On the Paley–Wiener theorem in the Mellin transform setting

In this paper we establish a version of the Paley–Wiener theorem of Fourier analysis in the frame of Mellin transforms. We provide two different proofs, one involving complex analysis arguments, namely the Riemann surface of the logarithm and Cauchy theorems, and the other one employing a Bernstein inequality here derived for Mellin derivatives.