Certain new subclasses of \(m\)-fold symmetric bi-pseudo-starlike functions using \(Q\)-derivative operator

1. Introduction

Let \(\mathcal{A}\) be the family of holomorphic functions, normalized by the conditions \(f(0)=f'(0)-1=0\) which is of the form

in the open unit disk \(\varOmega=\{z;z\in \mathbb{C}\;\text{and}\;|z|< 1\}\). We denote by \(\mathcal{G}\) the subclass of functions in \(\mathcal{A} \) which are univalent in \(\varOmega\) (for more details see [1]).

The Keobe-One Quarter Theorem [1] state that the image of \(\varOmega\) under all univalent function \(f\in \mathcal{A}\) contains a disk of radius \(\frac{1}{4}\). Hence all univalent function \(f\in \mathcal{A}\) has an inverse \(f^{-1}\) satisfy \(f^{-1}(f(z))\) and \(f(f^{-1}(\upsilon))=\upsilon\) \((|\upsilon|< r_0(f),\;r_0(f)\ge\frac{1}{4})\), where

A function \(f\in \mathcal{A}\) denoted by \(\varSigma\) is said to be bi-univalent in \(\varOmega\) if both \(f^{-1}(z)\) ans \(f(z)\) are univalent in \(\varOmega\) (see for details [2,3,4,5,6,7,8,9,10,11]).

A domain \(\varPsi\) is said to be \(m\)-fold symmetric if a rotation of \(\varPsi\) about the origin through an angle \(2\pi/m\) carries \(\varPsi\) on itself. Therefore, a function \(f(z)\) holomorphic in \(\varOmega\) is said to be \(m\)-fold symmetric if

\begin{equation*} f\left(e^{\frac{2\pi i}{m}}z\right) =e^{\frac{2\pi i}{m}}f(z).\end{equation*} A function is said to be \(m\)-fold symmetric if it has the following normalized form Let \(\mathfrak{S}_m\) the class of \(m\)-fold symmetric univalent functions in \(\varOmega\), that are normalized by (3), in which, the functions in the class \(\mathfrak{S}\) are \(one\)-fold symmetric. Similar to the concept of \(m\)-fold symmetric univalent functions, we introduced the concept of \(m\)-fold symmetric bi-univalent functions which is denoted by \(\varSigma_m\). Each of the function \(f\in \varSigma\) produces \(m\)-fold symmetric bi-univalent function for each integer \(m\in\mathcal{N}\).

The normalized form of \(f(z)\) is given as in (3) and the series expansion for \(f^{-1}(z)\), which has been investigated by Srivastava et al., [12], is given below:

Some of the examples of \(m\)-fold symmetric bi-univalent functions are \[\Biggl\{\frac{z^m}{1-z^m}\Biggr\}^{\frac{1}{m}},\] \[\left[-\log(1-z^{m})\right]^{\frac{1}{m}},\] and \[\Biggl\{\frac{1}{2}\log \left(\frac{1+z^m}{1-z^m}\right)^{\frac{1}{m}}\Biggr\}.\] For more details on \(m\)-fold symmetric analytic bi-univalent functions (see [5,12,13,14,15,16,17]).

Jackson [18,19] introduced the \(q\)-derivative operator \(\mathcal{D}_q\) of a function as follows;

and \(\mathcal{D}_qf(0)=f'(0)\). In case of \(g(z)=z^{k}\) for \(k\) is a positive integer, the \(q\)-derivative of \(f(z)\) is given by \begin{equation*} \mathcal{D}_{q}z^{k}=\frac{z^{k}-(zq)^{k}}{(q-1)z}=[k]_qz^{k-1}. \end{equation*} As \(q\longrightarrow1^{-}\) and \(k\in \mathcal{N}\), we get where \((z\neq 0,\;q\neq0)\). For more details on the concepts of \(q\)-derivative (see [5,20,21,22,23,24,25,26,27]).

Definition 1. [28] Let \(f(z)\in \mathcal{A}\), \(0\leq\chi< 1\) and \(\sigma\ge 1\) is real. Then \(f(z)\in L_{\sigma}(\chi)\) of \(\sigma\)-pseodu-starlike function of order \(\chi\) in \(\varOmega\) if and only if

Babalola [28] verified that, all pseodu-starlike function are Bazilevic of type \(\left(1-\frac{1}{\sigma}\right)\), order \(\chi^{\frac{1}{\sigma}}\) and univalent in \(\varOmega\).

