On the Taylor coefficients of a subclass of meromorphic univalent functions

Abstract

Let ${\mathcal{V}}_p(\lambda )$ be the collection of all functions $f$ defined in the unit disc $\ID$ having a simple pole at $z=p$ where $0<p<1$ and analytic in $\ID\setminus\{p\}$ with $f(0)=0=f^{\prime }(0)-1$ and satisfying the differential inequality $|(z/f(z){)}^2{f}^{\prime }(z)-1|<\lambda$ for $z\in \ID$, $0<\lambda \le 1$. Each $f\in {\mathcal{V}}_p(\lambda )$ has the following Taylor expansion: $$f(z)=z+\sum _{n=2}^{\infty }{a}_n(f)z^n,|z|<p.$$ In \cite{BF-3}, we conjectured that $$ |a_n(f)|\leq \frac{1-(λp^2)^n}{p^{n-1}(1-λp^2)}\quad \mbox{for}\quad n\geq3. $$ In the present article, we first obtain a representation formula for functions in the class ${\mathcal{V}}_p(\lambda )$. Using this representation, 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 $|a_{n+1}(f)-a_n(f)/p|,n\ge 2$.

8 pages