{\Lambda}_{\gamma}(\mathbb{R}^{n}),0< \gamma\leq1,m\in\mathbb{N}\) and \(0< \rho< n.\) If \(q_{1}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) satisfies (4) and (5) in Proposition 1 with \(q^{+}_{1}< n/\gamma,1/q_{1}(x)-1/q_{2}(x)=\gamma/n,\Omega\in L^{s}(\mathbb{S}^{n-1})(s>q^{+}_{2})\) with \(1\leq r'< q^{-}_{2}\). Then the commutator \([b^{m},\mu^{\rho}_{\Omega}]\) is bounded from \(L^{q_{1}(\cdot)}(\mathbb{R}^{n})\) to \(L^{q_{2}(\cdot)}(\mathbb{R}^{n}).\)

Lemma 11. [32] Let \(p(\cdot)\in \mathcal{P}(\Omega)\) abd \(h:\Omega\times \Omega\rightarrow \mathbb{R}\) is a measurable function (with respect to product measure) such that, \(y\in \Omega,h(\cdot,y)\in L^{p(\cdot)}(\Omega)\), then we have $$\left\|\int_{\Omega}h(\cdot,y)dy\right\|_{L^{p(\cdot)}(\Omega)}\leq C \int_{\Omega}\left\|h(\cdot,y)\right\|_{L^{p(\cdot)}(\Omega)}dy.$$

3. Main Results

Theorem 1. Let \(0< \rho< n,0< v\leq1.\) Suppose that \(p_{1}(\cdot)\in\mathcal{B}(\mathbb{R}^{n}),\Omega\in L^{s}(\mathbb{S}^{n-1}),s>(p_{1}')_{+}\) and \(q_{1}(\cdot),q_{2}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) with \((q_{2})_{-}\geq(q_{1})_{+}\). If \(-n\delta_{1}-v-n/s< \alpha< n\delta_{2}-v-n/s\) with \(\delta_{1},\delta_{2}\) as defined in Lemma 2, then the operator \(\mu^{\rho}_{\Omega}\) is bounded from \( \dot{K}_{p_{1}(\cdot)}^{\alpha,q_{1}(\cdot)}(\mathbb{R}^{n})\) to \(\dot{K}_{p_{1}(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n})\) and from \(\left({K}_{p_{1}(\cdot)}^{\alpha,q_{1}(\cdot)}(\mathbb{R}^{n})\right)\) to \(\left({K}_{p_{1}(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n})\right)\).

Proof. Let \(h(x)\in \dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})\). Rewrite \(h(x)=\sum_{j=-\infty}^{\infty}h(x)\chi_{j}=\sum_{j=-\infty}^{\infty}h_{j}(x).\) From Definition 2, we have \begin{equation*} \|\mu^{\rho}_{\Omega}(h)\|_{\dot{K}^{\alpha,q_{2}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}=\inf\left\{\eta>0 : \sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }|\mu^{\rho}_{\Omega}(h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}} \leq1\right\}. \end{equation*} Since
\( \left\|\left(\frac{2^{k\alpha}|\mu^{\rho}_{\Omega}(h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}} \leq\left\|\left( \frac{2^{k\alpha}|\sum^{\infty}_{j=-\infty}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\sum^{3}_{i=1}\eta_{1i}} \right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\)\\ \(\leq\left\|\left( \frac{2^{k\alpha}|\sum^{k-2}_{j=-\infty}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{11}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}+\left\|\left( \frac{2^{k\alpha}|\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{12}} \right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}} +\left\|\left( \frac{2^{k\alpha}|\sum^{\infty}_{j=k+2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{13}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}, \)
where \begin{eqnarray*} \quad\quad\quad\quad\quad&\eta_{11}&=\left\|\left\{2^{k\alpha }|\sum^{k-2}_{j=-\infty}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}| \right\}^{\infty}_{k=-\infty}\right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})},\\ &\eta_{12}&=\left\|\left\{2^{k\alpha }|\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|\right\}^{\infty}_{k=-\infty} \right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})},\\ &\eta_{13}&=\left\|\left\{2^{k\alpha }|\sum^{\infty}_{j=k+2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|\right\}^{\infty}_{k=-\infty} \right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})}, \end{eqnarray*} and $$\eta=\eta_{11}+\eta_{12}+\eta_{13}=\sum^{3}_{i=1}\eta_{1i}.$$ Thus, $$\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }|\mu^{\rho}_{\Omega}(h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C.$$ Meanwhile, $$\|\mu^{\rho}_{\Omega}(h)\|_{\dot{K}^{\alpha,q_{2}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}\leq C\eta= C \sum^{3}_{i=1}\eta_{1i}.$$ To show Theorem 1, we only need to estimate \(\eta_{11},\eta_{12}\text{and}\eta_{13}\leq C \|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}\). To do this, denote \(\eta_{10}=\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}.\)
Step 1.For \(\eta_{12}\). From Lemma 5, we get

where $${(q^{1}_{2})_{k}}= \left\{\begin{array}{ll} (q_{2})_{-},\quad\quad\quad\left\|\left(\frac{2^{k\alpha } |\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\ (q_{2})_{+},\quad\quad\quad\left\|\left(\frac{2^{k\alpha } |\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1. \end{array}\right.$$ So, by using the Lemma 6, Remark 2 and \(h(x)\in \dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})\), we have \(\left\|\frac{2^{k\alpha}|h\chi_{k}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}\leq1\) and \(\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha } |h\chi_{k}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^\frac{{p_{1}(\cdot)}}{{q_{1}(\cdot)}}}\leq1\). Hence Which, together with \((p_{1})_{+}\leq(p_{2})_{-}\leq(q^{1}_{2})_{k}\) and \(q_{*}= \min\limits_{k\in N}\frac{(q^{1}_{2})_{k}}{(q_{1})_{+}}\) gives; Step 2. Now, let us deal with \(\eta_{11}\). Since \begin{eqnarray*} &\qquad|\mu_{\Omega}^{\rho}(h_{j})(x)|& := \left(\int^{\infty}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\mathrm{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\ &&\leq \left(\int^{|x|}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\ &&\quad+\left(\int_{|x|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\ &&:=\eta_{11}'+\eta_{11}''. \end{eqnarray*} Now we estimate \(\eta_{11}'\text{and}\eta_{11}''\). For \(\eta_{11}'\), note that \(x\in A_{k},y\in A_{j}\) and \(j\leq k-2.\) Since \(|x-y|\sim|x|\) so by virtue of the Mean Value Theorem, we have Substituting the inequality (9) into \(\eta_{11}'\) and by virtue of Minkowski's inequality, we deduced that Similarly, we obtain Combining the inequality (11) with Lemma 1, we get Now, consider \(\tilde{p}_{1}'(\cdot)>1\) and \(1/p_{1}'(x)=1/\tilde{p}'_{1}(x)+1/s\). Since \(s>(p_{1}')_{+}\), so by virtue of Lemma 1 and Lemma 8, we get By using (12), (13), Lemmas 1, 2, 3, 5 and \(\left\|\frac{2^{j\alpha } |h\chi_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)q_{1}}}\leq1\), we get where $${(q^{2}_{2})_{k}}= \left\{\begin{array}{ll} (q_{2})_{-},\quad\quad\quad\left\|\left(\frac{2^{k\alpha } |\sum^{k-2}_{j=-\infty}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\ (q_{2})_{+},\quad\quad\quad\left\|\left(\frac{2^{k\alpha } |\sum^{k-2}_{j=-\infty}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1. \end{array}\right.$$ Which, together with \((q_{1})_{+}< 1\) and \((p_{1})_{+}\leq(p_{2})_{-}\leq(q^{2}_{2})_{k}\) gives;\\ \( \sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha } |\sum^{k-2}_{j=-\infty}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C \left\{\sum^{\infty}_{j=-\infty} \left\|\left(\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}\right)^{q_{1}(\cdot)} \right\|_{L^{p_{1}(\cdot)q_{1}(\cdot)}}\sum^{\infty}_{k=j+2}2^{(k-j)(\alpha+v+n/s-n\delta_{2})} \right\}^{q_{*}}\) where \(q_{*}= \min\limits_{k\in\mathbb{N}}\frac{(q^{1}_{2})_{k}}{(q_{1})_{+}}.\) Since \(\alpha< n\delta_{2}-(v+n/s)\), so if \((q_{1})_{+}\geq1\) and \((q^{2}_{2})_{k}\geq(q_{2})_{-}\geq(q_{1})_{+}\geq1\) then by using Remark 2% correct remark number and applying the generalized Hölder's inequality, we get where \(q_{*}= \min\limits_{k\in\mathbb{N}}\frac{(q^{2}_{2})_{k}}{(q_{1})_{+}}.\) Hence we have Step 3. Finally, we estimate \(\eta_{13}\). For each \(x\in A_{j}\) and \(j\geq k+2\), we have \begin{align*} |\mu_{\Omega}^{\rho}(h_{j})(x)|&:= \left(\int^{\infty}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\ &\leq \left(\int^{|y|}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\ &\quad+\left(\int_{|y|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\ &:=\eta_{13}'+\eta_{13}''.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \end{align*} The estimates of \(\eta_{13}'\) and \(\eta_{13}''\) can be obtained similarly as that of \(\eta_{11}'\) and \(\eta_{11}''\) in Step 2 and we get and Thus, we have Substituting (13) into (20), together with Lemmas 1, 2, 3, 5 and \(\left\|\frac{2^{j\alpha } |h\chi_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)q_{1}(\cdot)}}\leq1\), we get where $${(q^{3}_{2})_{k}}= \left\{\begin{array}{ll} (q_{2})_{-}, \left\|\left(\frac{2^{k\alpha } |\sum^{\infty}_{j=k+2}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\ (q_{2})_{+}, \left\|\left(\frac{2^{k\alpha } |\sum^{\infty}_{j=k+2}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1. \end{array}\right.$$ From above and by an argument similar to that of Step 2,we conclude The proof is completed.

