  In this article we consider the class $\mathcal{A}(p)$ which consists of
functions that are meromorphic in the unit disc $\ID$ having a simple pole at
$z=p\in (0,1)$ with the normalization $f(0)=0=f'(0)-1 $. First we prove some
sufficient conditions for univalence of such functions in $\ID$. One of these
conditions enable us to consider the class $\mathcal{V}_{p}(\lambda)$ that
consists of functions satisfying certain differential inequality which forces
univalence of such functions. Next we establish that
$\mathcal{U}_{p}(\lambda)\subsetneq \mathcal{V}_{p}(\lambda)$, where
$\mathcal{U}_{p}(\lambda)$ was introduced and studied in \cite{BF-1}. Finally,
we discuss some coefficient problems for $\mathcal{V}_{p}(\lambda)$ and end the
article with a coefficient conjecture.

Complex Variables