I believe the following statement is true, but I can't find a proof anywhere. Does someone know if there is a reference or if it has a name?
Let $f : \mathbb{R} \to \mathbb{R}$ be convex. Let $X$ be a random variable so that $f(X)$ is defined. Let $\gamma \in [0,1]$. If $\mathbb{E}[X]=0$, then $\mathbb{E}[f(\gamma X)] \leq \mathbb{E}[f(X)]$.
Setting $\gamma=0$ is simply Jensen's inequality.
The result is a corollary of Jensen's inequality. If $\gamma\in[0,1]$ then by convexity $$f(\gamma X)=f(\gamma X + (1-\gamma)0)\le \gamma f(X)+(1-\gamma)f(0).$$ Taking expectations, $$ {\mathbb E}[f(\gamma X)]\le \gamma {\mathbb E}[f(X)]+(1-\gamma) f(0). $$ But $f(0)=f({\mathbb E}[X])$, and by Jensen this last is at most ${\mathbb E}[f(X)]$. Finally $1-\gamma\ge0$, so $$(1-\gamma)f(0)\le(1-\gamma){\mathbb E}[f(X)],$$ and the result is proved. (So the result is both a generalization and a corollary of Jensen...)