Theorem 2. Suppose \(b\in \mathrm{BMO}(\mathbb{R}^{n}),m\in\mathbb{N},0< \rho< n,0< v\leq1.\) Further suppose that \(p_{1}(\cdot)\in\mathcal{B}(\mathbb{R}^{n}),\Omega\in L^{s}(\mathbb{S}^{n-1}),s>(p_{1}')_{+}\) and \(q_{1}(\cdot),q_{2}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) with \((q_{2})_{-}\geq(q_{1})_{+}\). If \(-n\delta_{1}-v-n/s< \alpha< n\delta_{2}-v-n/s\) with \(\delta_{1},\delta_{2}\) as defined in Lemma 2. Then the operator \([b^{m},\mu_{\Omega}^{\rho}]\) is bounded from \( \dot{K}_{p_{1}(\cdot)}^{\alpha,q_{1}(\cdot)}(\mathbb{R}^{n})\) to \(\dot{K}_{p_{1}(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n})\) and \(\left({K}_{p_{1}(\cdot)}^{\alpha,q_{1}(\cdot)}(\mathbb{R}^{n})\right)\) to \( \left({K}_{p_{1}(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n})\right)\).

Proof. Let \(h(x)\in \dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n}),b\in \mathrm{BMO}(\mathbb{R}^{n})\). We may write \(h(x)=\sum_{j=-\infty}^{\infty}h(x)\chi_{j}=\sum_{j=-\infty}^{\infty}h_{j}(x).\) By definition of \(\dot{K}^{\alpha,q(\cdot)}_{p(\cdot)}(\mathbb{R}^{n})\), we have $$\|[b^{m},\mu_{\Omega}^{\rho}](h)\|_{\dot{K}^{\alpha,q_{2}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})} =\inf\left\{\eta>0 : \sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }|[b^{m},\mu_{\Omega}^{\rho}](h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1\right\}.$$ Since \begin{eqnarray*}\left\|\left(\frac{2^{k\alpha}|[b^{m},\mu_{\Omega}^{\rho}](h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq\left\|\left(\frac{2^{k\alpha}|\sum^{k-2}_{j=-\infty} [b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\sum^{3}_{i=1}\eta_{2i}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\end{eqnarray*} \begin{eqnarray*}&\leq&\left\|\left(\frac{2^{k\alpha}|\sum^{\infty}_{j=-\infty} [b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{21}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}+\left\|\left(\frac{2^{k\alpha}|\sum^{k+2}_{j=k-2} [b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{22}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\\&&+\left\|\left(\frac{2^{k\alpha}|\sum^{\infty}_{j=k+2} [b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{23}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}. \end{eqnarray*} Let \begin{eqnarray*} &&\eta_{21}=\left\|\left\{2^{k\alpha }|\sum^{k-2}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|\right\}^{\infty}_{k=-\infty} \right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})},\\ &&\eta_{22}=\left\|\left\{2^{k\alpha }|\sum^{k+2}_{j=k-2}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|\right\}^{\infty}_{k=-\infty} \right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})},\\ &&\eta_{23}=\left\|\left\{2^{k\alpha }|\sum^{\infty}_{j=k+2}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|\right\}^{\infty}_{k=-\infty} \right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})}, \end{eqnarray*} where we put $$\eta=\eta_{21}+\eta_{22}+\eta_{23}=\sum^{3}_{i=1}\eta_{2i}.$$ Hence, $$\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }|[b^{m},\mu_{\Omega}^{\rho}](h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C.$$ So, it follows that $$\|[b^{m},\mu_{\Omega}^{\rho}](h)\|_{\dot{K}^{\alpha,q_{2}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}\leq C\eta= C \sum^{3}_{i=1}\eta_{1i}.$$ Hence, \(\eta_{21},\eta_{22}\text{and}\eta_{23}\leq C \|b\|_{\ast}\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})} \). Denoting that \(\eta_{10}= C\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}.\)\\ {\textbf Step 1.} We estimate \(\eta_{22}\). The proof of Theorem 2 is the same to that of Theorem 1 and we use the similar notation as in the proof \(\eta_{12}\) of Theorem 1. By Lemma 5 and \(\left(L^{p(\cdot)}(\mathbb{R}^{n}),L^{p(\cdot)}(\mathbb{R}^{n})\right)\)-boundedness of the operators \([b^{m},\mu_{\Omega}^{\rho}]\) , we directly arrive at $$\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha}|\sum^{k+2}_{j=k-2} [b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}\|b\|_{\ast}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C,$$ which, implies that

Step 2. Next we estimate \(\eta_{21}\). Since \begin{eqnarray*} &|[b^{m},\mu_{\Omega}^{\rho}](h_{j})(x)|&:= \left(\int^{\infty}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}[b(x)-b(y)]^{m}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\ &&\leq \left(\int^{|x|}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}[b(x)-b(y)]^{m}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\ &&\quad+\left(\int_{|x|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}[b(x)-b(y)]^{m}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\ &&:=\mathfrak{\eta'}_{22}+\mathfrak{\eta''}_{22}. \end{eqnarray*} Observe that \(|x-y|\approx|x|\) for each \(x\in A_{k},y\in A_{j}\) and \(j \leq k-2.\) From (9) and applying the Minkowski's and the generalized H{\"o}lder's inequality, we get Similarly, we consider \(\mathfrak{\eta''}_{22}\) Therefore, By (13) and Lemmas 6 and 7, we get From this, we deduced Applying Lemmas 1, 3, 4 and 5, we have \begin{align*} &\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha}|\sum^{k-2}_{j=-\infty} [b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}\|b\|^{m}_{\ast}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C \sum^{\infty}_{k=-\infty}\left\|\frac{2^{k\alpha}|\sum^{k-2}_{j=-\infty} [b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}\|b\|^{m}_{\ast}} \right\|^{(q^{2}_{2})_{k}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha } \sum^{k-2}_{j=-\infty}2^{-kn}2^{(k-j)(v+n/s)}\right.\left.\left\|\frac{| h_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}} \frac{1}{\|b\|^{m}_{\ast}} \|(b(\cdot)-b_{B_{j}})^{m}\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}\|\chi_{B_{j}}\|_{L^{p_{1}^{\prime}(\cdot)}} \right)^{(q^{2}_{2})_{k}} \end{align*} \begin{align*} \\ &\quad+ C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha } \sum^{k-2}_{j=-\infty}(k-j)^{m}2^{-kn}2^{(k-j)(v+n/s)}\right.\left.\left\|\frac{| h_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}} \|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}}\right)^{(q^{2}_{2})_{k}}\\ &\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha } \sum^{k-2}_{j=-\infty}(k-j)^{m}2^{-kn}2^{(k-j)(v+n/s)}\right.\left.\left\|\frac{| h_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}}\right)^{(q^{2}_{2})_{k}}\\ &\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha } \sum^{k-2}_{j=-\infty}(k-j)^{m}2^{(k-j)(v+n/s)}2^{-j\alpha} \left\|\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}} \frac{\|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}}}{ \|\chi_{B_{k}}\|_{L^{p^{\prime}_{1}(\cdot)}}}\right)^{(q^{2}_{2})_{k}}. \end{align*} Now, by Lemma 2, we have where $${(q^{2}_{2})_{k}}= \left\{\begin{array}{ll} (q_{2})_{-},\quad\quad\quad\left\|\left(\frac{2^{k\alpha } |\sum^{k-2}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}} \right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\ (q_{2})_{+},\quad\quad\quad\left\|\left(\frac{2^{k\alpha } |\sum^{k-2}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}} \right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1. \end{array}\right.$$ So, together with \((q_{1})_{+}< 1\), \((p_{1})_{+}\leq(p_{2})_{-}\leq(q^{2}_{2})_{k}\), along with Remark 1, gives where \(q_{*}= \min\limits_{k\in N}\frac{(q^{2}_{2})_{k}}{(q_{1})_{+}}.\) If \((q_{1})_{+} \leq 1\), then by Hölder's inequality and Remark 1, we have where \(q_{*}= \min\limits_{k\in N}\frac{(q^{2}_{2})_{k}}{(q_{1})_{+}}.\) This implies that Finally we estimate \(\eta_{23}\). For any \(x\in A_{j},j\geq k+2\), by the same argument as in \(\eta_{21}\), we obtain \begin{eqnarray*} |[b^{m},\mu_{\Omega}^{\rho}](h_{j})(x)| &&:= \left(\int^{\infty}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}[b(x)-b(y)]^{m}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\ \quad&&\leq \left(\int^{|y|}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}[b(x)-b(y)]^{m}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\ &&\quad+\left(\int_{|y|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}[b(x)-b(y)]^{m}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\ &&:=\mathfrak{\eta'}_{23}+\mathfrak{\eta''}_{23}. \end{eqnarray*} Noticing that \(j \geq k+2\). To estimate \(\eta_{23}'\) and \(\eta_{23}''\) we will use same method as that of \(\eta_{21}'\) and \(\eta_{21}''\) in Step 2. Since and Thus, From (13), by using Lemma 7 and Lemma 2, we get Hence, we plug the inequality (36) into (35) and obtain By Lemma 5 and the above inequality, we have \begin{align*} &\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha}|\sum^{\infty}_{j=k+2} [b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}\|b\|^{m}_{\ast}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C \sum^{\infty}_{k=-\infty}\left\|\frac{2^{k\alpha}|\sum^{\infty}_{j=k+2} [b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}\|b\|^{m}_{\ast}} \right\|^{(q^{2}_{2})_{k}}_{L^{p_{1}(\cdot)}}\notag\\ &\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha } \sum^{\infty}_{j=k+2}2^{-jn}2^{(k-j)(v+n/s)}\right.\left.\left\|\frac{| h_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}} \frac{1}{\|b\|^{m}_{\ast}} \|(b(\cdot)-b_{B_{j}})^{m}\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}} \|\chi_{B_{j}}\|_{L^{p_{1}^{\prime}(\cdot)}}\right)^{(q^{2}_{2})_{k}}\end{align*} where $${(q^{3}_{2})_{k}}= \left\{\begin{array}{ll} (q_{2})_{-},\quad\quad\quad\left\|\left(\frac{2^{k\alpha } |\sum^{k-2}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}} \right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\ (q_{2})_{+},\quad\quad\quad\left\|\left(\frac{2^{k\alpha } |\sum^{k-2}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}} \right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1. \end{array}\right.$$ Hence, by the similar argument to Theorem 1, we arrive at \(\eta_{23}\leq C\eta_{10}\|b\|_{\ast}\leq C\|b\|_{\ast}\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}\;.\) This completes the proof.

