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