The pre-Schwarzian norm estimate for analytic concave functions
type: Item
instance of: scholarly article
title: The pre-Schwarzian norm estimate for analytic concave functions
zbMATH Open document ID: 1476.30073
DOI: 10.1155/2015/814805
author: Young Jae Sim
author: Oh Sang Kwon
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2018-05-25T00:00:00Z
review text: Summary: Let \(\mathbb{D}\) denote the open unit disk and let \(\mathcal{S}\) denote the class of normalized univalent functions which are analytic in \(\mathbb{D}\). Let \(\mathrm{Co}(\alpha)\) be the class of concave functions \(f \in \mathcal{S}\), which have the condition that the opening angle of \(f(\mathbb{D})\) at infinity is less than or equal to \(\pi \alpha\), \(\alpha \in(1,2]\). In this paper, we find a sufficient condition for the Gaussian hypergeometric functions to be in the class \(\mathrm{Co}(\alpha)\). And we define a class \(\mathrm{Co}(\alpha, A, B)\), \((- 1 \leq B < A \leq 1)\), which is a subclass of \(\mathrm{Co}(\alpha)\) and we find the set of variabilities for the functional \((1 - | z |^2)(f''(z) / f'(z))\) for \(f \in \mathrm{Co}(\alpha, A, B)\). This gives sharp upper and lower estimates for the pre-Schwarzian norm of functions in \(\mathrm{Co}(\alpha, A, B)\). We also give a characterization for functions in \(\mathrm{Co}(\alpha, A, B)\) in terms of Hadamard product.
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C55
Mathematics Subject Classification ID: 30C80
Mathematics Subject Classification ID: 33C05
zbMATH DE Number: 6873265
Wikidata QID: Q59109855
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2015/814805
OpenAlex ID: W2027995064
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DBLP publication ID: journals/ijmmsc/SimK15
description: scientific article; zbMATH DE number 6873265