Theorem 3. Let \(b\in\dot{\Lambda}_{\gamma}(\mathbb{R}^{n}),0< \gamma\leq1,m\in\mathbb{N},0< \rho< n,0< v\leq1.\) Suppose that \(q^{+}_{1}< n/m\gamma,1/q_{1}(x)-1/q_{2}(x)=m\gamma/n,\Omega\in L^{s}(\mathbb{S}^{n-1})(s>q^{+}_{2})\) with \(1\leq r'< q^{-}_{2},\) \(p_{1}(\cdot)\in\mathcal{B}(\mathbb{R}^{n}),\Omega\in L^{s}(\mathbb{S}^{n-1}),s>(p_{1}')_{+}\) and \(q_{1}(\cdot),q_{2}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) with \((q_{2})_{-}\geq(q_{1})_{+}\). If \(-n\delta_{1}-v-n/s< \alpha< n\delta_{2}-v-n/s\) with \(\delta_{1},\delta_{2}\) as defined in Lemma 2, then the operator \([b^{m},\mu_{\Omega}^{\rho}]\) is bounded from \( \dot{K}_{p_{1}(\cdot)}^{\alpha,q_{1}(\cdot)}(\mathbb{R}^{n})\) to \(\dot{K}_{p_{1}(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n}\) and from \(\left({K}_{p_{1}(\cdot)}^{\alpha,q_{1}(\cdot)}(\mathbb{R}^{n})\right)\) to \(\left({K}_{p_{1}(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n})\right)\).

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

A real valued function \(\eta: I\rightarrow\mathbb{R}\) is said to be convex on \(I\), if the following inequality holds: The function \(\eta\) is said to be concave if reversed of inequality (1) holds.
A convex function is also equally defined by the well known Hadamard inequality stated as follows: \begin{equation*} \eta\left(\frac{\varsigma_{1}+\varsigma_{2}}{2}\right)\leq \frac{1}{\varsigma_{2}-\varsigma_{1}}\int^{\varsigma_{2}}_{\varsigma_{1}}\eta(\tau)d\tau\leq \frac{\eta(\varsigma_{1})+\eta(\varsigma_{2})}{2}, \end{equation*} where \(\eta:[\varsigma_{1},\varsigma_{2}]\rightarrow \mathbb{R}\) is a convex function.
In [1], Fejér gave the generalization of Hadamard inequality stated as follows: where \(\eta:[\varsigma_{1},\varsigma_{2}]\rightarrow \mathbb{R}\) is convex function and \(\kappa:[\varsigma_{1},\varsigma_{2}]\rightarrow \mathbb{R}\) is a positive, integrable and symmetric to \(\frac{\varsigma_{1}+\varsigma_{2}}{2}\).
The inequality (2) is well known as the Fejér-Hadamard inequality. The Hadamard and the Fejér-Hadamard inequalities have been analyzed by many researchers and produced frequently their generalizations, refinements and extensions (see, [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]).
In [19], Rashid et al., introduced the concept of exponentially \(m\)-convex functions defined as follows:

Definition 1. A real-valued function \(\eta: I\rightarrow\mathbb{R}\) is said to be exponentially \(m\)-convex, if the following inequality holds:

If we take \(m=1\) in (3), then exponentially convex function defined by Antczak in [20] is obtained, see also [6]. We recall that a real-valued function \(\eta: I\rightarrow\mathbb{R}\) is said to be exponentially convex, if the following inequality holds: Next we give the definition of generalized fractional integral operators containing Mittag-Leffler function in their kernels as follows:

Definition 2.[21] Let \(\omega,\vartheta,\theta,l,\rho,c\in \mathbb{C}\), \(\Re(\vartheta),\Re(\theta),\Re(l)>0\), \(\Re(c)>\Re(\rho)>0\) with \(p\geq0\), \(r>0\) and \(0< q\leq r+\Re(\vartheta)\). Let \(\eta\in L_{1}[\varsigma_{1},\varsigma_{2}]\) and \(u\in[\varsigma_{1},\varsigma_{2}].\) Then the generalized fractional integral operators \(\Upsilon_{\vartheta,\theta,l,\omega,\varsigma_{1}^{+}}^{\rho,r,q,c}\eta \) and \(\Upsilon_{\vartheta,\theta,l,\omega,\varsigma_{2}^{-}}^{\rho,r,q,c}\eta\) are defined by:

where \(E_{\vartheta,\theta,l}^{\rho,r,q,c}(\tau;p)\) is the generalized Mittag-Leffler function defined as follows: \begin{equation*} E_{\vartheta,\theta,l}^{\rho,r,q,c}(\tau;p)= \sum_{n=0}^{\infty}\frac{\beta_{p}(\rho+nq,c-\rho)}{\beta(\rho,c-\rho)} \frac{(c)_{nq}}{\Gamma(\vartheta n +\theta)} \frac{\tau^{n}}{(l)_{n r}}. \end{equation*}

In [22], Farid defined the following unified integral operators:

Definition 3. Let \(\eta, \kappa: [\varsigma_{1},\varsigma_{2}]\rightarrow \mathbb{R}\), \(0< \varsigma_{1}< \varsigma_{2}\) be the functions such that \(\eta\) be a positive and integrable and \(\kappa\) be a differentiable and strictly increasing. Also, let \(\frac{\gamma}{u}\) be an increasing function on \([\varsigma_{1},\infty)\) and \(\theta,l,\rho,c\in \mathbb{C}\), \(p,\vartheta,r\geq0\) and \(0< q\leq r+\vartheta\). Then for \(u\in[\varsigma_{1},\varsigma_{2}]\) the integral operators \(_{\kappa}\Upsilon_{\vartheta, \theta,l, \varsigma_{1}^{+}}^{\gamma, \rho,r,q,c}\eta\) and \(_{\kappa}\Upsilon_{\vartheta, \theta,l, \varsigma_{2}^{-}}^{\gamma, \rho,r,q,c}\eta\) are defined by:

If we take \(\gamma(u)=u^\theta\) in (7) and (8), then we get the following generalized fractional integral operators containing Mittag-Leffler function:

Definition 4. Let \(\eta, \kappa: [\varsigma_{1},\varsigma_{2}]\rightarrow \mathbb{R}\), \(0< \varsigma_{1}< \varsigma_{2}\) be the functions such that \(\eta\) be a positive and integrable and \(\kappa\) be a differentiable and strictly increasing. Also, let \(\vartheta, \theta,l,\omega,\rho,c\in \mathbb{C}\), \(p,\vartheta,r\geq0\) and \(0< q\leq r+\vartheta\). Then for \(u\in[\varsigma_{1},\varsigma_{2}]\) the integral operators \(_{\kappa}\Upsilon_{\vartheta, \theta,l,\omega, \varsigma_{1}^{+}}^{\rho,r,q,c}\eta\) and \(_{\kappa}\Upsilon_{\vartheta, \theta,l,\omega, \varsigma_{2}^{-}}^{ \rho,r,q,c}\eta\) are defined by:

Remark 1. (9) and (10) are the generalization of the following fractional integral operators:

1. By taking \(\kappa(u)=u\), the fractional integral operators (5) and (6), can be achieved.
2. By taking \(\kappa(u)=u\) and \(p=0\), the fractional integral operators defined by Salim-Faraj in [23], can be achieved.
3. By taking \(\kappa(u)=u\) and \(l=r=1\), the fractional integral operators defined by Rahman et al., in [24], can be achieved.
4. By taking \(\kappa(u)=u\), \(p=0\) and \(l=r=1\), the fractional integral operators defined by Srivastava-Tomovski in [25], can be achieved.
5. By taking \(\kappa(u)=u\), \(p=0\) and \(l=r=q=1\), the fractional integral operators defined by Prabhakar in [26], can be achieved.
6. By taking \(\kappa(u)=u\) and \(\omega=p=0\), the Riemann-Liouville fractional integral operators can be achieved.

From generalized fractional integral operator (9), we have \begin{align*} \left(_{\kappa}\Upsilon_{\vartheta, \theta,l, \omega, \varsigma_{1}^{+}}^{ \rho,r,q,c}1\right)(u;p)&=\int_{\varsigma_{1}}^{u}(\kappa(u)-\kappa(\tau))^{\theta-1} E_{\vartheta, \theta, l}^{\rho,r,q,c} (\omega(\kappa(u)-\kappa(\tau))^{\vartheta};p)d(\kappa(\tau))\nonumber\\&=\int_{\varsigma_{1}}^{u}(\kappa(u)-\kappa(\tau))^{\theta-1} \sum_{n=0}^{\infty}\frac{\beta_{p}(\rho+nq,c-\rho)}{\beta(\rho,c-\rho)} \frac{(c)_{nq}}{\Gamma(\vartheta n +\theta)} \frac{\omega^{n}(\kappa(u)-\kappa(\tau))^{\vartheta n}}{(l)_{n r}}d(\kappa(\tau))\nonumber\\&=\sum_{n=0}^{\infty}\frac{\beta_{p}(\rho+nq,c-\rho)}{\beta(\rho,c-\rho)} \frac{(c)_{nq}}{\Gamma(\vartheta n +\theta)} \frac{\omega^{n}}{(l)_{n r}}\int_{\varsigma_{1}}^{u}(\kappa(u)-\kappa(\tau))^{\vartheta n+\theta-1}d(\kappa(\tau))\\&=(\kappa(u)-\kappa(\varsigma_{1}))^{\theta}\sum_{n=0}^{\infty}\frac{\beta_{p}(\rho+nq,c-\rho)}{\beta(\rho,c-\rho)} \frac{(c)_{nq}}{\Gamma(\vartheta n +\theta)} \frac{\omega^{n}}{(l)_{n r}}\frac{(\kappa(u)-\kappa(\varsigma_{1}))^{\vartheta n}}{\vartheta n+\theta}\\&=(\kappa(u)-\kappa(\varsigma_{1}))^{\theta}\sum_{n=0}^{\infty}\frac{\beta_{p}(\rho+nq,c-\rho)}{\beta(\rho,c-\rho)} \frac{(c)_{nq}}{\Gamma(\vartheta n +\theta+1)} \frac{\omega^{n}(\kappa(u)-\kappa(\varsigma_{1}))^{\vartheta n}}{(l)_{n r}}\\&=(\kappa(u)-\kappa(\varsigma_{1}))^{\theta}E_{\vartheta, \theta+1, l}^{\rho,r,q,c} (\omega(\kappa(u)-\kappa(\varsigma_{1}))^{\vartheta};p). \end{align*} Hence \begin{equation*} \left(_{\kappa}\Upsilon_{\vartheta, \theta,l, \omega, \varsigma_{1}^{+}}^{ \rho,r,q,c}1\right)(u;p)=(\kappa(u)-\kappa(\varsigma_{1}))^{\theta}E_{\vartheta, \theta+1, l}^{\rho,r,q,c} (\omega(\kappa(u)-\kappa(\varsigma_{1}))^{\vartheta};p) \end{equation*} and similarly, from generalized fractional integral operator (10), we get \begin{equation*} \left(_{\kappa}\Upsilon_{\vartheta, \theta,l, \omega, \varsigma_{2}^{-}}^{ \rho,r,q,c}1\right)(u;p)=(\kappa(\varsigma_{2})-\kappa(u))^{\theta}E_{\vartheta, \theta+1, l}^{\rho,r,q,c} (\omega(\kappa(\varsigma_{2})-\kappa(u))^{\vartheta};p). \end{equation*} We will use the following notations in the article: In [27], Mehmood et al., proved the following Hadamard and Fejér-Hadamard inequalities for exponentially \(m\)-convex functions via generalized fractional integral operators (5) and (6).

