Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$.
Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e. $\mathcal{L}_X(s)=\int_0^\infty e^{-s t} d F(t)$
Can one conclude immediately that, as $s \to 0$, $\log \mathcal{L}_X(s) \approx -s \mathbb E X + o(s^2) $ ?
If not, suppose now that $X$ has finite moments of all orders. Can one now conclude that, as $s \to 0$, $\log \mathcal{L}_X(s) \approx -s \mathbb E X + o(s^2) $ ?