In Folland's Real Analysis, there is a statement after the Riesz-Thorin interpolation theorem:
Let $M(t)$ be the operator norm of $T$ as a map from $L^{p_{t}}(\mu)$ to $L^{q_{t}}(\nu)$. We have shown that $M(t)\leq M_{0}^{1-t}M_{1}^{t}$. If $0<s<t<u<1$ and $t=(1-\tau)s+\tau u$, the theorem may be applied again to show that $M(t)\leq M(s)^{1-\tau}M(u)^{\tau}$. In short, the conclusion is that $\log M(t)$ is a convex function of $t$.
I know that$$\begin{aligned}M(t)&\leq M_{0}^{1-t}M_{1}^{t}\\ &=M_{0}^{1-s+\tau s-\tau u}M_{1}^{s-\tau s+\tau u}\\ &=M_{0}^{1+\tau s-s-\tau}M_{0}^{\tau-\tau u}M_{1}^{s-s\tau}M_{1}^{\tau u}\\ &=(M_{0}^{1-s}M_{1}^{s})^{1-\tau}(M_{0}^{1-u}M_{1}^{u})^{\tau}.\end{aligned}$$But how can I get the conclusion $M(t)\leq M(s)^{1-\tau}M(u)^{\tau}$?