Theorem 1. Let \(\eta: [\varsigma_{1},m\varsigma_{2}]\subset \mathbb{R}\to\mathbb{R}\) be a function such that \(\eta\in L_{1}[\varsigma_{1},m\varsigma_{2}]\) with \(\varsigma_{1}< m\varsigma_{2}\). If \(\eta\) is exponentially \(m\)-convex function, then the following inequalities hold: \begin{align*} &e^{\eta\left( \frac{\varsigma_{1}+m\varsigma_{2}}{2}\right)} \zeta_{\theta,\bar{\omega},\varsigma_{1}^{+}}(m\varsigma_{2};p)\nonumber \leq\frac{\left(\Upsilon_{\vartheta,\theta,l,\bar{\omega},\varsigma_{1}^{+}}^{\rho,r,q,c}e^\eta\right)(m\varsigma_{2};p)+ m^{\theta+1}\left(\Upsilon_{\vartheta,\theta,l,\bar{\omega}m^\vartheta,\varsigma_{2}^{-}}^{\rho,r,q,c}e^\eta\right)\left(\frac{\varsigma_{1}}{m};p \right)}{2}\\\nonumber &\leq\frac{m^{\theta+1}}{2(m\varsigma_{2}-\varsigma_{1})}\left[\left(e^{\eta(\varsigma_{1})}-m^2e^{\eta(\frac{\varsigma_{1}}{m^2})}\right)\zeta_{\theta+1,\bar{\omega}m^\vartheta,\varsigma_{2}^{-}}\left(\frac{\varsigma_{1}}{m};p \right)\right.\left.+(m\varsigma_{2}-\varsigma_{1})\left(e^{\eta(\varsigma_{2})}+me^{\eta(\frac{\varsigma_{1}}{m^2})}\right)\zeta_{\theta,\bar{\omega}m^\vartheta,\varsigma_{2}^{-}}\left(\frac{\varsigma_{1}}{m};p \right)\right]\nonumber, \end{align*} where \(\bar{\omega}=\frac{\omega}{(m\varsigma_{2}-\varsigma_{1})^{\vartheta}}\).

Theorem 2. Let \(\eta: [\varsigma_{1},m\varsigma_{2}]\subset \mathbb{R}\to\mathbb{R}\) be a function such that \(\eta\in L_{1}[\varsigma_{1},m\varsigma_{2}]\) with \(\varsigma_{1}< m\varsigma_{2}\). If \(\eta\) is exponentially \(m\)-convex function, then the following inequalities hold: \begin{align*} &e^{\eta\left( \frac{\varsigma_{1}+m\varsigma_{2}}{2}\right)}\zeta_{\theta,\bar{\omega}2^\vartheta,\left(\frac{\varsigma_{1}+m\varsigma_{2}}{2}\right) ^{+}}(m\varsigma_{2};p)\leq \frac{\left(\Upsilon^{\rho,r,q,c}_{\vartheta,\theta,l,\bar{\omega}2^\vartheta,\left( \frac{\varsigma_{1}+m\varsigma_{2}}{2}\right) ^{+}}e^\eta\right)(m\varsigma_{2};p)+m^{\theta+1} \left(\Upsilon^{\rho,r,q,c}_{\vartheta,\theta,l,\bar{\omega}(2m)^\vartheta,\left( \frac{\varsigma_{1}+m\varsigma_{2}}{2m}\right) ^{-}}e^\eta\right)\left(\frac{\varsigma_{1}}{m};p \right)}{2}\\\nonumber &\leq \frac{m^{\theta+1}}{2(m\varsigma_{2}-\varsigma_{1})}\left[\left(e^{\eta(\varsigma_{1})}-m^2e^{\eta(\frac{\varsigma_{1}}{m^2})}\right)\zeta_{\theta+1,\bar{\omega}(2m)^\vartheta,\left( \frac{\varsigma_{1}+m\varsigma_{2}}{2m}\right) ^{-}}\left(\frac{\varsigma_{1}}{m};p \right)\right.+(m\varsigma_{2}-\varsigma_{1})\left(e^{\eta(\varsigma_{2})}+me^{\eta(\frac{\varsigma_{1}}{m^2})}\right) \end{align*} \begin{align*} &\;\;\;\times\left.\zeta_{\theta,\bar{\omega}(2m)^\vartheta,\left( \frac{\varsigma_{1}+m\varsigma_{2}}{2m}\right) ^{-}}\left(\frac{\varsigma_{1}}{m};p \right)\right], \end{align*} where \(\bar{\omega}=\frac{\omega}{(m\varsigma_{2}-\varsigma_{1})^{\vartheta}}.\)

Theorem 3. Let \(\eta :[\varsigma_{1},m\varsigma_{2}]\subset\mathbb{R}\to\mathbb{R}\) be a function such that \(\eta\in L_{1}[\varsigma_{1},m\varsigma_{2}]\) with \( \varsigma_{1}< m\varsigma_{2}\). Also, let \(\kappa : [\varsigma_{1},m\varsigma_{2}]\to\mathbb{R}\) be a function which is non-negative and integrable. If \(\eta\) is exponentially \(m\)-convex function and \(\eta(v)=\eta(\varsigma_{1}+m\varsigma_{2}-mv)\), then the following inequalities hold: \begin{align*} &e^{\eta\left( \frac{\varsigma_{1}+m\varsigma_{2}}{2}\right)} \left(\Upsilon^{\rho,r,q,c}_{\vartheta,\theta,l,\bar{\omega}m^\vartheta,\varsigma_{2}^{-}}e^\kappa\right)\left( \frac{\varsigma_{1}}{m};p\right) \leq \frac{(1+m)\left(\Upsilon^{\rho,r,q,c}_{\vartheta,\theta,l,\bar{\omega}m^\vartheta,\varsigma_{2}^{-}}e^\eta e^\kappa\right)\left( \frac{\varsigma_{1}}{m};p\right)}{2}\\\nonumber &\leq \frac{m}{2(m\varsigma_{2}-\varsigma_{1})}\left[\left(e^{\eta(\varsigma_{1})}-m^2e^{\eta(\frac{\varsigma_{1}}{m^2})}\right)\left(\Upsilon^{\rho,r,q,c}_{\vartheta,\theta+1,l,\bar{\omega}m^\vartheta,\varsigma_{2}^{-}}e^\kappa\right)\left(\frac{\varsigma_{1}}{m};p \right)\right.+(m\varsigma_{2}-\varsigma_{1})\left(e^{\eta(\varsigma_{2})}+me^{\eta(\frac{\varsigma_{1}}{m^2})}\right)\\ &\;\;\;\left.\times\left(\Upsilon^{\rho,r,q,c}_{\vartheta,\theta,l,\bar{\omega}m^\vartheta,\varsigma_{2}^{-}}e^\kappa\right)\left(\frac{\varsigma_{1}}{m};p \right)\right], \end{align*} where \(\bar{\omega}=\frac{\omega}{(m\varsigma_{2}-\varsigma_{1})^{\vartheta}}\).

In this article, we establish the Hadamard and the Fejér-Hadamard inequalities for exponentially \(m\)-convex functions by the generalized fractional integral operators (9) and (10) containing Mittag-Leffler function via a monotone function. These inequalities lead to produce results for generalized fractional integral operators given in Remark 1. In Section 2, we prove the Hadamard inequalities for generalized fractional integral operators (9) and (10) via exponentially \(m\)-convex functions. In Section 3, we prove the Fejér-Hadamard inequalities for these generalized fractional integral operators via exponentially \(m\)-convex functions. In whole paper, we will consider real parameters of the Mittag-Leffler function.

2. Hadamard inequalities for exponentially \(m\)-convex functions

Here we will give two versions of the generalized fractional Hadamard inequality.

Theorem 4. Let \(\eta,\kappa: [\varsigma_{1},m\varsigma_{2}]\subset\mathbb{R}\to\mathbb{R}\), \(0< \varsigma_{1}< m\varsigma_{2}\) be the real valued-functions. If \(\eta\) be a integrable and exponentially \(m\)-convex and \(\kappa\) be a differentiable and strictly increasing. Then for integral operators (9) and (10), the following inequalities hold:

where \(\bar{\omega}=\frac{\omega}{(m\kappa(\varsigma_{2})-\kappa(\varsigma_{1}))^{\vartheta}}\).

