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september 2017 Criteria for univalence, Integral means and Dirichlet integral for Meromorphic functions
Bappaditya Bhowmik, Firdoshi Parveen
Bull. Belg. Math. Soc. Simon Stevin 24(3): 427-438 (september 2017). DOI: 10.36045/bbms/1506477691

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.

Citation

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Bappaditya Bhowmik. Firdoshi Parveen. "Criteria for univalence, Integral means and Dirichlet integral for Meromorphic functions." Bull. Belg. Math. Soc. Simon Stevin 24 (3) 427 - 438, september 2017. https://doi.org/10.36045/bbms/1506477691

Information

Published: september 2017
First available in Project Euclid: 27 September 2017

zbMATH: 1377.30008
MathSciNet: MR3706811
Digital Object Identifier: 10.36045/bbms/1506477691

Subjects:
Primary: 30C45 , 30C70

Keywords: Concave function , Dirichlet finite integral , integral mean , meromorphic function , ‎starlike function

Rights: Copyright © 2017 The Belgian Mathematical Society

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Vol.24 • No. 3 • september 2017
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