A basic probability problem Is it possible to give reference (like is there  any name for it, where it is used, some simplification for its expression in terms of $E[X]$) for the quantity $\ln E[e^X]$. I know that it is $\geq E[X]$ (from Jensen's ineq.) 
 A: Perhaps a more interesting quantity is
$$m(t) = \log\mathbb{E}[e^{tX}].$$
This is known as the log moment generating function. It (and related quantities) are found in many fields, such as information theory, large deviations, and statistical physics (and many more). Notice that, when $m(t)$ makes sense, it is convex (by Hölder's):
\begin{align*}
m(\lambda t + (1-\lambda)t)) &= \log\mathbb{E}[e^{(\lambda t+(1-\lambda)t)X}] \\
&= \log\mathbb{E}[(e^{t X})^{\lambda}(e^{tX})^{1-\lambda}] \\
&\leq \log\left((\mathbb{E}[e^{tX}])^\lambda (\mathbb{E}[e^{tX}])^{1-\lambda}\right)\\
&= \lambda m(t) + (1-\lambda)m(t).
\end{align*}
The Legendre transform $m^*$ turns out to be an extremely important related quantity:
$$m^*(s) = \sup_{t \in T} [ts - m(t)].$$
As an example, one uses the Legendre transform in the process of obtaining the rate function for some class of processes. For instance, let $X_i$ be integrable i.i.d. and for simplicity impose the superstrong condition that $m(t) < \infty$ for all $t \in \mathbb{R}$. Define $S_n := \sum_{i=1}^n X_i.$ Then Cramer's theorem says that
$$-\lim_{n\rightarrow\infty} \frac{1}{n}\log \mathbb{P}(S_n \geq \alpha n) = m^*(\alpha).$$