Proof. Since \(\eta\) is exponentially \(m\)-convex function on \([\varsigma_{1},m\varsigma_{2}]\), for \(\tau\in[0,1]\), we have

Also, from exponentially \(m\)-convexity, we have Multiplying both sides of (14) with \(\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p)\) and integrating over \([0,1]\), we have Putting \(\kappa(u)=\tau\kappa(\varsigma_{1})+m(1-\tau)\kappa(\varsigma_{2})\) and \(\kappa(v)=(1-\tau)\frac{\kappa(\varsigma_{1})}{m}+\tau\kappa(\varsigma_{2})\) in (16), we get \begin{align*} &2{e^{\eta\left( \frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2}\right)}}\int_{\varsigma_{1}}^{\kappa^{-1}(m\kappa(\varsigma_{2}))}(m\kappa(\varsigma_{2})-\kappa(u))^{\theta-1} E_{\vartheta,\theta,l}^{\rho,r,q, c}(\bar{\omega} (m\kappa(\varsigma_{2})-\kappa(u))^{\vartheta}; p)d(\kappa(u))\\& \leq \int_{\varsigma_{1}}^{\kappa^{-1}(m\kappa(\varsigma_{2}))}(m\kappa(\varsigma_{2})-\kappa(u))^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q, c}(\bar{\omega} (m\kappa(\varsigma_{2})-\kappa(u))^{\vartheta}; p){e^{\eta(\kappa(u))}}d(\kappa(u))\nonumber \\& \;\;\; +m^{\theta+1}\int_{\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})}{m}\right)}^{\varsigma_{2}}\left(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m}\right)^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q, c}\left(m^{\vartheta}\bar{\omega}\left(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m}\right)^{\vartheta};p\right){e^{\eta(\kappa(v))}}d(\kappa(v))\nonumber. \end{align*} By using (9), (10) and (11), the first inequality of (13) is obtained. Now multiplying both sides of (15) with \(\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p)\) and integrating over \([0,1]\), we have Putting \(\kappa(u)=\tau\kappa(\varsigma_{1})+m(1-\tau)\kappa(\varsigma_{2})\) and \(\kappa(v)=(1-\tau)\frac{\kappa(\varsigma_{1})}{m}+\tau\kappa(\varsigma_{2})\) in (17), then by using (9), (10) and (12), the second inequality of (13) is obtained.

Corollary 1. Under the assumptions of Theorem 4 if we take \(m=1\), then we get following inequalities for exponentially convex function:

where \(\bar{\omega}=\frac{\omega}{(\kappa(\varsigma_{2})-\kappa(\varsigma_{1}))^{\vartheta}}\).

Remark 2.

1. If we take \(\kappa(u)=u\) in (13), then Theorem 1 is obtained.
2. If we take \(\kappa(u)=u\) and \(m=1\) in (13), then [15,Corollary 2.2] is obtained.
3. If we take \(\kappa(u)=u\) in (18), then [15,Corollary 2.2] is obtained.

In the following we give another version of the Hadamard inequality for generalized fractional integral operators.

Theorem 5. Let \(\eta,\kappa: [\varsigma_{1},m\varsigma_{2}]\subset\mathbb{R}\to\mathbb{R}\), \(0< \varsigma_{1}< m\varsigma_{2}\) be the real-valued functions. If \(\eta\) be a integrable and exponentially \(m\)-convex and \(\kappa\) be a differentiable and strictly increasing. Then for integral operators (9) and (10), the following inequalities hold: \begin{align*} &\notag 2{e^{\eta\left( \frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2}\right)}}_{\kappa}\zeta^{\theta}_{\bar{\omega}2^\vartheta,\left(\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2}\right)\right) ^{+}}(\kappa^{-1}(m\kappa(\varsigma_{2}));p)\leq \left(_{\kappa}\Upsilon^{\rho,r,q,c}_{\vartheta,\theta,l,\bar{\omega}2^\vartheta,\left(\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2}\right)\right) ^{+}}e^{\eta\circ\kappa}\right)(\kappa^{-1}(m\kappa(\varsigma_{2}));p)\\&\nonumber\;\;\;+m^{\theta+1} \left(_{\kappa}\Upsilon^{\rho,r,q,c}_{\vartheta,\theta,l,\bar{\omega}(2m)^\vartheta,\left(\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2m}\right)\right) ^{-}}e^{\eta\circ\kappa}\right)\left(\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})}{m}\right);p \right) \end{align*}

where \(\bar{\omega}=\frac{\omega}{(m\kappa(\varsigma_{2})-\kappa(\varsigma_{1}))^{\vartheta}}\).

Proof. Since \(\eta\) is exponentially \(m\)-convex function on \([\varsigma_{1},m\varsigma_{2}]\), for \(\tau\in[0,1]\), we have

Also, from exponentially \(m\)-convexity, we have Multiplying both sides of (20) with \(\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p)\) and integrating over \([0,1]\), we have Putting \(\kappa(u)=\frac{\tau}{2}\kappa(\varsigma_{1})+m\frac{(2-\tau)}{2}\kappa(\varsigma_{2})\) and \(\kappa(v)=\frac{\tau}{2}\kappa(\varsigma_{2})+\frac{(2-\tau)}{2}\frac{\kappa(\varsigma_{1})}{m}\) in (22), we get \begin{align*} &2{e^{\eta\left( \frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2}\right)}}\!\!\int_{\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2}\right)}^{\kappa^{-1}(m\kappa(\varsigma_{2}))}\!\!(m\kappa(\varsigma_{2})-\kappa(u))^{\theta-1} E_{\vartheta,\theta,l}^{\rho,r,q, c}(2^\vartheta\bar{\omega} (m\kappa(\varsigma_{2})-\kappa(u))^{\vartheta}; p)d(\kappa(u))\\& \leq \int_{\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2}\right)}^{\kappa^{-1}(m\kappa(\varsigma_{2}))}(m\kappa(\varsigma_{2})-\kappa(u))^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q, c}(2^\vartheta\bar{\omega} (m\kappa(\varsigma_{2})-\kappa(u))^{\vartheta}; p){e^{\eta(\kappa(u))}}d(\kappa(u))\nonumber \\&\;\;\; +m^{\theta+1}\!\!\int_{\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})}{m}\right)}^{\kappa^{-1}\!\left(\frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2m}\right)}\!\!\left(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m}\right)^{\theta-1}\!\!\!\!E_{\vartheta,\theta,l}^{\rho,r,q, c}\!\!\left(\!(2m)^{\vartheta}\bar{\omega}(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m})^{\vartheta};p\!\right)\!\!{e^{\eta(\kappa(v))}}d(\kappa(v))\nonumber. \end{align*} By using (9), (10) and (11), the first inequality of (19) is obtained.
Now multiplying both sides of (21) with \( \tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p)\) and integrating over \([0,1]\), we have Putting \(\kappa(u)=\frac{\tau}{2}\kappa(\varsigma_{1})+m\frac{(2-\tau)}{2}\kappa(\varsigma_{2})\) and \(\kappa(v)=\frac{\tau}{2}\kappa(\varsigma_{2})+\frac{(2-\tau)}{2}\frac{\kappa(\varsigma_{1})}{m}\) in (\ref{x}), then by using (9), (10) and (12), the second inequality of (19) is obtained.

Corollary 2. Under the assumptions of Theorem \ref{gh} if we take \(m=1\), then we get following inequalities for exponentially convex function: \begin{align*} &\nonumber 2{e^{\eta\left( \frac{\kappa(\varsigma_{1})+\kappa(\varsigma_{2})}{2}\right)}}_{\kappa}\zeta^{\theta}_{\bar{\omega}2^\vartheta,\left(\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})+\kappa(\varsigma_{2})}{2}\right)\right) ^{+}}(\varsigma_{2};p)\leq\!\! \left(\!\!_{\kappa}\Upsilon^{\rho,r,q,c}_{\vartheta,\theta,l,\bar{\omega}2^\vartheta,\left(\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})+\kappa(\varsigma_{2})}{2}\right)\right) ^{+}}e^{\eta\circ\kappa}\!\right)\!\!(\varsigma_{2};p)\!\end{align*}

where \(\bar{\omega}=\frac{\omega}{(\kappa(\varsigma_{2})-\kappa(\varsigma_{1}))^{\vartheta}}\).

Remark 3.

1. If we take \(\kappa(u)=u\) in (19), then Theorem 2 is obtained.
2. If we take \(\kappa(u)=u\) and \(m=1\) in (19), then [15,Corollary 2.2] is obtained.
3. If we take \(\kappa(u)=u\) in (24), then [15,Corollary 2.2] is obtained.

3. Fejér-Hadamard Inequalities for exponentially \(m\)-convex functions

Here we give two versions of the Fejér-Hadamard inequality.

Theorem 6. Let \(\eta,\kappa :[\varsigma_{1},m\varsigma_{2}]\subset\mathbb{R}\to\mathbb{R}\), \(0< \varsigma_{1}< m\varsigma_{2}\) be the real-valued functions. If \(\eta\) be a integrable, exponentially \(m\)-convex and \(\eta(\kappa(v))=\eta(\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})-m\kappa(v))\) and \(\kappa\) be a differentiable and strictly increasing. Also, let \(\gamma : [\varsigma_{1},m\varsigma_{2}]\to\mathbb{R}\) be a function which is non-negative and integrable. Then for integral operators (9) and (10), the following inequalities hold:

where \(\bar{\omega}=\frac{\omega}{(m\kappa(\varsigma_{2})-\kappa(\varsigma_{1}))^{\vartheta}}\).

