# LOGHARMONIC MAPPINGS WITH TYPICALLY REAL ANALYTIC COMPONENTS

• AbdulHadi, Zayid (Department of Mathematics American University of Sharjah) ;
• Alarifi, Najla M. (Department of Mathematics Imam Abdulrahman Bin Faisal University) ;
• Ali, Rosihan M. (School of Mathematical Sciences Universiti Sains Malaysia)
• Accepted : 2018.05.16
• Published : 2018.11.30

#### 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.

#### Acknowledgement

Supported by : Universiti Sains Malaysia

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