Wikipedia states (without reference) that the cumulant generating function of a random variable has the property
Its first derivative ranges monotonically in the open interval from the infimum to the supremum of the support of the probability distribution, and its second derivative is strictly positive everywhere it is defined, except for the degenerate distribution of a single point mass.
I think the strict positivitivity of second derivative boils down to the claim that $$ \mathbb{E}\left(e^{x Z}\right) \mathbb{E}\left(Z^2 e^{x Z}\right)>\mathbb{E}\left(Z e^{x Z}\right)^2 $$ whenever the expectations exist, where $Z$ is any non-degenerate random variable. Is this correct?
Is there any good reference for the properties of Cumulant Generating Functions?
Yes, the inequality is correct. A proof of the inequality is quite simple: Let $Q$ be the probability measure on the sample space of $Z$ defined by $\frac {dQ} {dP}=\frac {e^{zZ}} {Ee^{zZ}}$. Then the inequality says $(E_QZ)^{2} \leq E_Q Z^{2}$ and strict inequality holds except when $Z$ is a constant.