Let $X,Y,Z$ be three standard normal distribution. Let $\rho_{XY},\rho_{YZ},\rho_{XZ}$ be the correlation between those random variables.

Let $f()$ be a monotone, odd, bounded, and differentiable function. More precisely, $f(x)\geq f(y)$ iff $x\geq y$, $f(x)=-f(-x)$, $|f(x)| \leq c$. For example, tanh could be one such function.

If $\rho_{XY}=\rho_{YZ}=\rho_{XZ}=0$, i.e., $X,Y,Z$ are independent, then we have

$E(f(X)f'(Y)Z) = 0$.

Can we say anything about $E(f(X)f'(Y)Z)$ when $\rho_{XY},\rho_{YZ},\rho_{XZ}$ are not 0 but very small?

In other words, I am looking for something like

$E(f(X)f'(Y)Z) \rightarrow 0 $ as $\rho_{XY},\rho_{YZ},\rho_{XZ}\rightarrow 0$,

or perhaps

$|E(f(X)f'(Y)Z)| \leq g(\rho_{XY},\rho_{YZ},\rho_{XZ}) \sqrt{E(f^2(X)f'^2(Y)Z^2)} $, where

$g(\rho_{XY},\rho_{YZ},\rho_{XZ}) \rightarrow 0$ as $\rho_{XY},\rho_{YZ},\rho_{XZ}\rightarrow 0$