Logharmonic mappings with typically real analytic components
Bull. Korean Math. Soc. 2018 Vol. 55, No. 6, 1783-1789
Published online June 29, 2018
Printed November 30, 2018
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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