This problem popped up in my research.
Let $X$ be a positive and bounded ($X\in (0,B) \ a.s.$)random variable with degrees of freedom $d$ and noncentrality parameter $\lambda$. My goal is to find $\mathbb{E}(e^{g(X)})$, which I am guessing to be a pretty difficult task and thus a tight upper bound is good enough for me. The functional form of $g$ is quite complicated but I have found that $g$ is a positive, increasing and concave function on its domain. Now it is not clear to me whether $g$ is such that $\exp\circ g$ is concave, since if it is, I could just invoke Jensen's inequality to find an upper bound. If it is the case that $\exp\circ g$ is convex, I can use a Hoeffding kind of technique to find that $X=\alpha B,\ \alpha\in (0,1)$, which then can be used to find an upper bound as $\mathbb{E}(X)\cdot {e^{g(B)}}/{B}$. However, if things are not as straight forward as these cases, I am not sure what to do because evaluating the actual integral is pretty much not an option. So my question is,
Is there any general method (maybe similar to Hoeffding or Bennet) that can be used in a situation like the above to find a tight upper bound of $\mathbb{E}(e^{g(X)})$ ?
Note that my emphasis is on a tight bound. I can find an upper bound right away using the fact that $g(X)\le g(0)+Xg'(0)$, that is, using the concavity of $g$ and then using the monotonicity of $\exp$. But I am not sure if the resulting bound qualifies as a tight bound. Please share ideas and possible references to attack this problem. Thanks in advance.