Sufficient conditions for univalence and study of a class of meromorphic univalent functions
Bull. Korean Math. Soc. 2018 Vol. 55, No. 3, 999-1006
Published online March 8, 2018
Printed May 31, 2018
Bappaditya Bhowmik, Firdoshi Parveen
Indian Institute of Technology Kharagpur, Indian Institute of Technology Kharagpur
Abstract : In this article we consider the class $\mathcal{A}(p)$ which consists of functions that are meromorphic in the unit disc $\ID$ having a simple pole at $z=p\in (0,1)$ with the normalization $f(0)=0=f'(0)-1 $. First we prove some sufficient conditions for univalence of such functions in $\ID$. One of these conditions enable us to consider the class $\mathcal{V}_{p}(\lambda)$ that consists of functions satisfying certain differential inequality which forces univalence of such functions. Next we establish that $\mathcal{U}_{p}(\lambda)\subsetneq \mathcal{V}_{p}(\lambda)$, where $\mathcal{U}_{p}(\lambda)$ was introduced and studied in \cite{BF-1}. Finally, we discuss some coefficient problems for $\mathcal{V}_{p}(\lambda)$ and end the article with a coefficient conjecture.
Keywords : meromorphic functions, univalent functions, subordination, Taylor coefficients
MSC numbers : 30C45, 30C55
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