Jensen's integral inequality conditions

Asked Aug 17 '21 at 05:23

Active Aug 17 '21 at 08:00

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In Jensen's integral inequality $\phi\bigg(\dfrac{\int_A gf}{\int_A g}\bigg)\leq \dfrac{\int_A g\,\phi(f)}{\int_A g}$, let $\phi$ be convex on $(a,b)$, which contains the range of $f$. Assume $g$ is non-negative, and $\int_A g > 0$. My question is why is $\dfrac{\int_A gf}{\int_A g}$ guaranteed to be a quantity within $(a,b)$? I know it is within $[\inf(f), \sup(f)]$, but that may include $a,b$ which escape $(a,b)$?

The assumption on $f$ is Riemann integrable. Therefore $A$ could be taken as a closed interval on $\mathbb{R}$.

edited Aug 17 '21 at 08:00

asked Aug 17 '21 at 05:23

Jun Xu

1

What is your assumption on $g$? Why is $\int_A g \neq 0$? – Kavi Rama Murthy Aug 17 '21 at 05:31
Are there any assumptions on $f$? – Frank Lu Aug 17 '21 at 06:07
Where is the statement you quoting from? – Clement C. Aug 17 '21 at 08:14
If the quantity is $b$, then $f = b$ almost everywhere on the set where $g > 0$, so $f(x) = b$ for some $x$. Thus the quantity cannot be $b$. Similarly, it cannot be $a$. – Mason Dec 10 '22 at 00:42

Jensen's integral inequality conditions

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