# On the Taylor Coefficients of a Subclass of Meromorphic Univalent Functions

• Firdoshi Parveen
Article

## 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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© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018