Suppose for a random variable $X$ that $\mathbb{E}X=0$ and $\phi(\lambda):=\mathbb{E}e^{\lambda X}$ exists for all $\lambda\in (0,b)$ for some $b>0.$ Is $\phi$ necessarily non-decreasing in $(0,b)?$ Is it infinitely many times differentiable?
My guesses are NO and YES.