## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 24, Number 3 (2017), 427-438.

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

Bappaditya Bhowmik and Firdoshi Parveen

#### Abstract

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 $$ \Delta(r,z/f)=\displaystyle\iint_{|z|<r} |\left(z/f(z)\right)'|^2 \,dx\, dy, \quad(z=x+iy),~0<r\leq 1. $$ 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.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 24, Number 3 (2017), 427-438.

**Dates**

First available in Project Euclid: 27 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1506477691

**Mathematical Reviews number (MathSciNet)**

MR3706811

**Zentralblatt MATH identifier**

1377.30008

**Subjects**

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C70: Extremal problems for conformal and quasiconformal mappings, variational methods

**Keywords**

Meromorphic function Concave function Starlike function Dirichlet finite integral Integral mean

#### Citation

Bhowmik, Bappaditya; Parveen, Firdoshi. Criteria for univalence, Integral means and Dirichlet integral for Meromorphic functions. Bull. Belg. Math. Soc. Simon Stevin 24 (2017), no. 3, 427--438. https://projecteuclid.org/euclid.bbms/1506477691