General inequalities, consequences and applications
type: Item
instance of: scholarly article
title: General inequalities, consequences and applications
zbMATH Open document ID: 1097.26015
author: Arcadii Z. Grinshpan
published in: Advances in Applied Mathematics
publication date: 2005-05-04T00:00:00Z
review text: The author proves a variety of sharp inequalities of integral, polynomial, coefficient, binomial, exponential and other types. This is a continuation of author's recent work [``Inequalities generated by the gamma function'', Adv. Appl. Math. 29, No. 1, 115--125 (2002; Zbl 1026.26007)] on inequalities involving the binomial coefficients \(d_{n}(\alpha)\), defined by \((1-z)^{-\alpha}=\sum_{n=0}^{\infty}d_{n}(\alpha)z^n\). The author proves far reaching extensions of his earlier results. He focuses on a convolution inequality with weights generated by the gamma and beta functions, which can also be viewed as an inequality for fractional integrals. Then the author gives its bi-Hermitian equivalent and considers applications of this result in hypergeometric series, fractional integrals and univalent functions. A family of positive definite kernels and related transformations naturally arise in this study. The formulation in terms of positive definite matrices is also discussed.
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 26D20
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 33C05
Mathematics Subject Classification ID: 33C15
zbMATH DE Number: 2164931
zbMATH Keywords: general inequalities
zbMATH Keywords: binomial coefficients
zbMATH Keywords: gamma function
zbMATH Keywords: beta function
zbMATH Keywords: convolutions
zbMATH Keywords: combinatorial inequalities
zbMATH Keywords: bi-Hermitian forms
zbMATH Keywords: fractional integrals
zbMATH Keywords: integral transformations
zbMATH Keywords: hypergeometric series
zbMATH Keywords: confluent hypergeometric functions
zbMATH Keywords: positive definite matrices and kernels
zbMATH Keywords: Bernstein polynomials
zbMATH Keywords: univalent functions
MaRDI profile type: Publication
full work available at URL: j.aam.2004.05.001
OpenAlex ID: W2000487042
Wikidata QID: Q55951342
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DBLP publication ID: journals/aam/Grinshpan05
DOI: 10.1016/J.AAM.2004.05.001
description: scientific article; zbMATH DE number 2164931