Laplace transform of bounded variation function extends to entire function. Does it imply the integral of the LT is absolutely convergent

I have the following problem. Assume that $u$ is a bounded variation function on $(0,infty)$. We know that
$$h(z)=int_{0}^{infty}e^{-zy}u(y)dy$$
is absolutely convergent for $Re(z)>-1$ and $h$ can be extended to an entire function. Does this imply that $e^{-zy}u(y)$ is absolutely integrable on $(0,infty)$?

I would appreciate any hint and perhaps some reference to a body of literature on Laplace transforms of signed measure provided you are aware of such.

Thanks in advance!