Question, Let $f \in L^1(\alpha, \beta)$ , $\beta>0$ and $$ F(x)= \int_{\alpha}^{\beta} f(t)\exp(e^{xt}) \, dt $$ such that $\displaystyle \lim_{x\to +\infty}F(x)=0$. Does this imply that $f$ is identically zero almost everywhere?
A similar question posed by myself was answered using Phragmén-Lindelöf, but the difficulty here stems from the extremely fast growth of $\exp(\exp(z))$ along the positive real axis.
This I why I think we can't use the Phragmén–Lindelöf principle.
