International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 1, Pages 141-157
Spectral inequalities involving the sums and products of functions
Department of Mathematics, University of Malaya, Kuala Lumpur 22-11, Malaysia
Received 2 April 1975; Revised 11 December 1980
Copyright © 1982 Kong-Ming Chong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, the notation and denote the Hardy-Littlewood-Pólya spectral order relations for measurable functions defined on a fnite measure space with , and expressions of the form and are called spectral inequalities. If , it is proven that, for some , whenever , here and respectively denote decreasing and increasing rearrangement. With the particular case of this result, the Hardy-Littlewood-Pólya-Luxemburg spectral inequality for , is shown to be a consequence of the well-known but seemingly unrelated spectral inequality (where ), thus giving new proof for the former spectral inequality. Moreover, the Hardy-Littlewood-Pólya-Luxemburg spectral inequality is also tended to give and for not necessarily non-negative .