$\|f\|_2\le\|f\|_4^{\frac{2}{3}}\|f\|_1^{\frac{1}{3}}$

Question

I want to prove that $$ \|f\|_2\le\|f\|_4^{\frac{2}{3}}\|f\|_1^{\frac{1}{3}} $$ I proved it by Holder inequality. But this is an exercise under "Interpolation". So I guess it can be proved using interpolation. But I can't see any of the two common interpolation theorems apply.

Answer 1

The log-convexity of $L^p$-norms, used in the Riesz-Thorin Interpolation Theorem, allows us to interpolate $$ \|f\|_{p_\theta}\le\|f\|_{p_0}^{1-\theta}\|f\|_{p_1}^{\theta} $$ where $$ \frac1{p_\theta}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1} $$ Plug in $p_0=1$, $p_1=4$, and $\theta=\frac23$