Suppose $e^{ax}f(x)$ is convex for all $a \in \mathbb R$ then $ x \to [f(x)]^a$ is convex for all $a>0$

Question

I am trying to prove the P. Montel's theorem which states that a positive function $f$ is log convex if and only if $e^{ax}f(x)$ is convex for all $x \in \mathbb{R}$

I am having doubt in proving the following fact:

Suppose $e^{ax}f(x)$ is convex for all $a \in \mathbb R$ then $ x \to [f(x)]^a$ is convex for all $a>0$

Please give me some hint/reference to prove this fact.

Answer 1

In french: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&ved=2ahUKEwiXvZeOye3gAhWOAWMBHX3JCGgQFjACegQIBxAC&url=https%3A%2F%2Fwww.agregorio.net%2Fdownloads%2Fpdf%2Fdev%2FGamma_Log_Cvx.pdf&usg=AOvVaw14y4RWswRJ1uMMxqxO8Gip

$f$ is log-convex

iff $\forall a>0, a\log f$ is convex

iff $\forall a>0, \log f^a$ is convex,

which implies $\forall a>0,$ $f^a$ is convex.