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International Journal of Mathematics and Mathematical Sciences
Volume 5, Issue 1, Pages 141-157
http://dx.doi.org/10.1155/S0161171282000143

Spectral inequalities involving the sums and products of functions

Department of Mathematics, University of Malaya, Kuala Lumpur 22-11, Malaysia

Received 2 April 1975; Revised 11 December 1980

Copyright © 1982 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, the notation and denote the Hardy-Littlewood-Pólya spectral order relations for measurable functions defined on a fnite measure space (X,Λ,μ) with μ(X)=a, and expressions of the form fg and fg are called spectral inequalities. If f,gL1(X,Λ,μ), it is proven that, for some b0, log[b+(δfιg)+]log[b+(fg)+]log[b+(δfδg)+] whenever log+[b+(δfδg)+]L1([0,a]), here δ and ι respectively denote decreasing and increasing rearrangement. With the particular case b=0 of this result, the Hardy-Littlewood-Pólya-Luxemburg spectral inequality fgδfδg for 0f, gL1(X,Λ,μ) is shown to be a consequence of the well-known but seemingly unrelated spectral inequality f+gδf+δg (where f,gL1(X,Λ,μ)), thus giving new proof for the former spectral inequality. Moreover, the Hardy-Littlewood-Pólya-Luxemburg spectral inequality is also tended to give (δfιg)+(fg)+(δfδg)+ and (δfδg)(fg)(δfιg) for not necessarily non-negative f,gL1(X,Λ,μ).