Proof. Multiplying both sides of (14) with \(\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p)e^{\gamma( (1-\tau)\frac{\kappa(\varsigma_{1})}{m}+\tau\kappa(\varsigma_{2}))}\) and integrating over \([0,1]\), we have

Putting \(\kappa(v)=(1-\tau)\frac{\kappa(\varsigma_{1})}{m}+\tau\kappa(\varsigma_{2})\) in (26), we get \begin{align*} &2{e^{\eta\left( \frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2}\right)}}\!\!\!\int_{\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})}{m}\right)}^{\varsigma_{2}}\!\!\left(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m}\right)^{\theta-1} \!\!\!\!E_{\vartheta,\theta,l}^{\rho,r,q, c}\left(\!\!m^{\vartheta}\bar{\omega}(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m})^{\vartheta};p\right)\!\!{e^{\gamma(\kappa(v))}}d(\kappa(v))\\&\leq\!\!\!\int_{\kappa^{-1}\!\left(\!\frac{\kappa(\varsigma_{1})}{m}\!\right)}^{\varsigma_{2}}\!\!\!\!\left(\kappa(v)\!-\!\frac{\kappa(\varsigma_{1})}{m}\!\right)^{\theta-1}\!\!\!\!\!E_{\vartheta,\theta,l}^{\rho,r,q, c}\!\!\left(\!\!m^{\vartheta}\bar{\omega}(\kappa(v)\!-\!\!\frac{\kappa(\varsigma_{1})}{m})^{\vartheta};p\!\!\right)\!\! {{e^{\eta(\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})-m\kappa(v))}}}{\!e^{\gamma(\kappa(v))}}\!d(\kappa(v))\\&\nonumber\;\;\;+m\int_{\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})}{m}\right)}^{\varsigma_{2}}\left(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m}\right)^{\theta-1} E_{\vartheta,\theta,l}^{\rho,r,q, c}\left(m^{\vartheta}\bar{\omega}(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m})^{\vartheta};p\right){e^{\eta(\kappa(v))}}{e^{\gamma(\kappa(v))}}d(\kappa(v)). \end{align*} By using (10) and given condition \(\eta(\kappa(v))=\eta(\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})-m\kappa(v))\), the first inequality of (25) is obtained. Now multiplying both sides of (15) with \(\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p)e^{\gamma( (1-\tau)\frac{\kappa(\varsigma_{1})}{m}+\tau\kappa(\varsigma_{2}))}\) and integrating over \([0,1]\), we have \begin{align*} &\int_{0}^{1}\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p)e^{{\eta(\tau\kappa(\varsigma_{1})+m(1-\tau)\kappa(\varsigma_{2}))}}e^{\gamma( (1-\tau)\frac{\kappa(\varsigma_{1})}{m}+\tau\kappa(\varsigma_{2}))} d\tau\\ \nonumber&\;\;\;+m\int_{0}^{1}\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p)e^{{\eta((1-\tau)\frac{\kappa(\varsigma_{1})}{m}+\tau\kappa(\varsigma_{2}))}}e^{\gamma( (1-\tau)\frac{\kappa(\varsigma_{1})}{m}+\tau\kappa(\varsigma_{2}))}d\tau.\\&\leq \left(e^{\eta(\kappa(\varsigma_{1}))}-m^2e^{\eta\left(\frac{\kappa(\varsigma_{1})}{m^2}\right)}\right)\int_{0}^{1}\tau^{\theta}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p)e^{\gamma( (1-\tau)\frac{\kappa(\varsigma_{1})}{m}+\tau\kappa(\varsigma_{2}))}d\tau\\\nonumber &\;\;\;+m\left(e^{\eta(\kappa(\varsigma_{2}))}+me^{\eta\left(\frac{\kappa(\varsigma_{1})}{m^2}\right)}\right) \int_{0}^{1}\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p)e^{\gamma( (1-\tau)\frac{\kappa(\varsigma_{1})}{m}+\tau\kappa(\varsigma_{2}))}d\tau. \end{align*} From above the second inequality of (25) is achieved.

Corollary 3. Under the assumptions of Theorem 6 if we take \(m=1\), then we get following inequalities for exponentially convex function:

where \(\bar{\omega}=\frac{\omega}{(\kappa(\varsigma_{2})-\kappa(\varsigma_{1}))^{\vartheta}}\).

Remark 4.

1. If we take \(\kappa(u)=u\) in (25), then Theorem 3 is obtained.
2. If we take \(\kappa(u)=u\) and \(m=1\) in (25), then [15,Corollary 2.8] is obtained.
3. If we take \(\kappa(u)=u\) in (27), then [15,Corollary 2.8] is obtained.

In the following we give another generalized fractional version of the Fejér-Hadamard inequality.

Theorem 7. Let \(\eta,\kappa :[\varsigma_{1},m\varsigma_{2}]\subset\mathbb{R}\to\mathbb{R}\), \(0< \varsigma_{1}< m\varsigma_{2}\) be the real-valued functions. If \(\eta\) be a integrable, exponentially \(m\)-convex and \(\eta(\kappa(v))=\eta(\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})-m\kappa(v))\) and \(\kappa\) be a differentiable and strictly increasing. Also, let \(\gamma : [\varsigma_{1},m\varsigma_{2}]\to\mathbb{R}\) be a function which is non-negative and integrable. Then for integral operators (9) and (10), the following inequalities hold:

where \(\bar{\omega}=\frac{\omega}{(m\kappa(\varsigma_{2})-\kappa(\varsigma_{1}))^{\vartheta}}\).

Proof. Multiplying both sides of (20) with \(\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p) {e^{\gamma\left(\frac{\tau}{2}\kappa(\varsigma_{2})+\frac{(2-\tau)}{2}\frac{\kappa(\varsigma_{1})}{m}\right)}}\) and integrating over \([0,1]\), we have \begin{align*} &\notag 2{e^{\eta\left( \frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2}\right)}}\int_{0}^{1}\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p) {e^{\gamma\left(\frac{\tau}{2}\kappa(\varsigma_{2})+\frac{(2-\tau)}{2}\frac{\kappa(\varsigma_{1})}{m}\right)}} d\tau\\\nonumber & \leq \int_{0}^{1}\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p){e^{\eta\left( \frac{\tau}{2}\kappa(\varsigma_{1})+m\frac{(2-\tau)}{2}\kappa(\varsigma_{2})\right)}} {e^{\gamma\left(\frac{\tau}{2}\kappa(\varsigma_{2})+\frac{(2-\tau)}{2}\frac{\kappa(\varsigma_{1})}{m}\right)}} d\tau \end{align*}

Putting \(\kappa(v)=\frac{\tau}{2}\kappa(\varsigma_{2})+\frac{(2-\tau)}{2}\frac{\kappa(\varsigma_{1})}{m}\) in (29), we get \begin{align*} &2{e^{\eta\left( \frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2}\right)}}\int_{\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})}{m}\right)}^{\left(\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2m}\right)\right)}\left(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m}\right)^{\theta-1} E_{\vartheta,\theta,l}^{\rho,r,q, c}\left((2m)^{\vartheta}\bar{\omega}(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m})^{\vartheta};p\right){e^{\gamma(\kappa(v))}}d(\kappa(v))\\&\leq\int_{\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})}{m}\right)}^{\left(\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2m}\right)\right)}\left(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m}\right)^{\theta-1} E_{\vartheta,\theta,l}^{\rho,r,q, c}\left((2m)^{\vartheta}\bar{\omega}(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m})^{\vartheta};p\right)\\&\;\;\;\times {{e^{\eta(\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})-m\kappa(v))}}}{e^{\gamma(\kappa(v))}}d(\kappa(v))\!\!+\!m\!\!\int_{\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})}{m}\right)}^{\left(\kappa^{-1}\left(\frac{\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})}{2m}\right)\right)}\!\!\left(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m}\right)^{\theta-1} \\&\;\;\;\times E_{\vartheta,\theta,l}^{\rho,r,q, c}\left((2m)^{\vartheta}\bar{\omega}(\kappa(v)-\frac{\kappa(\varsigma_{1})}{m})^{\vartheta};p\right){e^{\eta(\kappa(v))}}{e^{\gamma(\kappa(v))}}d(\kappa(v)). \end{align*} By using (10) and given condition \(\eta(\kappa(v))=\eta(\kappa(\varsigma_{1})+m\kappa(\varsigma_{2})-m\kappa(v))\), the first inequality of (28) is obtained. Now multiplying both sides of (21) with \(\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p) {e^{\gamma\left(\frac{\tau}{2}\kappa(\varsigma_{2})+\frac{(2-\tau)}{2}\frac{\kappa(\varsigma_{1})}{m}\right)}}\) and integrating over \([0,1]\), we have \begin{align*} &\int_{0}^{1}\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p)e^{\eta\left( \frac{\tau}{2}\kappa(\varsigma_{1})+m\frac{(2-\tau)}{2}\kappa(\varsigma_{2})\right)} {e^{\gamma\left(\frac{\tau}{2}\kappa(\varsigma_{2})+\frac{(2-\tau)}{2}\frac{\kappa(\varsigma_{1})}{m}\right)}} d\tau\\ \nonumber&\;\;\;+m\int_{0}^{1}\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p)e^{\eta\left(\frac{\tau}{2}\kappa(\varsigma_{2})+\frac{(2-\tau)}{2}\frac{\kappa(\varsigma_{1})}{m}\right)} {e^{\gamma\left(\frac{\tau}{2}\kappa(\varsigma_{2})+\frac{(2-\tau)}{2}\frac{\kappa(\varsigma_{1})}{m}\right)}}d\tau.\\&\leq \left(e^{\eta(\kappa(\varsigma_{1}))}-m^2e^{\eta\left(\frac{\kappa(\varsigma_{1})}{m^2}\right)}\right)\int_{0}^{1}\tau^{\theta}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p) {e^{\gamma\left(\frac{\tau}{2}\kappa(\varsigma_{2})+\frac{(2-\tau)}{2}\frac{\kappa(\varsigma_{1})}{m}\right)}}d\tau\\\nonumber &\;\;\;+m\left(e^{\eta(\kappa(\varsigma_{2}))}+me^{\eta\left(\frac{\kappa(\varsigma_{1})}{m^2}\right)}\right) \int_{0}^{1}\tau^{\theta-1}E_{\vartheta,\theta,l}^{\rho,r,q,c}(\omega \tau^{\vartheta};p) {e^{\gamma\left(\frac{\tau}{2}\kappa(\varsigma_{2})+\frac{(2-\tau)}{2}\frac{\kappa(\varsigma_{1})}{m}\right)}}d\tau. \end{align*} From above the second inequality of (28) is achieved.

