convexity of log of moment generating function

Question

Why is log of a moment generating function of random variable Z is convex? that is

$\log \mathbb{E}[\exp(\lambda.Z)]$

My logic says since expectation is linear so it is in particular convex and exponential is convex therefore $\mathbb{E}[\exp(\lambda.Z)]$ is convex but how to know if apllying log doesnt affect convexity?

Answer 1

Holder's inequality says: $$ E(UV) \le (E|U|^p)^{1/p} (E|V|^q)^{1/q} $$ for any $1 < p,q < \infty$ with $\frac1p+\frac1q = 1$.

Put $U = \exp((1-\theta)\lambda_0 Z)$, $V = \exp(\theta\lambda_1 Z)$, $p=\frac1{1-\theta}$, $q=\frac1\theta$ for any $0<\theta<1$, take logs of both sides, and you get $$ \log E(\exp(((1-\theta)\lambda_0+\theta\lambda_1)Z)) \le (1-\theta) \log E(\exp(\lambda_0 Z)) + \theta \log E(\exp(\lambda_1 Z)) .$$