Open Journal of Mathematical Analysis
ISSN: 2616-8111 (Online) 2616-8103 (Print)
DOI: 10.30538/psrp-oma2021.0081
Certain new subclasses of \(m\)-fold symmetric bi-pseudo-starlike functions using \(Q\)-derivative operator
Timilehin Gideon Shaba
Department of Mathematics, University of Ilorin, P. M. B. 1515, Ilorin, Nigeria.; shabatimilehin@gmail.com
Abstract
Keywords:
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
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 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 formThe 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:
Jackson [18,19] introduced the \(q\)-derivative operator \(\mathcal{D}_q\) of a function as follows;
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
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
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
Proof. Using inequalities (1) and (9), we get
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
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
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
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
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
Proof. Using inequalities (31) and (32), we get
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.References
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