Let $X$ and $Y$ be discrete random variables. Is there a known class of joint distributions $p(x,y)$ which satisfies the following property:
$$\mathbb{E}\left[ e^{\lambda X} e^{\lambda Y} \right]< \mathbb{E}\left[ e^{\lambda X} \right]\mathbb{E}\left[ e^{\lambda Y} \right]$$ for some $\lambda>0$?
Suppose $X, W$ are independent, $X$ has nonzero variance, and $Y=W-X$. Then $$ E[e^{\lambda X} e^{\lambda Y}] < E[e^{\lambda X}] E[e^{\lambda Y}] \quad \forall \lambda >0$$ This is because (by Jensen's inequality) : \begin{align} E[e^{\lambda X}] &> e^{\lambda E[X]}\\ E[e^{\lambda Y}] &= E[e^{\lambda W}]E[e^{-\lambda X}] > E[e^{\lambda W}]e^{-\lambda E[X]} \end{align} and so (since the right-hand-sides of the above inequalities are positive): $$ E[e^{\lambda X}]E[e^{\lambda Y}] > E[e^{\lambda W}] = E[e^{\lambda (X+Y)}]$$