Weighted Geometric Mean Inequalities Over Cones in $R^N$

Volume 4, Issue 4, 2003

Article 68

WEIGHTED GEOMETRIC MEAN INEQUALITIES OVER CONES IN R^N

BABITA GUPTA, PANKAJ JAIN, LARS-ERIK PERSSON AND ANNA WEDESTIG

DEPARTMENT OF MATHEMATICS,
SHIVAJI COLLEGE (UNIVERSITY OF DELHI),
RAJA GARDEN,
DELHI-110 027 INDIA.
E-Mail: babita74@hotmail.com

DEPARTMENT OF MATHEMATICS,
DESHBANDHU COLLEGE (UNIVERSITY OF DELHI),
NEW DELHI-110019, INDIA.
E-Mail: pankajkrjain@hotmail.com

DEPARTMENT OF MATHEMATICS,
LULEA UNIVERSITY OF TECHNOLOGY,
SE-971 87 LULEA , SWEDEN.
E-Mail: larserik@sm.luth.se

E-Mail: annaw@sm.luth.se

Received 7 November, 2002; Accepted 20 March, 2003.
Communicated by: B. Opic

ABSTRACT. Let $0<p\leq q<\infty .$ Let

be a measurable subset of the unit sphere in $\mathbb{R}^{N},$ let $E=\left\{ \mathbf{x}\in \mathbb{R}^{N}: \mathbf{x}=s\sigma ,0\leq s<\infty ,\sigma \in A\right\}$ be a cone in $\mathbb{R}^{N}$ and let $S_{\mathbf{x}}$ be the part of

with 'radius' $\leq \left\vert \mathbf{x}\right\vert .$ A characterization of the weights

and

is given such that the inequality

$\displaystyle \left( \int_{E}\left( \exp \left( \frac{1}{\left\vert S_{\mathbf{... ...\left( \int_{E}f^{p}( \mathbf{x})u(\mathbf{x})d\mathbf{x}\right) ^{\frac{1}{p}}$

holds for all $f\geq 0$ and some positive and finite constant

The inequality is obtained as a limiting case of a corresponding new Hardy type inequality. Also the corresponding companion inequalities are proved and the sharpness of the constant

is discussed.

Key words:
Inequalities, Multidimensional inequalities, Geometric mean inequalities, Hardy type inequalities, Cones in $R^{N}$, Sharp constant.

2000 Mathematics Subject Classification:
26D15, 26D07.

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