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INEQUALITIES OF THE HILBERT TYPE IN $\mathbb{R}^{n}$ WITH NON-CONJUGATE EXPONENTS

Published online by Cambridge University Press:  04 February 2008

Aleksandra Čižmešija ,
Ivan Perić  and
Predrag Vuković
Aleksandra Čižmešija
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia (cizmesij@math.hr)
Ivan Perić
Affiliation:
Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia (iperic@pbf.hr)
Predrag Vuković
Affiliation:
Teacher Training College Čakovec, Ante Starčevića 55, 40000 Čakovec, Croatia (predrag.vukovic@vus-ck.hr)
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Abstract

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In this paper we state and prove a new general Hilbert-type inequality in $\mathbb{R}^{n}$ with $k\geq2$ non-conjugate exponents. Using Selberg's integral formula, this result is then applied to obtain explicit upper bounds for the doubly weighted Hardy–Littlewood–Sobolev inequality and some further Hilbert-type inequalities for $k$ non-negative functions and non-conjugate exponents.

Keywords

Primary 26D1026D15inequalitiesHilbert's inequalitydoubly weighted Hardy–Littlewood–Sobolev inequalitySelberg's integral formulagamma functionbeta function
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 51 , Issue 1 , February 2008 , pp. 11 - 26
Copyright
Copyright © Edinburgh Mathematical Society 2008