Corollary 4. Under the assumptions of Theorem 7 if we take \(m=1\), then we get following inequalities for exponentially convex function:

where \(\bar{\omega}=\frac{\omega}{(\kappa(\varsigma_{2})-\kappa(\varsigma_{1}))^{\vartheta}}\).

4. Concluding remarks

Here we have proved two generalized fractional versions of the Hadamard inequality as well as two generalized fractional versions of the Fejér-Hadamard inequality. For proving these fractional inequalities we have utilized exponentially \(m\)-convex functions and generalized integral operators containing Mittag-Leffler functions in their kernels. The presented results hold for exponentially convexity and well known fractional integral operators given in Remark 1. Reader can obtain desired fractional Hadamard and fractional Fejér-Hadamard inequalities from this paper.

Acknowledgments

The research work of the Ghulam Farid is supported by Higher Education Commission of Pakistan under NRPU 2016, Project No. 5421.

Autho Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflict of Interests

The authors declare no conflict of interest.

1. Introduction

At the present stage of the development of society, which is characterized by the intensive introduction of information systems in virtually areas of activity, issues related to the assessment of the risks that occur during their operation are of particular importance. When analyzing and assessing risks, issues related to the definition of distribution laws are of the greatest importance. The given work is devoted to the construction of distribution laws.

In the modeling of information systems, risk is a random variable and is described by a probability distribution on a given set [1, 2, 3]. In contrast to experiments conducted in physics, where the possibility of their multiple conduct, the conditions of the functioning of information systems are characterized by a constant impact of negative external influences and are constantly changing [4], and consequently the repetition of the experiment under the same conditions is practically impracticable. The laws of probability distribution of risk events, as a rule, do not correspond to the law of the normal Gaussian distribution [5, 6].

2. Construction of continuous distribution laws with the maximum entropy

Entropy coefficient is often used [7, 8] with the classification of distribution laws of random continuous value (RV) with number characteristics. In the formula (1), \(\sigma = \sqrt{\mu_2}\) is standard deviation, and \(\mu_2\) is the second central power moment for this distribution law; value \(H\) is the entropy, which is defined as: where \(p(x)\) is the density of probability distribution (PDD) SV. Entropy coefficient has the maximum value for Gaussian law is \( \delta_e =2.066\); for uniform law is \(\delta_e =1.73\) and for Koshi distribution is \(\delta_e = 0\) etc.
The entropy value does not depend on shift parameter, to simple computation let's consider, that it is equal to zero. Firstly we need to find distribution law from unilateral laws of distribution of unlimited RV, for which entropy value (2) reaches the maximum with the following limitations imposed on probability density \(p(x)\): where \(\beta\) is scale parameter and \(\nu\) is value of maximum existing primary direct moment. Here and next we'll consider positive power moment as a direct moment in accordance with (3) and negative power moment as a reverse moment. To find the extremum we'll use the method of indefinite Lagrange multipliers [9]. We need to maximize by inserting Lagrange multipliers \(\lambda_1\) and \(\lambda_2\) and considering the limitations (3). Equating the result of variation integrand expression in (4) when \(p(x) = 0\), we'll take the equation relatively to \(p(x)\) : So, the density \(p(x)\) which satisfy (3) and maximizes H can be found from the equation (5): By substituting (6) in (3) and integrating, we have From (7), we find that \(\lambda_2 = \dfrac{-1}{\beta^\nu}\) and ~\(exp(\lambda_1 - 1) = \dfrac{v}{\beta \Gamma (\tfrac{1}{\nu})}\). Consequently where \(\bar{A} (z) \) is gamma function. From (8), it follows that if only the first beginning direct moment \(\nu = 1\) exists than exponential law has the maximum entropy; if there are two moments \((\nu = 2 )\) then unilateral Gaussian law and if all direct moments exist \((\nu \rightarrow \infty )\) than unilateral uniform law. Indeed, the limiting moment (8) with \((\nu \rightarrow \infty )\) is a unilateral uniform law \(p(x) = \beta^-1,~ 0 < x < \beta\). So, if all direct moments exist then uniform law has the maximum entropy from unilateral distribution laws of RV. Analogically, for bilateral symmetry laws of distribution of RV, it can be shown that if the first \(\nu\) of absolute central direct moments then the probability density has the maximum entropy:

From (9), it follows that if only first absolute central moment exists \(( \nu = 1) \) then Laplace distribution has the biggest entropy; if there are two moments \(( \nu = 2)\) then Gaussian law and if all direct moments exist \((\nu \rightarrow \infty )\) then uniform law. Indeed, the limiting case for (9) is a uniform law \(p(x) = 0.5 \beta^-1,~ -\beta < x < \beta \). So, if all direct moments exist then uniform law has the biggest entropy from bilateral symmetry distribution laws of RV. Considered private cases of bilateral laws with the maximum entropy coincide with already known laws (Laplace and Gaussian) which have maximum entropy that confirms the correctness of received results.

From analysis of the received expressions (8) and (9), it follows that for increasing the amount of information about evaluating parameters of distribution laws with big length (with long "tails") with the help of a method of moments is necessary to use direct moments of lesser order, including fractional order. If the parameters of distribution laws with lesser length are used then it is necessary to use direct moment of higher order.

Let's find from unilateral distribution laws of unlimited RV distribution law with which entropy value H reaches maximum with the following limitations imposed on probability density \(p(x)\): where \(\nu \) is value of maximum existing beginning reverse moment. Considering the entropy is defined by an expression: where \(y^{-2} p(\frac{1}{y})\) is a probability density RV \(\eta\) which is reverse to \(\xi\) and has the probability density \(p(x)\). As a result of using the method of indefinite Lagrange numerators, we'll receive following expression for distribution law with the maximum entropy: The limiting case for (12) with \(\nu \rightarrow \infty\) (all reverse moments exist) is a unilateral distribution law of limitations down from RV \(p(x) = \frac{1}{\beta} x^2, \frac{1}{\beta} < x < \infty\).
Let's define the bilateral distribution laws of RV for which entropy value \(H\) reaches the maximum with the following limitations imposed on probability density \(p(x)\) where \(\nu\) is the value of maximum existing primary direct exponential moment. Considering the entropy \(H\) is defined by the expression; By using the method of indefinite Lagrange numerators we'll receive the following expression for distribution law with the maximum entropy; The limiting case for (15) when \(\nu \rightarrow \infty\) (all direct exponential moments exist) is a distribution law of bordered above RV \(p(x) = \frac{exp(x)}{\beta}, -\infty < x < ln(\beta)\). Now let's find such distribution law from bilateral distribution laws of unlimited RV for which the value of entropy H reaches maximum with the following limitations imposed on probability density \(p(x)\): where \(\nu\) is the value of maximum existing primary reverse exponential moment. Considering an entropy \(H\) is defined by the expression: As a result of using the method of indefinite Lagrange numerators, we'll receive the following expression for distribution law with the maximum entropy: The limiting case for (18) when \(\nu \rightarrow \infty\) (all direct exponential moments exist) is a distribution law of bordered above RV \(p(x) = \frac{exp(-x)}{\beta}, -ln(\beta) < x< \infty \).
From the analysis of expressions (15) and (18), it follows that exponential transformation of RV leads to transformation of form parameter \(\nu\) in scale parameter and \(\beta\) parameter in shift parameter.
Finally let's define such distribution law from unilateral distribution laws of unlimited RV for which the value of entropy H reaches maximum with the following limitations imposed on probability density \(p(x)\): where \(\nu\) is the value of maximum existing primary direct logarithmic moment. Considering an entropy \(H\) is defined by the expression: As a result of using the method of indefinite Lagrange numerators, we'll receive the following expression for distribution law with the maximum entropy: From (21), it follows that if only two absolute logarithmic moments exist (\(\nu = 2\) ) then logarithmic normal law has the biggest entropy. If \(\nu \rightarrow \infty\) (all absolute primary moments exist) then (21) is transforming in Shannon law for limitations from above and down of RV \(p(x) = 0.5/\beta x,~ exp(-\beta) < x < exp(\beta)\). It is necessary to notice that with logarithmic transformation of RV scale parameter transforms in form parameter and shift parameter transforms in scale parameter.
In general case, if RV \(\eta\) connected with RV \(\eta \square\) by a ratio \(y =f(x)\) and known PDD \(p(y)\) of continuous RV \(\xi\) , then PDD \(p(x)\) can be found by a method of functional transformation with the help of expression: Considering (22), the entropy \begin{align*} H = - \int_{\Omega} p(y) ln (p(y)) dy \end{align*} takes the form where \(q(x) = \left|\dfrac{dy}{dx}\right|^{-1}\) and \(\Omega\) is the areas of existence RV \(\eta\) and \(\xi\) respectively.

