On the Taylor coefficients of a subclass of meromorphic univalent functions

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 $f(0)=0=f'(0)-1$ and satisfying the differential inequality $|(z/f(z))^2 f'(z)-1|< \lambda$ for $z\in \ID$, $0<\lambda\leq 1$. Each $f\in\mathcal{V}_p(\lambda)$ has the following Taylor expansion: $$f(z)=z+\sum_{n=2}^{\infty}a_n(f) z^n, \quad |z|<p.$$ In \cite{BF-3}, we conjectured that $$|a_n(f)|\leq \frac{1-(\lambda p^2)^n}{p^{n-1}(1-\lambda p^2)}\quad \mbox{for}\quad n\geq3.$$ 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 $|a_n(f)|,\,n\geq 3$ and for $|a_{n+1}(f)-a_n(f)/p|,\,n\geq 2$.