Complete monotonicity of log-functions

Abstract In this article we investigate the property of complete monotonicity within a special family $$\mathcal {F}_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> of functions in s variables involving logarithms. The main result of this work provides a linear isomorphism between $$\mathcal {F}_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> and the space of real multivariate polynomials. This isomorphism identifies the cone of completely monotone functions with the cone of non-negative polynomials. We conclude that the cone of completely monotone functions in $$\mathcal {F}_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> is semi-algebraic. This gives a finite time algorithm to decide whether a function in $$\mathcal {F}_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> is completely monotone.