Sufficient conditions for univalence and study of a class of meromorphic univalent functions

In this article we consider the class $\mathcal{A}(p)$ which consists of functions that are meromorphic in the unit disc $\text{ID}$ having a simple pole at $z=p\in (0,1)$ with the normalization $f(0)=0=f^{\mathrm{\prime }}(0)-1$. First we prove some sufficient conditions for univalence of such functions in $\text{ID}$. 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 ${\mathcal{U}}_p(\lambda )⊊{\mathcal{V}}_p(\lambda )$, 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.