Spectral inequalities involving the sums and products of functions

Kong-Ming Chong

International Journal of Mathematics and Mathematical Sciences, 1982, vol. 5, 1-17

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 ∈ L 1 ( X , Λ , μ ) , it is proven that, for some b ≥ 0 , log [ b + ( δ f ι g ) + ] ≺ ≺ log [ b + ( f g ) + ] ≺ ≺ log [ b + ( δ f δ g ) + ] whenever log + [ b + ( δ f δ g ) + ] ∈ L 1 ( [ 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 f g ≺ ≺ δ f δ g for 0 ≤ f , g ∈ L 1 ( X , Λ , μ ) is shown to be a consequence of the well-known but seemingly unrelated spectral inequality f + g ≺ δ f + δ g (where f , g ∈ L 1 ( 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 ( δ f ι g ) + ≺ ≺ ( f g ) + ≺ ≺ ( δ f δ g ) + and ( δ f δ g ) − ≺ ≺ ( f g ) − ≺ ≺ ( δ f ι g ) − for not necessarily non-negative f , g ∈ L 1 ( X , Λ , μ ) .

Date: 1982
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:906852

DOI: 10.1155/S0161171282000143

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