Refinements of two fractional versions of Hadamard inequalities for Caputo fractional derivatives and related results

1. Introduction

Convex functions are important in the theory of mathematical inequalities, many well-known results are direct implications of these functions. The concepts of various types of new convex functions are obtained from the definition of convex functions. Their generalizations, extensions and refinements proved very beneficial in mathematical analysis, mathematical statistics, optimization theory, graph theory.

The main objective of this paper is to obtain refinements of two versions of the Hadamard inequality for Caputo fractional derivatives of convex functions proved [1,2]. The refinements of their error estimations are also established. For this purpose we employ the definition of strongly convex functions. In the following we give the definitions of functions utilized for getting the new and related results.

Definition 1. [3] A function \(\psi:I\rightarrow\mathbb{R}\), where \(I\) is an interval in \(\mathbb{R}\), is said to be convex if undermentioned inequality holds:

Strongly convex function was introduced by Polyak in [4].

Definition 2. Let \(D\) be a convex subset of \(\mathbb{X}\), \((\mathbb{X},||.||)\) be a normed space. A function \(\psi:D\subset \mathbb{X}\rightarrow\mathbb{R}\) is called strongly convex function with modulus \(C\) if it satisfies

\(\forall\, x_{0}, y_{0}\in {D}, z\in[0,1]\) and \(C>0\). If we take \(C=0\) in (2) we get inequality (1).

Many authors have been inventing the properties and applications of strongly convex function for more information, [5,6,7,8,9,10]. In this paper we will use this definition for normed space \(\mathbb{R}\).

A well-known inequality named the Hadamard inequality is an equivalent interpretation of convex function. It is given as follows:

Theorem 1. Let \(\psi:I \rightarrow\mathbb{R}\) be a convex function on interval \(I\subset\mathbb{R}\) and \(x_{0}, y_{0}\in I\) where \(x_{0} < y_{0}\). Then the undermentioned inequality holds:

If order in (3) is reversed, then it holds for concave function.

In the literature of mathematical inequalities the Hadamard inequality appears in various forms for convex and related functions. In [11] it was studied for Riemann-Liouville fractional integrals and after it many researchers started to get its versions for different kinds of fractional integral operators and functions. In [1,2], we have derived such inequalities for Caputo fractional derivatives. For the fractional integral inequalities we refer readers to [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. The Caputo fractional derivatives is defined as follows.

Definition 3. [26] Let \(\beta>0\) and \(\beta\notin \{1, 2, 3, . . . \}\), \(n = [\beta]+1\), \(\psi \in AC^n[x_{0}, y_{0}]\). Then Caputo fractional derivatives of order \(\beta\) are defined as:

and If \(\beta = n \in \{1, 2, 3, . . . \}\) and usual derivative of order \(n\) exists, then \(({^C}D^{\beta}_{x_{0}+}\psi)(x)=\psi^{(n)}(x)\), whereas \(({^C}D^{\beta}_{y_{0}-}\psi)(x)=(-1)^n\psi^{(n)}(x)\). In particular we have where \(n=1\) and \(\beta=0\).

Next, we give the results which are directly related to the findings of this paper. Farid et al., [1] proved the undermentioned Hadamard inequality for Caputo fractional derivatives:

Theorem 2. Let \(\psi: [x_{0}, y_{0}]\rightarrow\mathbb{R}\) be the function with \(\psi\in C^n[x_{0}, y_{0}]\) and \(0\leq x_{0}< y_{0}\). Also let \(\psi^{(n)}\) be positive and convex function on \([x_{0}, y_{0}].\) Then the undermentioned inequality holds for Caputo fractional derivatives:

Lemma 1. [1] For Caputo fractional derivatives the following identity holds:

Farid et al., [1] also proved the undermentioned inequality for Caputo fractional derivatives:

Theorem 3. Let \(0\leq x_{0}< y_{0}\), \(\psi\in C^{n+1}[x_{0}, y_{0}]\) and also let \(|\psi^{(n+1)}|\) be convex. Then for Caputo fractional derivatives we have:

Kang et al., [2] proved the undermentioned version of the Hadamard inequality for Caputo fractional derivatives:

Theorem 4. Under assumptions of Theorem 2, for Caputo fractional derivatives we have:

Lemma 2. [2] For Caputo fractional derivatives the following identity holds:

Kang et al., [2] also proved the undermentioned inequalities for Caputo fractional derivatives:

Theorem 5. Let \(\psi: [x_{0},y_{0}]\rightarrow\mathbb{R}\) be a differentiable mapping on \((x_{0}, y_{0})\) with \(\psi\in C^{n+1}[x_{0},y_{0}]\) and \(x_{0}< y_{0}\). If \(|\psi^{(n+1)}|^q\) is convex on \([x_{0}, y_{0}]\) for \(q\geq1,\) then for Caputo fractional derivatives we have:

Theorem 6. [2] Under assumptions of Theorem 3, for Caputo fractional derivatives we have:

where \(\frac{1}{p}+\frac{1}{q}=1.\)

The paper is organized in the manner that, in Section 2 we give two versions of the Hadamard inequality via Caputo fractional derivatives of strongly convex functions. These inequalities give refinements of Theorems 2 and 4. In Section 3, by employing identities stated in Lemmas 1 and 2, the refinements of Theorems 3 and 6 are presented.

