Criteria for univalence, integral means and Dirichlet integral for meromorphic functions. (English) Zbl 1377.30008

Summary: Let \(\mathcal{A}(p)\) be the class consisting of functions \(f\) that are holomorphic in \(\mathbb D\setminus \{p\}\), \(p\in (0,1)\) 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 \(\mathbb D\). We obtain the exact value for \(\displaystyle\max_ {f\in \Sigma(p)}\Delta(r,z/f)\), where the Dirichlet integral \(\Delta(r,z/f)\) is given by \[ \Delta(r,z/f)=\int\int_{|z|<r}|(z/f(z))'|^2dx dy,\quad (z=x+iy),0<r\leq 1. \] 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.


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
Full Text: arXiv Euclid