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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Wiley
1982-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171282000143 |
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| author | Kong-Ming Chong |
| author_facet | Kong-Ming Chong |
| author_sort | Kong-Ming Chong |
| collection | DOAJ |
| description | 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). |
| format | Article |
| id | doaj-art-9d449da0d42645cab21f475136947fea |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1982-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-9d449da0d42645cab21f475136947fea2025-02-03T01:21:19ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251982-01-015114115710.1155/S0161171282000143Spectral inequalities involving the sums and products of functionsKong-Ming Chong0Department of Mathematics, University of Malaya, Kuala Lumpur 22-11, MalaysiaIn 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).http://dx.doi.org/10.1155/S0161171282000143equimeasurable rearrangementsspectral inequalitiesconvex functionsdiscrete measurenon-atomic measuremartingale convergence theorem. |
| spellingShingle | Kong-Ming Chong Spectral inequalities involving the sums and products of functions International Journal of Mathematics and Mathematical Sciences equimeasurable rearrangements spectral inequalities convex functions discrete measure non-atomic measure martingale convergence theorem. |
| title | 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 |
| title_full_unstemmed | Spectral inequalities involving the sums and products of functions |
| title_short | Spectral inequalities involving the sums and products of functions |
| title_sort | spectral inequalities involving the sums and products of functions |
| topic | equimeasurable rearrangements spectral inequalities convex functions discrete measure non-atomic measure martingale convergence theorem. |
| url | http://dx.doi.org/10.1155/S0161171282000143 |
| work_keys_str_mv | AT kongmingchong spectralinequalitiesinvolvingthesumsandproductsoffunctions |