Let $Omega subseteq mathbb{R}^2$ be a nice open bounded, connected domain, having Lebesgue measure $m(Omega)=1$.
Let $F:(0,infty) to (0,infty)$ be a $C^2$ strictly convex function. Suppose that $F”$ is an everywhere positive strictly decreasing function, and that $lim_{x to infty} F”(x)=0$.
Let $Y_n:Omega to mathbb (0,infty)$ be continuous, with constant expectations $int_{Omega} Y_n=c>0$, and suppose that
$$lim_{n to infty} int_{Omega} F(Y_n)F(int_{Omega} Y_n)=0.$$
Is $lim_{n to infty} int_{Omega} (Y_nc)^2=0$?
If we replace $Omega$ with an arbitrary probability space $X$, and only require $Y_n:X to (0,infty)$ to be measurable, then the answer can be negative, as the following example shows:
Set $F(x) = e^{x}$. For $n in {1, 2, 3, …}$ define
$$ Y_n := left{begin{array}{ll}
1 – frac{1}{sqrt{n}} & mbox{ with prob $11/n$}\
1+ frac{n1}{sqrt{n}} & mbox{ with prob $1/n$}
end{array}right.$$
Then

$E(Y_n)=1$ for all $n in {1, 2, 3, …}$.

$lim_{nrightarrowinfty} E(F(Y_n)) = F(1)$.

$lim_{nrightarrowinfty} E((Y_n1)^2)=1$.
(This example is taken from here.)
The question is whether by forcing $Y_n$ to be continuous, the answer changes.