Turán type inequalities for Krätzel functions

Complete monotonicity, Laguerre and Turán type inequalities are established for the so-called Krätzel function Zρν, defined byZρν(u)=∫0∞tν−1e−tρ−utdt, where u>0 and ρ,ν∈R. Moreover, we prove the complete monotonicity of a determinant function of which entries involve the Krätzel function.