Let $X$ be a r.v. with MGF $M(t)$, which exists for $t \in (-t_0,t_0)$, where $t_0>0$. To show that $$E|X|^n < n!s^{-n}[M(s)+M(-s)]$$
I've reduced the problem to showing that $$2\{1+\frac{s^2}{2!}E|X|^2+\frac{s^4}{4!}E|X|^4+\cdots\}>\frac{s^n}{n!}E|X|^n$$
It should be easy, but I'm stuck at this step. I know Cauchy-Schwartz (in general, Holder) inequality and the result that $\{E|X|^n\}^{1/n}$ is non-decreasing in $n$, but cannot find a way to apply them. Thanks in advance.
Source : Rohatgi, Saleh. p.$91$. Problem $9$.
I do not how you arrived at that inequality but I will give simple proof without the reduction you want to use. For any real number x we have $e^{x}+e^{-x} >\frac {|x|^{n}} {n!}$. (Just consider the cases $x>0$ and $x \leq 0$ to verify this). Hence $M(s)+M(-s) =E(e^{sX}+e^{-sX}) > E\frac {|X|^{n}s^{n}} {n!}$ Just multiply both sides by $\frac {n!} {s^{n}}$.