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.
Zayid AbdulHadi, Najla M. Alarifi, Rosihan M. Ali
American University of Sharjah, Imam Abdulrahman Bin Faisal University, University Sains Malaysia
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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