Is it true that $E[e^{tX}] \le e^{E[t^2X^2/2]}$? I've seen some one use the inequality
$E_X[e^{tX}] \le e^{E_X[t^2X^2/2]}$.
However, I don't know if it is true and can not prove it.
Thanks for help.
 A: It seems that this inequality could in practice appear in relation with the normal distribution (although this is not restrictive of course).
The LHS denotes the moment generating function (MGF) of a random variable, as a function of any real $t$, and the MGF of a normal $N(\mu, \sigma^2)$ distribution is
$$MGF(t) = \exp\left \{\frac 12 t^2\sigma^2+t\mu \right\}$$
The RHS equals 
$$RHS =\exp\left \{E(t^2X^2/2)\right\}=\exp\left \{\frac {t^2}{2}E(X^2)\right\} = \exp\left \{\frac {t^2}{2}\left[\operatorname{Var}(X) + \left(E(X)\right)^2 \right]\right\}$$
and if $X$ is normal, we have
$$RHS = \exp\left \{\frac {t^2}{2}\left(\sigma^2 + \mu^2 \right)\right\}  $$
So in case of $X$ being normal the examined inequallity is written as
$$\exp\left \{\frac 12 t^2\sigma^2+t\mu \right\} \le\;?\;\exp\left \{\frac {t^2}{2}\left(\sigma^2 + \mu^2 \right)\right\}$$
and simplifying 
$$\Rightarrow e^{t\mu } \le\;?\;e^{t^2\mu^2/2} \Rightarrow t\mu \le\;?\; \frac 12 t^2\mu^2$$
Now it is easy to formulate the conditions under which this inequality will or will not hold (always for the case of $X$ being a normal random variable).