2. Main results

The following results gives the refinement of the Hadamard inequality for Caputo fractional derivatives stated in Theorem 2.

Theorem 7. Let \(\psi: [x_{0},y_{0}]\rightarrow\mathbb{R}\) be the positive function such that \(\psi\in C^{n}[x_{0},y_{0}]\) and \(0\leq{x_{0}}< y_{0}\). If \(\psi^{(n)}\) is strongly convex function with modulus \(C\), then for Caputo fractional derivatives we have:

with \({\beta}>0\).

Proof. Since \(\psi^{(n)}\) is strongly convex function with modulus \(C\), for \(x,y\in[x_{0},y_{0}]\), we have

Let \(x=x_{0}z+(1-z)y_{0}\) and \(y=y_{0}z+(1-z)x_{0}\), \(z\in[0,1]\). Then we have Multiplying (16) with \(z^{n-\beta-1}\) on both sides and making integration over \([0,1]\) we get By using change of variables and computing the last integral, from (17) we get Further it takes the following form Since \(\psi^{(n)}\) is strongly convex function with modulus \(C\), for \(z\in[0,1],\) then one has Multiplying (20) with \(z^{n-\beta-1}\) on both sides and making integration over \([0,1]\) we get By using change of variables and computing the last integral, from (21) we get \begin{align} &\frac{1}{y_{0}-x_{0}}\int_{x_{0}}^{y_{0}}\psi^{(n)}(x)\left(\frac{y_{0}-x}{y_{0}-x_{0}}\right)^{n-\beta-1}\!\!\!dx+\frac{1}{y_{0}-x_{0}}\!\!\!\int_{x_{0}}^{y_{0}}\!\!\!\psi^{(n)}(y)\left(\frac{y-x_{0}}{y_{0}-x_{0}}\right)^{n-\beta-1}\!\!\!dy\notag\\ &\nonumber\leq \frac{\psi^{(n)}(x_{0})+\psi^{(n)}(y_{0})}{n-\beta}-\frac{2C(y_{0}-x_{0})^2}{({n-\beta+1})({n-\beta+2})}. \end{align} Further, it takes the following form Inequalities (19) and (22) constituted the required inequality.

Remark 1. If \(C=0\) in (14), then we will get the fractional Hadamard inequality stated in Theorem 2, otherwise its refinement is obtained.

The upcoming result is the refinement of another version of the Hadamard inequality for Caputo fractional derivatives stated in Theorem 4.

Theorem 8. Let \(\psi: [x_{0},y_{0}]\rightarrow\mathbb{R}\) be the positive function such that \(\psi\in C^{n}[x_{0},y_{0}]\) and \(0\leq{x_{0}}< y_{0}\). If \(\psi^{(n)}\) is a strongly convex function with modulus \(C\), then the undermentioned inequality for Caputo fractional derivatives holds:

with \({\beta}>0\).

Proof. Let \(x=x_{0}\frac{z}{2}+y_{0}\left(\frac{2-z}{2}\right)\) and \(y=x_{0}\left(\frac{2-z}{2}\right)+y_{0}\frac{z}{2}\), \(z\in[0,1]\) in (15), then we have

Multiplying (24) with \(z^{n-\beta-1}\) on both sides and making integration over \([0,1]\) we get By using change of variables and computing the last integral, from (25) we get Further, it takes the following form Since \(\psi^{(n)}\) is strongly convex function and \(z\in[0,1]\), we have the following inequality: Multiplying (28) with \(z^{n-\beta-1}\) on both sides and making integration over \([0,1]\) we get By using change of variables and computing the last integral, from (29) we get Further, it takes the following form From (27) and (31), (23) can be obtained.

Remark 2. If \(C=0\) in (23), then we will get the fractional Hadamard inequality stated in Theorem 4, otherwise its refinement is obtained.

