Spectral inequalities involving the sums and products of functions

Chong, K.-M. (1982) Spectral inequalities involving the sums and products of functions. International Journal of Mathematics and Mathematical Sciences, 5 (1). pp. 141-157. ISSN 0161-1712, DOI https://doi.org/10.1155/S0161171282000143.

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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 f≺g and f≺≺g are called spectral inequalities. If f,g∈L1(X,Λ,μ), it is proven that, for some b≥0, 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 fg≺≺δfδg for 0≤f, g∈L1(X,Λ,μ) is shown to be a consequence of the well-known but seemingly unrelated spectral inequality f+g≺δf+δg (where f,g∈L1(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,g∈L1(X,Λ,μ).

Item Type:	Article
Funders:	UNSPECIFIED
Uncontrolled Keywords:	Equimeasurable rearrangements; Spectral Inequalities; Convex functions; Discrete measure; Non-atomic measure; Martingale convergence theorem
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Institute of Mathematical Sciences
Depositing User:	Ms. Juhaida Abd Rahim
Date Deposited:	07 Jul 2017 02:42
Last Modified:	07 Jul 2017 02:42
URI:	http://eprints.um.edu.my/id/eprint/17453

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