# 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.