Generalization of an inequality for integral transforms with kernel and related results
type: Item
instance of: scholarly article
title: Generalization of an inequality for integral transforms with kernel and related results
zbMATH Open document ID: 1223.26041
DOI: 10.1155/2010/948430
author: S. Iqbal
author: Yong Zhou
author: J. Pečarić
published in: Journal of Inequalities and Applications
publication date: 2011-02-08T00:00:00Z
review text: The authors establish a generalization of the inequality introduced by \textit{D. S. Mitrinović} and \textit{J. E. Pečarić} [Bull. Pol. Acad. Sci., Math. 36, No.~9--10, 645--648 (1988; Zbl 0758.26011)]. Specifically, they prove that \[ \int^ b _ a r(x)\phi\Big(\frac{u_ 1 (x)}{u_ 2 (x)}\Big)dx \leq \int^ b _ a q(x)\phi\Big(\frac{v_ 1 (x)}{v_ 2 (x)}\Big)dx, \] where \[ q(x)=v_ 2 (x)\int^ b _ a \frac{r(x)K(t,x)}{u_ 2 (t)}dt \quad u_ 2(t)\neq 0, \] \(\phi\) is a convex function and \(u_ i \in {\mathcal U}(\nu,K)\), \(i=1,2\). Mean-value theorems of Cauchy type for the above inequality are established and proved by taking its difference. Furthermore, the positive semidefiniteness of the matrices generated by the difference of the inequality which implies the exponential convexity and logarithmic convexity is also proved. Also, new means of Cauchy type are defined and the monotonicity of these means is proved. Several consequences of the results obtained are presented and discussed.
reviewed by: James A. Oguntuase
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26E70
zbMATH DE Number: 5847785
zbMATH Keywords: convex function
zbMATH Keywords: mean-value theorem
zbMATH Keywords: monotonicity
zbMATH Keywords: fractional derivative
Wikidata QID: Q59252730
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2010/948430
OpenAlex ID: W2060278390
cites work: Q4014476
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description: scientific article; zbMATH DE number 5847785