3. Error bounds of fractional Hadamard inequalities

In this section we give refinements of the error bounds of fractional Hadamard inequalities for Caputo fractional derivatives:

Theorem 9. Let \(\psi: [x_{0},y_{0}]\rightarrow\mathbb{R}\) be the function such that \(\psi\in C^{n+1}[x_{0}, y_{0}]\) and \(0\leq x_{0} < y_{0}\). If \(|\psi^{(n+1)}|\) is a strongly convex function on \([x_{0},y_{0}]\), then the undermentioned inequality for Caputo fractional derivatives holds:

with \({\beta}>0\).

Proof. By applying Lemma 1 and the strong convexity of \(|\psi^{(n+1)}|\), we find

In the following we compute integrals appearing on the right side of inequality (33): By putting the values from (34) and (35) in (33), we get (32).

Remark 3. If \(C=0\) in (32), we get the fractional Hadamard inequality which is stated in Theorem 3, otherwise its refinement is obtained.

By using Lemma 2, we give the following error bounds of Caputo fractional derivative inequality (23).

Theorem 10. Let \(\psi:[x_{0},y_{0}]\rightarrow\mathbb{R}\) be the function such that \(\psi\in C^{n+1}[x_{0}, y_{0}]\) and \(0\leq x_{0}< y_{0}\). If \(|\psi^{(n+1)}|^q\) is strongly convex function on \([x_{0},y_{0}]\) for \(q\geq1\), then the undermentioned inequality for Caputo fractional derivatives holds:

Proof. By applying Lemma 2 using power mean inequality, we have \begin{align*} \nonumber &\left|\frac{2^{n-\beta-1}\Gamma({n-\beta+1})}{(y_{0}-x_{0})^{n-\beta}}\left[(^{C}{D_{(\frac{x_{0}+y_{0}}{2})^+}^{\beta}}\psi)(y_{0})+(-1)^n(^{C}{D_{(\frac{x_{0}+y_{0}}{2})^-}^{\beta}}\psi)(x_{0})\right]\right. \left.-\psi^{(n)}\left(\frac{x_{0}+y_{0}}{2}\right)\right|\\ &\leq\frac{y_{0}-x_{0}}{4}\left[\int_{0}^{1}z^{n-\beta}\left|\psi^{(n+1)}\left(x_{0}\frac{z}{2}+y_{0}\left(\frac{2-z}{2}\right)\right)\right|dz\right. \left.+\int_{0}^{1}z^{n-\beta}\left|\psi^{(n+1)}\left(x_{0}\left(\frac{2-z}{2}\right)+y_{0}\frac{z}{2}\right)\right|dz\right]\\ &\leq\frac{y_{0}-x_{0}}{4}\left(\int_{0}^{1}z^{n-\beta}dz\right)^{1-\frac{1}{q}}\left[\left(\int_{0}^{1}z^{n-\beta}\left|\psi^{(n+1)}\left(x_{0}\frac{z}{2}+y_{0}\left(\frac{2-z}{2}\right)\right)\right|^qdz\right)^{\frac{1}{q}}\right.\\ &\;\;\;\left.+\left(\int_{0}^{1}z^{n-\beta}\left|\psi^{(n+1)}\left(x_{0}\left(\frac{2-z}{2}\right)+y_{0}\frac{z}{2}\right)\right|^qdz\right)^{\frac{1}{q}}\right], \end{align*} Now, applying strong convexity of \(|\psi^{(n+1)}|\), we have \begin{align*} &\leq\frac{y_{0}-x_{0}}{4(n-\beta+1)^{\frac{1}{p}}}\left[\left(|\psi^{(n+1)}(x_{0})|^q\int_{0}^{1}\frac{z^{n-\beta+1}}{2}dz+|\psi^{(n+1)}(y_{0})|^q\int_{0}^{1}\frac{2z^{n-\beta}-z^{n-\beta+1}}{2}dz\right.\right.\\ &\;\;\;\left.\left.-\frac{C(y_{0}-x_{0})^2}{4}\int_{0}^{1}(2z^ {n-\beta+1}-z^{n-\beta+2})dz\right)^\frac{1}{q}+\left(|\psi^{(n+1)}(x_{0})|^q\int_{0}^{1}\frac{2z^{n-\beta}-z^{n-\beta+1}}{2}dz\right.\right.\\& \;\;\;\left.\left.+|\psi^{(n+1)}(y_{0})|^q\int_{0}^{1}\frac{z^{n-\beta+1}}{2}dz-\frac{C(y_{0}-x_{0})^2}{4}\int_{0}^{1}(2z^{n-\beta+1}-z^{n-\beta+2})dz\right)^\frac{1}{q}\right] \\&=\frac{y_{0}-x_{0}}{4(n-\beta+1)^{\frac{1}{p}}}\left(\frac{1}{2(n-\beta+1)({n-\beta+2})}\right)^\frac{1}{q}\left[\left(({n-\beta+1})|\psi^{(n+1)}(x_{0})|^q\right.\right.\\ &\;\;\;\left.\left.+({n-\beta+3})|\psi^{(n+1)}(y_{0})|^q-\frac{C(y_{0}-x_{0})^2({n-\beta+1})({n-\beta+4})}{2({n-\beta+3})}\right)^\frac{1}{q}\right.\\ &\;\;\;\left.+\left(({n-\beta+3})|\psi^{(n+1)}(x_{0})|^q+({n-\beta+1})|\psi^{(n+1)}(y_{0})|^q\right.\right.\left.\left.-\frac{C(y_{0}-x_{0})^2({n-\beta+1})({n-\beta+4})}{2({n-\beta+3})}\right)^\frac{1}{q}\right], \end{align*} inequality (36) is obtained.

