Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2023; 60(5): 1253-1263

Online first article August 22, 2023      Printed September 30, 2023

https://doi.org/10.4134/BKMS.b220643

Copyright © The Korean Mathematical Society.

Certain properties of the class of univalent functions with real coefficients

Milutin Obradovic, Nikola Tuneski

Bulevar Kralja Aleksandra 73; Karpo\v{s} II b.b., 1000 Skopje

Abstract

Let $\mathcal U^+$ be the class of analytic functions $f$ such that $\frac{z}{f(z)}$ has real and positive coefficients and $f^{-1}$ be its inverse. In this paper we give sharp estimates of the initial coefficients and initial logarithmic coefficients for $f$, as well as, sharp estimates of the second and the third Hankel determinant for $f$ and $f^{-1}$. We also show that the Zalcman conjecture holds for functions $f$ from $\mathcal U^+$.

Keywords: Univalent, real coefficients, logarithmic coefficients, coefficient estimates, Hankel determinant, Zalcman conjecture

MSC numbers: 30C45, 30C50, 30C55