Lemma 1. [1] Let the function \(\omega\in \mathcal{P}\) be given by the following series \(\omega(z)=1+\omega_1z+\omega_2z^2+\cdots\quad(z\in \varOmega).\) The sharp estimate given by \(|\omega_n|\leq2\quad(n\in \mathcal{N})\) holds true.

In [29] Girgaonkar et al., introduced a new subclasses of holomorphic and bi-univalent functions as follows:

Definition 2. A function \(f(z)\) given by (1) is said to be in the class \(\mathcal{M}_{\varSigma}(\chi)\;(0< \chi\leq1,(z,\upsilon)\in\varOmega)\) if \( f\in\mathcal{E}\), \(|\arg(f'(z))^{\sigma}|< \frac{\chi\pi}{2} \) and \( |\arg(g'(\upsilon))^{\sigma}|< \frac{\chi\pi}{2}, \) where \(g(\upsilon)\) is given by (2).

Definition 3. A function \(f(z)\) given by (1) is said to be in the class \(\mathcal{M}_{\varSigma}(\psi)\;(0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) if \( \vartheta\in\varSigma\), \( \Re[(f'(z))^{\sigma}]>\psi \) and \( \Re[(g'(\upsilon))^{\sigma}]>\psi,\) where \(g(\upsilon)\) is given by (2).

In this current research, we introduced two new subclasses denoted by \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\chi)\) and \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\psi)\) of the function class \(\varSigma_m\) and obtain estimates coefficient \(|\rho_{m+1}|\) and \(|\rho_{2m+1}|\) for functions in these two new subclasses.

2. Main 4esults

Definition 4. A function \(f(z)\) given by (3) is said to be in the class \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\chi)\;(m\in \mathcal{N}, 0< q< 1, \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) if

and where \(g(\upsilon)\) is given by (2).

Remark 1. We have the class \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{\sigma}_{\varSigma,1}(\chi)=\mathcal{M}^{\sigma}_{\varSigma}(\chi)\) which was introduced and studied by Girgaonkar et al., [29].

Remark 2. We have the class \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{1}_{\varSigma,1}(\chi)=\mathcal{M}_{\varSigma}(\chi)\) which was introduced and studied by Srivastava et al., [11].

Theorem 1. Let \(f(z)\in \mathcal{M}^{q,\sigma}_{\varSigma,m}(\chi)\), \((m\in \mathcal{N}, 0< q< 1, \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) be given (3). Then

and

Proof. Using inequalities (1) and (9), we get

and respectively, where \(\tau(z)\) and \(\varsigma(\upsilon)\) in \(\mathcal{P}\) are given by the following series and Clearly, \begin{equation*} [\tau(z)]^{\chi}=1+\chi\tau_{m}z^{m}+\left(\chi\tau_{2m}+\frac{\chi(\chi-1)}{2}\tau_{m}^2\right)z^{2m}+\cdots, \end{equation*} and \begin{equation*} [\varsigma(\upsilon)]^{\chi}=1+\chi\varsigma_{m}\upsilon^{m}+\left(\chi\varsigma_{2m}+\frac{\chi(\chi-1)}{2}\varsigma_{m}^2\right)\upsilon^{2m}+\cdots. \end{equation*} Also \begin{equation*} (\mathcal{D}_{q}f(z))^{\sigma}=1+\sigma[m+1]_q\rho_{m+1}z^{m}+\left(\sigma[2m+1]_q\rho_{2m+1}+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2\right)z^{2m}+\cdots, \end{equation*} and \begin{align*} (\mathcal{D}_{q}g(\upsilon))^{\sigma}=&1-\sigma[m+1]_q\rho_{m+1}\upsilon^{m}-\sigma[2m+1]_q\rho_{2m+1}\upsilon^{2m}\\&+\Biggl(\sigma(m+1)[2m+1]_q\rho_{m+1}^2+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2\Biggr)\upsilon^{2m}+\cdots \end{align*} Comparing the coefficients in (12) and (13), we have From (16) and (18), we obtain and Further from (17), (19) and (21), we obtain that \begin{equation*} \sigma(\sigma-1)\chi[m+1]_q^2\rho_{m+1}^2+(m+1)\sigma\chi[2m+1]_q\rho_{m+1}^2-(\chi-1)\sigma^2[m+1]_q^2\rho_{m+1}^2=\chi^2(\tau_{2m}+\varsigma_{2m}). \end{equation*} Therefore, we have By applying Lemma 1 for the coefficients \(\tau_{2m}\) and \(\varsigma_{2m}\), then we have \begin{equation*} |\rho_{m+1}|\leq\frac{2\chi}{\sqrt{(m+1)\sigma\chi[2m+1]_q-(\chi-\sigma)\sigma[m+1]_q^2}}. \end{equation*} Also, to find the bound on \(|\rho_{2m+1}|\), using the relation (19) and (17), we obtain It follows from (20), (21) and (23), Applying Lemma 1 for the coefficients \(\tau_{m},\tau_{2m},\varsigma_{m},\varsigma_{2m}\), then we have \begin{equation*} |\rho_{2m+1}|\leq\frac{2\chi}{\sigma[2m+1]_q}+\frac{2(m+1)\chi^2}{\sigma^2[m+1]_q^2}. \end{equation*} Choosing \(q\longrightarrow1^{-1}\) in Theorem 1, we get the following result:

