Here is an interpolation inequality of the norms: if $a\leq b\leq c$ and suppose $\lambda$ satisfies $$\frac{1}{b}=\frac{\lambda}{a}+\frac{1-\lambda}{c}$$
then $$||u||_b \leq ||u||_a^\lambda||u||_c^{1-\lambda}$$
I tried taking logarithm of both sides and tried to apply Holder without success. How might we prove this?