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December 7

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if

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If for two functions and we have that:

(1)

where is an arbitrary interval on the real line, then

(2)

This follows in a straightforward way from the Bogoliubov inequality. It seems to be a reasonable powerful tool to get to sharp bounds for certain integrals or summations, but it doesn't seem to be used a lot for this purpose in mathematics. I was wondering if a more straightforward proof can be given for the special case of integrals and summations.

This can be used to obtain sharp lower bounds for integrals of the form by writing down a function containing parameters, and then imposing the constraint (1) to eliminate one parameter and then maximizing w.r.t. the remaining parameters. A simple example is to take and for and the positive real line, which yields the inequality .

Count Iblis (talk) 00:11, 7 December 2020 (UTC)[reply]