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# Multidimensional Exponential Inequalities with Weights

Published:2010-04-06
Printed: Jun 2010
• Dah-Chin Luor,
Department of Applied Mathematics, I-Shou University, Ta-Hsu, Kaohsiung 84008, Taiwan
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## Abstract

We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality $$\Bigl(\int_E \Phi(T_k f(x))^q u(x)\,dx\Bigr)^{1/q} \le C \Bigl (\int_E \Phi(f(x))^p v(x)\,dx\Bigr )^{1/p},$$ 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: multidimensional inequalities, geometric mean operators, exponential inequalities, star-shaped regions
 MSC Classifications: 26D15 - Inequalities for sums, series and integrals 26D10 - Inequalities involving derivatives and differential and integral operators

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