Let $(X_n)_{n \geq 0}$ be a sequence of integrable ($\mathbb{E} |X_n| < \infty$) random variables and denote by $l_n(t)$ the Laplace transforms of $X_n$. Similarly, let $X$ be a r.v. and $l(t)$ its Laplace transform.
It is known (Levy continuity theorem for Laplace transforms) that $l_n(t) \to l(t)$ for every $t>0$ iff $X_n \Longrightarrow X$ (where with the notation $X_n \Longrightarrow X$ I mean that $X_n$ converges weakly/in distribution to $X$).
Denote by $g'$ the first derivative of $g$. I was wondering whether additional hypotheses such $l_n'(t) \to l'(t)$ for every $t>0$ (or maybe simply $l'_n(0) \to l(0)'$) imply that $X_n \stackrel{L^1}{\to} X$.