Bulletin of the Belgian Mathematical Society - Simon Stevin

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

Bappaditya Bhowmik and Firdoshi Parveen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 27 September 2017

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Meromorphic function Concave function Starlike function Dirichlet finite integral Integral mean


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

Export citation