Prove $s^\lambda t^{(1-\lambda)}\leq \lambda s +(1-\lambda)t$

Question

Prove $s^\lambda t^{(1-\lambda)}\leq \lambda s +(1-\lambda)t$, for all $s,t\geq 0$ and $\lambda \in (0,1)$

Answer 1

Note that the function $\phi \to e^\phi$ is strictly convex; that is, $$ e^{\lambda x+ (1-\lambda)y} < \lambda e^{x}+ (1-\lambda) e^y, $$ for any $x, y \in \mathbb{R}$. Putting $s=e^x$ and $t=e^y$ gives the required result.