generalized Jensen's inequality for monotonicity order of variances

Question

Is there any Jensen-type inequality of the form $\mathbf{E} f(X) \le \mathbf{E}f(Y)$ for convex $f$ given $\mathbf{E}X = \mathbf{E}Y$ and some kind of "variance monotonicity relation" like $\mathbf{Var}(X) \le \mathbf{Var}(Y)$?

The standard Jensen inequality would be the boundary case where $X = \mathbf{E}Y$ is deterministic, and the other boundary case would be "pushing out the mass" of $Y$ to the endpoints of its domain, as in this proof of Hoeffding's lemma https://stats.stackexchange.com/questions/21075/understanding-proof-of-a-lemma-used-in-hoeffding-inequality.

Answer 1

From conditional Jensen's inequality we also have $f(E[X|G]) \leq E[f(X)|G]$ a.s. and taking expectations would give $Ef(Y) \leq Ef(X)$ with $Y=E[X|G]$. Certainly $EX = EY$ but we can have $\mathrm{Var}(X) > \mathrm{Var}(Y)$ or $\mathrm{Var}(X) < \mathrm{Var}(Y)$ depending on the choice of $X,G$. It follows that no such bound is possible without assuming more.