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I am interested in the average behaviour of the log of a function.

I know the average of the function over the range of interest: [tex]F = \frac{1}{(b-a)} \int_a^b f(x) dx.[/tex]

I also know that [tex]f(x)[/tex] is convex and bounded from below by [tex]1.[/tex]

I want to know the average [tex]\frac{1}{(b-a)} \int_a^b \log( f(x) ) dx.[/tex]

In particular, under what circumstances this would be equal to the log of the average, [tex]\log(F)[/tex], up to a constant term, if [tex]F = \frac{1}{(b-a)}[/tex] and [tex](b-a)[/tex] tends to zero.

Many thanks for any help.

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# Average of log

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