Moment-generating function of a random variable with mean 0 and variance 1

Question

Consider a random variable $X$ with mean $0$ and variance $1$. Given $0 < d < 1$, do we have

$$E(e^{d X}) \leq 1 + O(d^2)$$

(it's true when $|X| \leq 1$)

Answer 1

Consider $X \sim N(0,1)$.

We know that $\mathbb{E}[e^{dX}]=e^{d^2/2}$, which is clearly not $O(d^2)$. So the answer is no, based on this counter-example.