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. Preprints2020, 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.
Cite as:
Sugawa, T.; Wang, L. On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order. Preprints2020, 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
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
