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On the Taylor Coefficients of a Subclass of Meromorphic Univalent Functions

  • Bappaditya Bhowmik
  • Firdoshi Parveen
Article
  • 28 Downloads

Abstract

Let \({\mathcal {V}}_p(\lambda )\) be the collection of all functions f defined in the unit disc \({{\mathbb {D}}}\) having a simple pole at \(z=p\) where \(0<p<1\) and analytic in \({{\mathbb {D}}}\setminus \{p\}\) with \(f(0)=0=f'(0)-1\) and satisfying the differential inequality \(|(z/f(z))^2 f'(z)-1|< \lambda \) for \(z\in {{\mathbb {D}}}\), \(0<\lambda \le 1\). Each \(f\in {\mathcal {V}}_p(\lambda )\) has the following Taylor expansion:
$$\begin{aligned} f(z)=z+\sum _{n=2}^{\infty }a_n(f) z^n, \quad |z|<p. \end{aligned}$$
We recently conjectured that
$$\begin{aligned} |a_n(f)|\le \frac{1-(\lambda p^2)^n}{p^{n-1}(1-\lambda p^2)}\quad \text{ for }\quad n\ge 3, \end{aligned}$$
while investigating functions in the class \({\mathcal {V}}_p(\lambda )\). In the present article, we first obtain a representation formula for functions in this class. Using this, we prove the aforementioned conjecture for \(n=3,4,5\) whenever p belongs to certain subintervals of (0, 1). Also we determine non sharp bounds for \(|a_n(f)|,\,n\ge 3\) and for \(|a_{n+1}(f)-a_n(f)/p|,\,n\ge 2\).

Keywords

Meromorphic functions Univalent functions Subordination Taylor coefficients 

Mathematics Subject Classification

30C45 30C50 30C55 30C80 

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

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