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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)
  • Received : 2017.12.11
  • 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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