Generalization of Jensen's inequality

Question

Let $X=(X_1,\dots,X_n)$ be a $\mathbb R^n$-valued random vector such that $E(|X_i|)<\infty$ for all $i$. Let $f: \mathbb R^n \to \mathbb R$ be a convex function.

Jensen's inequality tells us that $E(f(X_1,\dots,X_n))$ exists (in $]-\infty,\infty]$) and that $$E(f(X_1,\dots,X_n)) \ge f(E(X_1),\dots,E(X_n)).$$

So if we replace each $X_i$ by its expectation $E(X_i)$ we get something smaller. Does this still hold if we substitute only some of the $X_i$ by their expectations?

Question: Does it hold that $E(f(X_1,\dots,X_n)) \ge E(f(E(X_1),X_2\dots,X_n))$?

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Here are my thoughts:

Using the conditional Jensen's inequality we get that \begin{align*} E(f(X_1,\dots,X_n)) &= E(E(f(X_1,\dots,X_n)|X_2,\dots,X_n))\\ &\ge E(f(E(X_1|X_2,\dots,X_n),X_2\dots,X_n)) \end{align*} holds whenever $E(|X_1||X_2,\dots,X_n)$ is a.s. finite.

If $X_1,\dots,X_n$ are independent it follows that $$E(f(X_1,\dots,X_n)) \ge E(f(E(X_1),X_2\dots,X_n))$$ and we can iterate this to get $$E(f(E(X_1),X_2\dots,X_n)) \ge E(f(E(X_1),E(X_2),X_3\dots,X_n)),$$ etc.

But what if $X_1, \dots, X_n$ are not independent?

Answer 1

Let $X_1$ be any non-constant random variable, and let $X_2=-X_1$.

For $f(x,y)=(x+y)^2$, we have

$Ef(X_1,X_2)=E((X_1+X_2)^2)=0$

and

$E(f(EX_1,X2))=E((EX_1+X_2)^2)=E((X_2-EX_2)^2)=Var(X_2)>0$.