How to obtain the inequality $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p} $ from Jensen's inequality?

Question

Let $f$ be a positive function with $\lVert f \rVert_{L^2}=1$. Let $p= 2n/(n-2)$.

How to obtain $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p}$ from Jensen's inequality?

Here all norms and integrals are over a compact manifold $M$ of dimension $n$.

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For context: this is a part of a proof of logarithmic Sobolev inequality. The log-Sobolev inequality follows by estimating the right hand side by $C\int |\nabla f|^2$.

Answer 1

By Jensen's inequality $\int |f|^2 \log |f|=\int |f|^2 \cdot \frac{1}{p-2}\log |f|^{p-2} = \frac{1}{p-2}\cdot\int |f|^2\log |f|^{p-2} \leq \frac{1}{p-2}\log (\int |f|^{p-2}\cdot |f|^2) = \frac{1}{p-2}\cdot \frac{p}{2}\log (\int|f|^p)^\frac{2}{p}=\frac{1}{p-2}\cdot \frac{p}{2} \log ||f||_p^2$

because $\frac{1}{p-2}=\frac{n-2}{4}, \frac{p}{2}=\frac{n}{n-2},$ so $\frac{1}{p-2}\cdot \frac{p}{2}=\frac{n}{4}$. We are done.