Applying Jensen's inequality to Real and Complex Analysis Exercise 3.5

Question

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!

Answer 1

\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*}