Let $X$ be a random variable taking values from $\mathbb{R}^+$. If $\mathbb{E}[a^{X}]$ is given for some positive constant $a$, then what can be said about $\mathbb{E}[X]$? I am particularly looking for some statement stronger than what Jensen's inequality suggests, i.e., $\mathbb{E}[a^X] \ge a^{\mathbb{E}[X]}$.
Is it possible to calculate or bound $\mathbb{E}[X]$ if nothing is known about distribution of $X$?