Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

Version 1 : Received: 13 November 2020 / Approved: 16 November 2020 / Online: 16 November 2020 (15:08:35 CET)

How to cite: Sugawa, T.; Wang, L. On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order. Preprints 2020, 2020110428. https://doi.org/10.20944/preprints202011.0428.v1. Sugawa, T.; Wang, L. On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order. Preprints 2020, 2020110428. https://doi.org/10.20944/preprints202011.0428.v1.

## Abstract

We consider the inverse function $z=g(w)$ of a (normalized) starlike function $w=f(z)$ of order $\alpha$ on the unit disk of the complex plane with $0<\alpha<1.$ Krzy{\. z}, Libera and Z\l otkiewicz obtained sharp estimates of the second and the third coefficients of $g(w)$ in their 1979 paper. Prokhorov and Szynal gave sharp estimates of the fourth coefficient of $g(w)$ as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of $g(w)$ together with explicit forms of the extremal functions.

## Keywords

starlike function of order $\alpha$; inverse function; coefficient estimates

## Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

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