Spectral inequalities involving the sums and products of functions

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)