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Multidimensional Exponential Inequalities with Weights

Published online by Cambridge University Press:  20 November 2018

Dah-Chin Luor
Dah-Chin Luor*
Affiliation:
Department of Applied Mathematics, I-Shou University, Ta-Hsu, Kaohsiung 84008, Taiwan e-mail: [email protected]
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Abstract

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We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality

$${{\left( \int_{E}{\Phi {{\left( {{T}_{k}}f\left( x \right) \right)}^{q}}u\left( x \right)dx} \right)}^{1/q}}\,\le \,C{{\left( \int_{E}{\Phi {{\left( f\left( x \right) \right)}^{p}}v\left( x \right)dx} \right)}^{1/p}},$$

where $0\,<\,p,\,q\,<\,\infty $, $\Phi $ is a logarithmically convex function, and ${{T}_{k}}$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin–Cochran-Lee type inequalities.

Keywords

26D1526D10multidimensional inequalitiesgeometric mean operatorsexponential inequalitiesstar-shaped regions
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 2 , 01 June 2010 , pp. 327 - 339
Copyright
Copyright © Canadian Mathematical Society 2010

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