Let $\mathbf{X}$ be a random vector in $\mathbb{R}^{n}$ and $f: \mathbb{R}^{n} \to \mathbb{R}$ a convex function. Jensen's inequality gives that $$f(\mathbb{E}[\mathbf{X}]) \leq \mathbb{E} f(\mathbf{X}).$$ I'm interested in approximating $f(\mathbb{E}[\mathbf{X}])$ with $\mathbb{E} f(\mathbf{X})$, which requires a lower bound as well. What further conditions can be placed on $f$ such that there exists $\gamma > 0$ with $$ \gamma\, \mathbb{E} f(\mathbf{X}) \leq f(\mathbb{E}[\mathbf{X}]) \leq \mathbb{E} f(\mathbf{X})?$$
I know that this can't hold for all convex $f$ (see for example this question). Instead, I'm looking for a condition which restricts the set of possible functions so that the property holds. My hope is that this restricted set of functions is still large enough to be interesting.