3. Distributions arising in the analysis of the sequence of independent tests with three outputs

Next, consider the development of a probabilistic model of a sequence of independent trials with three outcomes which becomes particularly important in the formation of estimates of the information security of information processing systems [10].
During the test, it is taken into account that its result is either event A or the opposite event C. The probability of event A in any test is independent of the outcomes of all other tests (the tests are independent) and equal to the probability (this is ensured by the same set of conditions for each test). This scheme of tests was first considered by J. Bernoulli and bears his name [11, 12, 13, 14]. The probability \(P_A(k)\) of the fact that event \(A\) in \(N\) tests will come precisely \(k\) times (\(k = 1,2 , \dots , N\)) is defined by Bernoulli's formula [13, 14, 15]: which represents binomial distribution. For \(N = 1\), it transforms to Bernoulli's distribution. The limiting case of binomial distribution when \(p \rightarrow 0\) and \(N \rightarrow \infty\) and product \(Np\) aims to some positive constant value \(\lambda\) (i.e., \(Np \rightarrow \infty \)) is Poisson's distribution [13, 14, 15]. If sequence of tests with Bernoulli's scheme continues to appear m "failures" then the number of successes \(k\) obeys to negative binomial distribution where \(\Gamma(m)\) is the gamma function. Main purpose of this work is to invent sequence probability model of independent tests with three outputs and with it's help receive formulas analogue to (24), (26) and (27) for defining the probabilities of coming coinciding events. Let it be produced \(N\) of independent tests. Every test can end with three outputs: either event \(A\) with the probability \(p_1\) will come, or event \(B\) with the probability \(p_2\) will come, or event \(C\) with the probability \((1 - p_1 - p_2)\) will come. Let's match random discrete value to random output of every test which takes three values: -1 if event A happened; 0 if event \(C\) happened and 1 if event \(B\) happened. Positive or negative output of every test we'll consider as a "success" and zero output - "failure". In this the probability of coming events A, C and B in every test can be found by an expression where \( 0 < p_1 < 1, 0 < p_2 < 1, p_1 + p_2 < 1\). This distribution of probabilities, analogically to Bernoulli's distribution (25), can be called bilateral Bernoulli's distribution. Let's find characteristic function for distribution (28), using ratio [15] Using (28), we'll get Since ongoing tests are independent so characteristic function \(\theta_N (j, \vartheta)\) of distribution laws \(P(k)\) in \(N\) tests will be equal to expression: In this probability distribution \(P(k)\) in \(N\) tests can be found by the formula: Let's find obvious expression for probability distribution \(P(k)\) in \(N\) tests by substituting (31) in (32) and integrating where \(B(i, k ) = \frac{0.5 (1+ (-1)^{ + |k|})}{ \Gamma (0.5(i - k) + 1 ) \eta (0.5 (i + k) + 1) }\). Expression (10) can be simplified for five private cases:

1. If \( p_1 = p_2 = p < 0.5\), then
   \begin{align}\label{equ34} P(k) = (1 - 2p) ^N \times \sum_{i = |k|}^{N} \frac{N!}{(N - i)!} \times \biggl(\frac{p}{1 - 2p} \biggr)^i \times \frac{0.5[1+ (-1)^{i + |k|}]}{\Gamma [0.5 (i + k) + 1] \Gamma [0.5 (i -k) + 1]} \end{align}

   (34)
   \item If \(p_1 = (1 - p)^2, p_2 = p^2\), then
   \begin{align}\label{equ35} P(k) = \frac{(2N)!}{(N - k)! (N+ k)!} \times p^(N + k) (1 - p)^{(N - k)}, ~ k = -N, ~ -(N - 1), \dots , N \end{align}

   (35)
   Probability distribution (35), just like distribution (24), is a binomial distribution with not-zero shift parameter.
2. Let's view limiting case for distribution (33), when probability of coming value \(C\) is aims to zero, i.e., \((p_1 + p_2) \rightarrow 1\). In this case every test will end in two outputs: either coming of event \(A\) with the probability \((1 p)\), or event \(B\) with the probability \(p\). Those outputs can be matched discrete random value, which takes two values: -1, if event \(A\) happened and 1, if event \(B\) happened. In this probability distribution (33), the result can be transformed to distribution:
   \begin{align}\label{equ36} P(k) = (0.5 N! [1 + (-1)^{N + |k|}]) \times (\Gamma [0.5 (N + k ) + 1] \Gamma [0.5 (N - k ) + 1 ]^-1) \times \biggl( \frac{p}{1 - p} \biggr)^{0.5k} (p (1 - p))^{0.5N} \end{align}

   (36)
3. Let's view the second limiting case for distribution (33), when probability of coming event \(A\) aims to zero, i.e. \(p_1 \rightarrow 0\). In this case every test will end in two outputs: either coming of event \(C\) with a probability \((i - p)\), or event \(B\) with a probability \(p\). Those outputs can be matched random discrete value, which takes two values: 0, if event \(C\) happened and 1, if event \(B\) happened. This probability distribution (33) as a result of limiting transition transforms is the binomial distribution (24) and that's why received probability distribution (33) can be called generalized Bernoulli's formula, or bilateral binomial distribution.
4. Let's view the third limiting case for distribution (33), when \(p_1 \rightarrow 0, ~ p_2 \rightarrow 0, ~ N \rightarrow \infty\), and products \(Np_1, ~ Np_2\) aim to some positive constant values \(\lambda_1\), \(\lambda_2\) (i.e. \(Np_1 \rightarrow \lambda_1, ~ Np_2 \rightarrow \lambda_2 \) ). This probability distribution (33) in result of limiting transition transforms is the probability distribution either
   \begin{align}\label{equ37} P(k) = exp ( -\lambda_1 - \lambda_2) \biggl( \sqrt{\frac{\lambda_2}{\lambda_1}} \biggr)^k \times \sum_{i = |k|}^{\infty} \frac{0.5[1+ (-1)^{i + |k|}] \sqrt{\lambda_1 \lambda_2}^i}{\Gamma [0.5 (i + k) + 1] \Gamma [0.5 (i -k) + 1]} \end{align}

   (37)
   or
   \begin{align}\label{equ38} P(k) = exp ( -\lambda_1 - \lambda_2) \times \biggl( \sqrt{\frac{\lambda_2}{\lambda_1}}\biggr)^k I_{|k|} (2 \sqrt{\lambda_1 \lambda_2}), ~ -\infty < k < \infty \end{align}

   (38)
   where \(I_\nu(z)\) is the modified Bessel's function.

If parameter \(\lambda_1 \rightarrow 0\), and parameter \(\lambda_2 \rightarrow \lambda\), then distribution (37) or (38) transforms in Poisson's distribution (26). That's why probability distribution (37) or (38) can be called bilateral Poisson's distribution. Characteristic function for it is; The first, second, third and fourth order for distribution (33) can be found from the expressions \begin{eqnarray} m_ 1 &=& N (p_2 -p_1); \\ \notag \end{eqnarray} To compute these moments, we need asymmetry coefficient \(K_a\) and excess coefficient \(K_e\), which are given as; Expressions (40), (41) and (42) for moments are significantly simplified for private distribution cases (33). So, for distribution (33), we have: In this case For distribution (45), we have: \begin{eqnarray} m_1 &= & N (2p -1); \\ \notag M_2 &= &2N p( 1- p); \\ \notag \end{eqnarray} where For distribution (36), we have where For expression (37) or (38), we have where Probability \(P_B(k)\) of fact, that event \(B\) in \(N\) tests will come \(k\) times can be found from formula (33), or from it's private cases (34), (35), (36), (37) or (38). In this we suppose that \(P_B(k) = P(k), k = 1, 2, \dots , N\).
Probability \(P_A(k)\) of fact, that event \(A\) in \(N\) tests will come \(k\) times can be also found from formula (33), or it's private cases (34), (35), (36), (37) or (38). In this we suppose that \(P_A(k) = P(k), k = -1, -2, \dots , -N\).
Probability \(P_C\) of coming event \(C\) in \(N\) tests can be found using formula (33), or it's private cases (35), (36), (37) or (38). In that we suppose, that \(P_C = P(0)\). Probability \(P_C\) matches to probability of fact, that in \(N\) cases events \(A\) and \(B\) won't come.
Let's view the example. Two symmetric coins are being thrown for ten rimes. In every throw three outputs are possible: two "eagles" with probability 0.25; two "tails of coin" with probability 0.25 and "eagle and tail of coin" with probability 0.5. It's necessary to find: 1) probability of fact, that precisely five times two "eagles" drop; 2) probability \(P_{tt}\) of fact, that precisely three times two "tails of coin" drop; 3) probability \(P_{et}\) of fact, that precisely five times two "eagles" and three "tales of coin" drop. In the match with example's condition we have \begin{align*} p_1 = p_2 = p = 0.25,\,\,\,\,\,\,\,\,\,\,\,\, N = 10; ~ p_{ee} = P_A (-5),\,\,\,\,\,\,\,\,\,\,\,\, P_{tt} = P_B (3),\,\,\,\,\,\,\,\,\,\,\,\, P_{et } = P_A (-5) P_B (3). \end{align*} i.e., \(p_1 = p_2\), then we use expression (9) as a counting formula. With it's help we find, that either or where \(F(k) = F_1 (0.5 ( m + |k|)),~ 0.5 ( m+ |k| + 1), ~ 1+ |k|, ~ 4 p_1 p_2 \) is the Hypergeometric Gaussian function. Characteristic function of distribution (54) or (55) has the form Primary moment of the first order and central moments of the second, the third and the fourth orders for expressions (54) or (55) are defined by expressions \begin{eqnarray} m_1 &=& \frac{m( p_2 - p_1)}{1 - p_1 - p_2},\notag \end{eqnarray} Let's view limiting case for distribution (31) or (32), when probability \(p_1 \rightarrow 0\), and probability \(p_2 = p\). In the probability distribution (31) or (32) as a result of limiting transaction transforms in negative binomial distribution (4). That's why received probability distribution (31) or (32) can be called bilateral negative binomial distribution.
Choosing from bilateral binomial, Poisson's and negative binomial distributions, we can use following properties of those distributions: Binomial - \(K_a M_2 < 1\), Poisson's - \(K_e M_2 = 1 \), Negative binomial - \(K_e M_2 > 1\).
So, there was developed probability model for sequence of independent tests with three outputs, were received expressions for it's general number characteristics, and also for calculating the probabilities of coming matched events precisely k times. It was shown, that limiting cases of received bilateral distributions are binomial, negative binomial and Poisson's distributions.

4. Conclusion

The following results are obtained in this paper

• Generalized expressions for one-way and two-way continuous distribution laws with maximum entropy depending on the number of existing power, exponential or logarithmic moments. With their help, one can more reasonably choose the a priori distribution under the conditions of a priori uncertainty in the analysis of the risks of information systems. From the analysis of expression (23) and its particular cases (2), (11), (14), (17), (20) at the appropriate values \(q(x)\) it follows that in the general case the entropy depends also on the type of moments used to determine the numerical characteristics of the distribution law.
• Probabilistic model for a sequence of independent trials with three outcomes, which acquire special significance in the formation of information security assessments of information systems. Expressions for its basic numerical characteristics are obtained. It is shown that the limiting cases of the obtained two-way distributions are the binomial, negative binomial and Poisson distributions.

Acknowledgments

The authors would like to express their gratitude to University of Lagos for providing the enabling environment to conduct this research work.