Remark 4. If \(C=0\) in (36), then we will get inequality stated in Theorem 5, otherwise its refinement is obtained.

Theorem 11. Let \(\psi: [x_{0},y_{0}]\rightarrow\mathbb{R}\) be a differentiable mapping on \((x_{0},y_{0})\) with \(\psi\in C^{n+1}[x_{0}, y_{0}]\) and \(x_{0}< y_{0}\). If \(|\psi^{(n+1)}|^q\) is a strongly convex function on \([x_{0},y_{0}]\) for \(q>1\), then the undermentioned inequality for Caputo fractional derivatives holds:

Proof. By applying Lemma 2 and with the help of modulus property, we get \begin{align*} &\left|\frac{2^{n-\beta-1}\Gamma({n-\beta+1})}{(y_{0}-x_{0})^{n-\beta}}\left[(^{C}{D_{(\frac{x_{0}+y_{0}}{2})^+}^{\beta}}\psi)(y_{0})+(-1)^n(^{C}{D_{(\frac{x_{0}+y_{0}}{2})^-}^{\beta}}\psi)(x_{0})\right]-\psi^{(n)}\left(\frac{x_{0}+y_{0}}{2}\right)\right|\\ &\leq\frac{y_{0}-x_{0}}{4}\left[\int_{0}^{1}z^{n-\beta}\left|\psi^{(n+1)}\left(\frac{z}{2}x_{0}+y_{0}\frac{(2-z)}{2}\right)\right|dz+\int_{0}^{1}z^{n-\beta}\left|\psi^{(n+1)}\left(\frac{2-z}{2}x_{0}+\frac{z}{2}y_{0}\right)\right|dz\right]. \end{align*} Now applying Hölder's inequality, we get \begin{align*} &\left|\frac{2^{n-\beta-1}\Gamma({n-\beta+1})}{(y_{0}-x_{0})^{n-\beta}}\left[(^{C}{D_{(\frac{x_{0}+y_{0}}{2})^+}^{\beta}}\psi)(y_{0})+(-1)^n(^{C}{D_{(\frac{x_{0}+y_{0}}{2})^-}^{\beta}}\psi)(x_{0})\right]-\psi^{(n)}\left(\frac{x_{0}+y_{0}}{2}\right)\right|\\ &\leq\frac{y_{0}-x_{0}}{4}\left(\frac{1}{np-\beta p+1}\right)^{\frac{1}{p}}\left[\left(\int_{0}^{1}\left|\psi^{(n+1)}\left(\frac{z}{2}x_{0}+y_{0}\frac{(2-z)}{2}\right)\right|dz\right)^{\frac{1}{q}}\right.\\ &\;\;\;\left.+\left(\int_{0}^{1}z^{n-\alpha}\left|\psi^{(n+1)}\left(\frac{2-z}{2}x_{0}+\frac{z}{2}y_{0}\right)\right|dz\right)^{\frac{1}{q}}\right]. \end{align*} Using strong convexity of \(|\psi^{(n+1)}|^q\), we get

Here we have used the fact that \((a_{1}+b_{1})^q\geq(a_{1})^q+(b_{1})^q\), where \(q>1\), \(a_{1}, b_{1}\geq 0\). This completes the proof.

Remark 5. If \(C=0\) in (37), then we will get inequality stated in Theorem 6, otherwise its refinement is obtained.

4. Concluding Remarks

This paper provides refinements of fractional versions of the Hadamard inequalities for Caputo fractional derivatives using strongly convex functions. Also, refinements of error bounds of the Hadamard inequalities are given by using well established identities. Further, the authors are working for refinements of other well known fractional integral inequalities by employing strongly convexities of other kinds.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 11871202, 61673169, 11701176, 11626101, 11601485). The research work of Atiq Ur Rehman and Ghulam Farid is supported by the Higher Education Commission of Pakistan with Project No. 7962 and Project No. 5421 respectively.

Author Contributions

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

Conflicts of interest

''The authors declare no conflict of interest.''