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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 thisdoi:10.1155/s0161171282000143 fatcat:v5stwzdd6rar3btglpgcod2tka