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,...
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1982
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my.um.eprints.174532017-07-07T02:42:45Z http://eprints.um.edu.my/17453/ Spectral inequalities involving the sums and products of functions Chong, K.-M. QA Mathematics 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). Hindawi Publishing Corporation 1982 Article PeerReviewed application/pdf en http://eprints.um.edu.my/17453/1/ChongKM_%281982%29.pdf Chong, K.-M. (1982) Spectral inequalities involving the sums and products of functions. International Journal of Mathematics and Mathematical Sciences, 5 (1). pp. 141-157. ISSN 0161-1712 http://dx.doi.org/10.1155/S0161171282000143 doi:10.1155/S0161171282000143 |
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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 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). |
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Chong, K.-M. |
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Chong, K.-M. |
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Chong, K.-M. |
title |
Spectral inequalities involving the sums and products of functions |
title_short |
Spectral inequalities involving the sums and products of functions |
title_full |
Spectral inequalities involving the sums and products of functions |
title_fullStr |
Spectral inequalities involving the sums and products of functions |
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Spectral inequalities involving the sums and products of functions |
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spectral inequalities involving the sums and products of functions |
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Hindawi Publishing Corporation |
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1982 |
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http://eprints.um.edu.my/17453/1/ChongKM_%281982%29.pdf http://eprints.um.edu.my/17453/ http://dx.doi.org/10.1155/S0161171282000143 |
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