Bounding the moment generating function at $2$ from the expected value and the value at $1$

Question

Let us suppose that $X$ is a random variable with expected value $0$ and whose moment generating function at $1$ is $m_X(1)=z\ge1$. As a function of $z$, what is the smallest possible value of $m_X(2)$?

Answer 1

Based on what you have, we can see: $$ M'_X(0)=0\\ M_X(0)=1\\ M_X(1)=z. $$ Because $M_X(t)$ is convex you can say: $$ M_X(2)\geq 2z-1 $$ where $y=(z-1)t+1$ is the line joining points $(0,1)$ and $(1,z)$. The equality is obtained if $X$ is a constant equal to $0$ which is possible because $M'_X(0)=0$.