Corollary 1. Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma,m}(\chi)\), \((m\in \mathcal{N}, \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) be given (3). Then

and Choosing \(m=1\) (0ne-fold case) in Theorem 1, we get the following result:

Corollary 2. Let \(f(z)\in \mathcal{M}^{q,\sigma}_{\varSigma}(\chi)\), \((0< q< 1, \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) be given (1). Then

and Choosing \(q\longrightarrow1^{-1}\) in Corollary 2, we get the following result:

Corollary 3. [29] Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma}(\chi)\), \(( \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) be given (1). Then

and

Remark 3. For one-fold case, we have \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{q,1}_{\varSigma,1}(\chi)=\mathcal{M}_{\varSigma}(\chi)\), and we can get the results of Srivastava et al., [11].

Definition 5. A function \(f(z)\) given by (3) is said to be in the class \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\psi)\;(m\in \mathcal{N}, 0< q< 1, \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) if

and where \(g(\upsilon)\) is given by (2).

Remark 4. We have the class \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{\sigma}_{\varSigma,1}(\psi)=\mathcal{M}^{\sigma}_{\varSigma}(\chi)\) which was introduced and studied by Girgaonkar et al., [29].

Remark 5. We have the class \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{1}_{\varSigma,1}(\psi)=\mathcal{M}_{\varSigma}(\chi)\) which was introduced and studied by Srivastava et al., [11].

Theorem 2. Let \(f(z)\in \mathcal{M}^{q,\sigma}_{\varSigma,m}(\psi)\), \((m\in \mathcal{N}, 0< q< 1, \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (3). Then

and

Proof. Using inequalities (31) and (32), we get

and here \(\tau(z)\) and \(\varsigma(\upsilon)\) in \(\mathcal{P}\) are given by the following series \begin{equation*} \tau(z)=1+\tau_mz^m+\tau_{2m}z^{2m}+\tau_{3m}z^{3m}+\cdots, \end{equation*} and \begin{equation*} \varsigma(\upsilon)=1+\varsigma_m\upsilon^m+\varsigma_{2m}\upsilon^{2m}+\varsigma_{3m}\upsilon^{3m}+\cdots. \end{equation*} Clearly, \begin{equation*} \psi+(1-\psi)\tau(z)=1+(1-\psi)\tau_{m}z^{m}+(1-\psi)\tau_{2m}z^{2m}+\cdots, \end{equation*} and \begin{equation*} \psi+(1-\psi)\varsigma(\upsilon)=1+(1-\psi)\varsigma_{m}\upsilon^{m}+(1-\psi)\varsigma_{2m}\upsilon^{2m}+\cdots. \end{equation*} Also \begin{equation*} (\mathcal{D}_{q}f(z))^{\sigma}=1+\sigma[m+1]_q\rho_{m+1}z^{m}+\left(\sigma[2m+1]_q\rho_{2m+1}+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2\right)z^{2m}+\cdots, \end{equation*} and \begin{align*} (\mathcal{D}_{q}g(\upsilon))^{\sigma}=&1-\sigma[m+1]_q\rho_{m+1}\upsilon^{m}-\sigma[2m+1]_q\rho_{2m+1}\upsilon^{2m}\\&+\Biggl(\sigma(m+1)[2m+1]_q\rho_{m+1}^2+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2\Biggr)\upsilon^{2m}+\cdots. \end{align*} Now comparing the coefficients in (35) and (36), we get From (37) and (39), we obtain and Also, from (38) and (40), we get Applying the Lemma 1 for the coefficients \(\tau_{m},\tau_{2m},\varsigma_{m},\varsigma_{2m}\), we find that \begin{equation*} |\rho_{m+1}|\leq2\sqrt{\frac{(1-\psi)}{\sigma(\sigma-1)[m+1]_q^2+(m+1)\sigma[2m+1]_q}}. \end{equation*} Also, to find the bound on \(|\rho_{2m+1}|\), using the relation (40) and (38), we obtain or equivalently By substituting the value of \(\rho_{m+1}^2\) from (42), we have Applying the Lemma 1 for the coefficients \(\tau_{m},\tau_{2m},\varsigma_{m},\varsigma_{2m}\), we get \begin{equation*} |\rho_{2m+1}|\leq\frac{2(1-\psi)}{\sigma[2m+1]_q}+\frac{2(m+1)(1-\psi)^2}{2\sigma^2[m+1]_q^2}. \end{equation*} Also, by using (43) and (45), and applying Lemma 1 we obtain \begin{equation*} |\rho_{2m+1}|\leq\frac{2(m+1)(1-\psi)}{\sigma(\sigma-1)[m+1]_q^2+(m+1)\sigma[2m+1]_q}+\frac{2(1-\psi)}{\sigma[2m+1]_q}. \end{equation*} This complete the proof.

