Chong, K.M. (1982) Spectral inequalities involving the sums and products of functions. International Journal of Mathematics and Mathematical Sciences, 5 (1). pp. 141157. ISSN 01611712

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Abstract
In this paper, the notation ≺ and ≺≺ denote the HardyLittlewoodPólya 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 HardyLittlewoodPólyaLuxemburg spectral inequality fg≺≺δfδg for 0≤f, g∈L1(X,Λ,μ) is shown to be a consequence of the wellknown 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 HardyLittlewoodPólyaLuxemburg spectral inequality is also tended to give (δfιg)+≺≺(fg)+≺≺(δfδg)+ and (δfδg)−≺≺(fg)−≺≺(δfιg)− for not necessarily nonnegative f,g∈L1(X,Λ,μ).
Item Type:  Article 

Uncontrolled Keywords:  Equimeasurable rearrangements; Spectral Inequalities; Convex functions; Discrete measure; Nonatomic 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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