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 $\text{ID}$ having a simple pole at $z=p$ where $0<p<1$ and analytic in $\text{ID}\setminus \{p\}$ with $f(0)=0=f^{\mathrm{\prime }}(0)-1$ and satisfying the differential inequality $\mathrm{\mid }(z\mathrm{/}f(z){)}^2{f}^{\mathrm{\prime }}(z)-1\mathrm{\mid }<\lambda$ for $z\in \text{ID}$, $0<\lambda \le 1$. Each $f\in {\mathcal{V}}_p(\lambda )$ has the following Taylor expansion: $$f(z)=z+\sum _{n=2}^{\mathrm{\infty }}{a}_n(f)z^n,\mathrm{\mid }z\mathrm{\mid }<p\mathrm{.}$$ In \cite{BF-3}, we conjectured that $$\mathrm{\mid }{a}_n(f)\mathrm{\mid }\le \frac{1-(\lambda p^2{)}^n}{p^{n-1}(1-\lambda p^2)}forn\ge 3.$$ 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 $\mathrm{\mid }{a}_n(f)\mathrm{\mid },n\ge 3$ and for $|a_{n+1}(f)-a_n(f)/p|,,n\geq 2$.