Tbilisi Mathematical Journal

Initial coefficient bounds for a subclass of $m$-fold symmetric bi-univalent functions

H. M. Srivastava, S. Sivasubramanian, and R. Sivakumar

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Abstract

Let $\Sigma$ denote the class of functions $$f(z)=z+\sum_{n=2}^{\infty}a_nz^n$$ belonging to the normalized analytic function class $\mathcal{A}$ in the open unit disk $\mathbb{U}$, which are bi-univalent in $\mathbb{U}$, that is, both the function $f$ and its inverse $f^{-1}$ are univalent in $\mathbb{U}$. The usual method for computation of the coefficients of the inverse function $f^{-1}(z)$ by means of the relation $f^{-1}\big(f(z)\big)=z$ is too difficult to apply in the case of $m$-fold symmetric analytic functions in $\mathbb{U}$. Here, in our present investigation, we aim at overcoming this difficulty by using a general formula to compute the coefficients of $f^{-1}(z)$ in conjunction with the residue calculus. As an application, we introduce two new subclasses of the bi-univalent function class $\Sigma$ in which both $f(z)$ and $f^{-1}(z)$ are $m$-fold symmetric analytic functions with their derivatives in the class $\mathcal{P}$ of analytic functions with positive real part in $\mathbb{U}$. For functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for $|a_{m+1}|$ and $|a_{2m+1}|$.

Article information

Source
Tbilisi Math. J., Volume 7, Issue 2 (2014), 1-10.

Dates
Received: 9 June 2014
Accepted: 13 October 2014
First available in Project Euclid: 12 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1528768970

Digital Object Identifier
doi:10.2478/tmj-2014-0011

Mathematical Reviews number (MathSciNet)
MR3313050

Zentralblatt MATH identifier
1304.30026

Subjects
Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Secondary: 30C50: Coefficient problems for univalent and multivalent functions 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination

Keywords
Analytic function Univalent functions Bi-Univalent functions $m$-Fold symmetric functions $m$-Fold symmetric bi-univalent functions

Citation

Srivastava, H. M.; Sivasubramanian, S.; Sivakumar, R. Initial coefficient bounds for a subclass of $m$-fold symmetric bi-univalent functions. Tbilisi Math. J. 7 (2014), no. 2, 1--10. doi:10.2478/tmj-2014-0011. https://projecteuclid.org/euclid.tbilisi/1528768970


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