# Characterisation of completely monotone functions in terms of Mellin transform

A smooth function $f(x)$ of variable $x=(x_1,\ldots,x_n)>0$ is called completely monotone if for any multiindex $\alpha \in \mathbb{N}^n_0$ the equality holds: $$(-1)^{|\alpha|} \frac{\partial f^{|\alpha|}(x)}{\partial x^\alpha} \geqslant 0$$ for any $x>0$. Is there a characterisation of completely monotone functions in terms of their Mellin transforms $(\mathcal{M}f)(z)$ defined as $$(\mathcal M f)(z) = \int\limits_{\mathbb{R}^n_+} x^{z-I} f(x) \; \mathrm dx$$ where $I = (1,\ldots,1)$, $z=(z_1,\ldots,z_n)$?

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