here is my question:

I have a function $f(x,\epsilon_1, \dots, \epsilon_n)$ that depends on a decision $x$ I make and a certain amount of random variables $\epsilon_i$.

I define the two following functions:

$g(x) = \mathbf{Var}[(f(x,\epsilon_1, \dots, \epsilon_n))^2]$

and

$h(x) = \mathbf{Var}[|f(x,\epsilon_1, \dots, \epsilon_n))|]$.

I would like to know if the following implication holds:

$$ g \text{ is convex} \Rightarrow h \text{ is convex }.$$

Thank you for your help!