In this proof that $p \mapsto \|f\|_p$ is an increasing function of $p > 0$ (for fixed $f \in L^p$ on a finite measure space) the approach is to show that $\frac{d}{dp} \|f\|_p \geq 0$, which ends up being equivalent to $$ \int |f|^p \ln|f|^p \geq \ln \int |f|^p. $$ It's stated in the final paragraph that one can approximate the integrals by finite sums and use the concavity of $\ln$ to prove that inequality. I'm having trouble with this though. Can someone sketch this out in more detail to help me understand?
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This is not true in general https://math.stackexchange.com/questions/1397097/how-to-show-that-lp-norm-is-monotone-increasing?rq=1 – Math1000 May 25 '18 at 15:58
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True; let's assume we're working with the Lebesgue measure on a compact interval. – Jon Warneke May 25 '18 at 16:00
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First, lets assume that we have a finite measure space with measure $\mu(\Omega)=1$. Because if $\mu(\Omega)>1$ then the claim is wrong, using $f=1$ as a counterexample.
Then the last equation from the linked pdf $$ \int |f|^p \ln |f|^p \geq \int |f|^p \ln \int |f|^p $$ is true because the function $g(x)= x \ln(x)$ is convex and then we can directly apply Jensen's Inequality.
harfe
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