Criteria for univalence, Integral means and Dirichlet integral for Meromorphic functions

Abstract

Let $\mathcal{A}(p)$ be the class consisting of functions $f$ that are holomorphic in $\ID\setminus \{p\}$, $p\in (0,1)$ possessing a simple pole at the point $z=p$ with nonzero residue and normalized by the condition $f(0)=0=f'(0)-1$. In this article, we first prove a sufficient condition for univalency for functions in $\mathcal{A}(p)$. Thereafter, we consider the class denoted by $\Sigma(p)$ that consists of functions $f \in \mathcal{A}(p)$ that are univalent in $\ID$. We obtain the exact value for $\ds\max_ {f\in \Sigma(p)}\Delta(r,z/f)$, where the Dirichlet integral $\Delta(r,z/f)$ is given by $$ \Delta(r,z/f)=\ds\iint_{|z|<r} |\left(z/f(z)\right)'|^2 \,dx\, dy, \quad(z=x+iy),~0<r\leq 1. $$ We also obtain a sharp estimate for $\Delta(r,z/f)$ whenever $f$ belongs to certain subclasses of $\Sigma(p)$. Furthermore, we obtain sharp estimates of the integral means for the aforementioned classes of functions.