Choosing \(q\longrightarrow1^{-1}\) in Theorem 2, we get the following result:

Corollary 4. Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma,m}(\psi)\), \((m\in \mathcal{N}, \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (3). Then \begin{equation*} |\rho_{m+1}|\leq\left \{ \begin{array}{cc} 2\sqrt{\frac{(1-\psi)}{\sigma(\sigma-1)[m+1]^2+(m+1)\sigma[2m+1]}} & 0\leq\psi\leq\frac{m}{1+2m},\\ \frac{2(1-\psi)}{\sigma[m+1]} & \frac{m}{1+2m}\leq\psi< 1, \end{array} \right. \end{equation*} and \begin{equation*} |\rho_{2m+1}|\leq\frac{2(m+1)(1-\psi)}{\sigma(\sigma-1)[m+1]^2+(m+1)\sigma[2m+1]}+\frac{2(1-\psi)}{\sigma[2m+1]}. \end{equation*} For one fold case, Corollary 4, yields the following Corollary:

Corollary 5. Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma}(\psi)\), \(( \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (1). Then \begin{equation*} |\rho_{2}|\leq\left \{ \begin{array}{cc} \sqrt{\frac{2(1-\psi)}{\sigma(2\sigma+1)}} & 0\leq\psi\leq\frac{1}{3},\\ \frac{(1-\psi)}{\sigma} & \frac{1}{3}\leq\psi< 1, \end{array} \right. \end{equation*} and \begin{equation*} |\rho_{3}|\leq\frac{(1-\psi)(2\sigma-3\psi+3)}{3\sigma^2}. \end{equation*}

Remark 6. Corollary 5 gives above is the improvement of the estimates for coefficients on \(|\rho_{2}|\) and \(|\rho_{3}|\) investigated by Girgaonkar et al., [29].

Corollary 6. [29] Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma}(\psi)\), \(( \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (1). Then \begin{equation*} |\rho_{2}|\leq\sqrt{\frac{2(1-\psi)}{\sigma(2\sigma+1)}}, \end{equation*} and \begin{equation*} |\rho_{3}|\leq\frac{(1-\psi)(2\sigma-3\psi+3)}{3\sigma^2}. \end{equation*} Taking \(\sigma=1\) in Corollary 7, we get the following result:

Corollary 7. [11] Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma}(\psi)\), \(( \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (1). Then \begin{equation*} |\rho_{2}|\leq\sqrt{\frac{2(1-\psi)}{3}}, \end{equation*} and \begin{equation*} |\rho_{3}|\leq\frac{(1-\psi)(5-3\psi)}{3}. \end{equation*}

3. Conclusion

In this present paper, two new subclasses indicated by \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\chi)\) and \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\psi)\) of function class of \(\mathcal{E}_m\) was obtained and worked on. Also, the estimates coefficients for \(|\rho_{m+1}|\) and \(|\rho_{2m+1}|\) of functions in these classes are determined.

Conflicts of Interest

The author declares no conflict of interest.