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2014 Initial coefficient bounds for a subclass of $m$-fold symmetric bi-univalent functions
H. M. Srivastava, S. Sivasubramanian, R. Sivakumar
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Tbilisi Math. J. 7(2): 1-10 (2014). DOI: 10.2478/tmj-2014-0011

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 $|a_{2m+1}|$.

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H. M. Srivastava. S. Sivasubramanian. R. Sivakumar. "Initial coefficient bounds for a subclass of $m$-fold symmetric bi-univalent functions." Tbilisi Math. J. 7 (2) 1 - 10, 2014. https://doi.org/10.2478/tmj-2014-0011

Information

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

zbMATH: 1304.30026
MathSciNet: MR3313050
Digital Object Identifier: 10.2478/tmj-2014-0011

Subjects:
Primary: 30C45
Secondary: 30C50, 30C80

Rights: Copyright © 2014 Tbilisi Centre for Mathematical Sciences

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Vol.7 • No. 2 • 2014
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