Title: 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 $f(0)=0=f'(0)-1$ 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 $|a_{n+1}(f)-a_n(f)/p|,\,n\geq
2$.