# Proving $||u||_b \leq ||u||_a^\lambda||u||_c^{1-\lambda}$ [duplicate]

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?