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December 2016 Advances on the coefficient bounds for m-fold symmetric bi-close-to-convex functions
Jay M. Jahangiri, Samaneh G. Hamidi
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Tbilisi Math. J. 9(2): 75-82 (December 2016). DOI: 10.1515/tmj-2016-0021

Abstract

In 1955, Waadeland considered the class of m-fold symmetric starlike functions of the form $f_{m}(z)=z+\sum_{n=1}^{\infty }a_{mn+1}z^{mn+1}$; $m\geq 1$; $|z|\lt1$ and obtained the sharp coefficient bounds $|a_{mn+1}|\leq\left[ (2/m+n-1)!\right] /\left[ (n!)(2/m-1)!\right] $. Pommerenke in 1962, proved the same coefficient bounds for m-fold symmetric close-to-convex functions. Nine years later, Keogh and Miller confirmed the same bounds for the class of m-fold symmetric Bazilevic functions. Here we will show that these bounds can be improved even further for the m-fold symmetric bi-close-to-convex functions. Moreover, our results improve those corresponding coefficient bounds given by Srivastava et al that appeared in 7(2) (2014) issue of this journal. A function is said to be bi-close-to-convex in a simply connected domain if both the function and its inverse map are close-to-convex there.

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Jay M. Jahangiri. Samaneh G. Hamidi. "Advances on the coefficient bounds for m-fold symmetric bi-close-to-convex functions." Tbilisi Math. J. 9 (2) 75 - 82, December 2016. https://doi.org/10.1515/tmj-2016-0021

Information

Received: 16 August 2015; Accepted: 10 September 2016; Published: December 2016
First available in Project Euclid: 12 June 2018

zbMATH: 1354.30006
MathSciNet: MR3571020
Digital Object Identifier: 10.1515/tmj-2016-0021

Subjects:
Primary: 30C45
Secondary: 30C50

Rights: Copyright © 2016 Tbilisi Centre for Mathematical Sciences

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Vol.9 • No. 2 • December 2016
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