Turán type inequalities for Krätzel functions

Complete monotonicity, Laguerre and Turán type inequalities are established for the so-called Krätzel function $Z_ρ^ν,$ defined by $$Z_ρ^ν(u)=\int_0^{\infty}t^{ν-1}e^{-t^ρ-\frac{u}{t}}\dt,$$ where $u>0$ and $ρ,ν\in\mathbb{R}.$ Moreover, we prove the complete monotonicity of a determinant function of which entries involve the Krätzel function.

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