Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc. 2018; 55(6): 1783-1789

Online first article June 29, 2018      Printed November 30, 2018

https://doi.org/10.4134/BKMS.b171072

Copyright © The Korean Mathematical Society.

Logharmonic mappings with typically real analytic components

Zayid AbdulHadi, Najla M. Alarifi, Rosihan M. Ali

American University of Sharjah, Imam Abdulrahman Bin Faisal University, University Sains Malaysia

Abstract

This paper treats the class of normalized logharmonic mappings $f(z)=zh(z)\overline{g(z)}$ in the unit disk satisfying $\varphi(z)=zh(z)g(z)$ is analytically typically real. Every such mapping $f$ admits an integral representation in terms of its second dilatation function and a function of positive real part with real coefficients. The radius of starlikeness and an upper estimate for arclength are obtained. Additionally, it is shown that $f$ maps the unit disk into a domain symmetric with respect to the real axis when its second dilatation has real coefficients.

Keywords: logharmonic mappings, typically real functions, radius of starlikeness, arclength

MSC numbers: Primary 30C35, 30C45

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