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

Bhowmik, Bappaditya; Parveen, Firdoshi

Bull. Belg. Math. Soc. - Simon Stevin 24, No. 3, 427-438 (2017).

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.

MSC:

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

Keywords:

concave function; star-like function; Dirichlet integral