Estimate exponent of sum knowing the estimation of each term

Question

Had such a question which seems interesting: Assume $\alpha$ and $\beta$ are constants. How is it possible to estimate $\mathbb{E}(\left.e^{\alpha X+\beta Y}\right|\mathcal{F})$ where $X$ and $Y$ and $\mathcal{G}$-measurable random variable, $\mathcal{F}\subset\mathcal{G}$ and we know both $\mathbb{E}(\left.e^{X}\right|\mathcal{F})$ and $\mathbb{E}(\left.e^{Y}\right|\mathcal{F})$?

Answer 1

First, it seems that $\mathcal G$ does not play any role since the requirement on it is fulfilled by the collection of all subsets.

Second, in general, we cannot express $\mathbb E\left[U^2\mid\mathcal F\right]$ as a function of $\mathbb E\left[U\mid\mathcal F\right]$ (consider the case where $U$ is independent of $\mathcal F$: knowing the expectation of a random variable does not allow to derive the second moment) hence the information is not enough in the case $a=b=1$.