Let $f:[0,\infty)\to \mathbb{R}$ be a piecewise differentiable function with $f(0)=0$ and $f'(t)$ of bounded variation. Its Laplace transform $\mathcal{L}f$ converges for $\Re s > 0$. Assume it can be continued meromorphically to $\Re s > -\sigma_0$ for some small $\sigma_0>0$, with a pole at $0$ (and nowhere else).
What would be a natural sufficient condition (or even: a necessary and sufficient condition) on $f$ to ensure that $$\mathcal{L}f(\sigma+ i t) \ll_{\sigma} \frac{1}{t^{1+\epsilon}}$$ for $\sigma>-\sigma_0$?
(For $\sigma>0$, such decay (with $\epsilon=1$) follows immediately from the assumption that $f'(t)$ is of bounded variation and $f(0)=0$.)
