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I am wondering about the following modified mellin transforms and if they are absolutely converging or not.

Let $\phi$ be some holomorphic function such that for all $y \in \mathbb{R}$ we have $\phi(\sigma \pm iy) \le C e^{\alpha|y|}$ where $\alpha < \pi$ and $a < \sigma < b$

Then $f(x) = \frac{1}{2\pi i}\int_{\sigma -i \infty}^{\sigma+i\infty}\Gamma(s)\phi(s)x^{-s}\,dx$ is a well defined function on $(0,\infty)$. And furthermore, we are guaranteed the convergence of the integral $\int_0^\infty f(x) x^{s-1}\,dx =\phi(s)$ for at least $a < \Re(s) < b$

This is because the Gamma function decays exponentially to zero at imaginary infinities of order $\pi$ and because of Mellin's inversion theorem.

However, I would like to know if we can strengthen this result to show that $\int_0^\infty |f(x)|x^{\sigma-1}\,dx< \infty$ for $a < \sigma < b$ I think this should be true, only because it's true for polynomials and exponentials that satisfy the conditions, and I think rationals.

My question is just how could I prove this?

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