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I'm attempting Rudin Real and Complex Analysis 3.5 (a) and trying to apply Jensen's Inequality.

In the solution given here, I wasn't sure why Jensen's inequality implies that $g(p):=\frac 1p\log\int_{\Omega}|f|^pd\mu-\int_{\Omega}\log|f|d\mu \ge 0$. Since by the vanilla version of Jensen's inequality (concave), we should have $\log\int_{\Omega}|f|d\mu-\int_{\Omega}\log|f|d\mu \ge 0$.

I'm curious where the $\frac 1p$ and the $p$-power in the integrand come from. Thanks for any help!

nekodesu
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1 Answers1

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\begin{align*} \int\log|f|d\mu&=\int\dfrac{1}{p}\log|f|^{p}d\mu\\ &=\dfrac{1}{p}\int\log|f|^{p}d\mu\\ &\leq\dfrac{1}{p}\log\int|f|^{p}d\mu. \end{align*}

user284331
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