Lower bound on an MGF value from another

A random variable $X$ has moment generating function $M(\theta)=\mathbb{E}[e^{\theta X}]$, and $M(1)=7$. What is the least value $M(4)$ can take?

$2401$. Put $Y=e^X>0$, so $M(1)=\mathbb{E}[Y]$ and $M(4)=\mathbb{E}[Y^4]$. Since $y\mapsto y^4$ is convex, Jensen gives $\mathbb{E}[Y^4]\ge(\mathbb{E}Y)^4=7^4=2401$, with equality when $X$ is the constant $\ln7$.