Spectral inequalities involving the sums and products of functions

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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

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