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http://dx.doi.org/10.4134/BKMS.b171072

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)
Publication Information
Bulletin of the Korean Mathematical Society / v.55, no.6, 2018 , pp. 1783-1789 More about this Journal
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;
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