Maximum ratio of L^p norms

Question

Let $u$ be an essentially bounded function on a compact interval $X$ of the real line. Show that the ratio $$ \frac{\|u\|_{p+1}^{p+1}}{\|u\|_{p}^{p}} = \frac{\int_X |u|^{p+1}}{\int_X |u|^{p}} $$ achieves its maximum as $p \to \infty$.

Answer 1

Just apply the Cauchy-Schwarz inequality: $$\left(\int_{X}\left|u\right|^{p+1}\right)^{2}=\left(\int_{X}\left|u\right|^{p/2+1}\left|u\right|^{p/2}\right)^{2}\leq\int_{X}\left|u\right|^{p+2}\int_{X}\left|u\right|^{p}.$$

Answer 2

Let $f(p):=\frac{\|u\|_{p+1}^{p+1}}{\|u\|_p^p}$. By interpolation inequality: $$\|u\|_{p+1}\leq \|u\|_p^{\lambda}\|u\|_{p+2}^{1-\lambda} $$ where $$\frac{1}{p+1}=\frac{\lambda}{p}+\frac{1-\lambda}{p+2} \iff \lambda = \frac{p}{2(p+1)},\;1-\lambda=\frac{p+2}{2(p+1)}$$ Hence $$\|u\|_{p+1}^{2(p+1)}\leq \|u\|_{p}^{2\lambda(p+1)}\|u\|_{p+2}^{2(1-\lambda)(p+1)}\Rightarrow \frac{\|u\|_{p+1}^{p+1}}{\|u\|_p^p}\leq \frac{\|u\|_{p+2}^{p+2}}{\|u\|_{p+1}^{p+1}}\Rightarrow f(p)\leq f(p+1)$$ and $f$ is increasing.