Turán inequalities and complete monotonicity for a class of entire functions (Q1980941)

The authors investigate analytical properties of the class of entire functions defined by \[ \mathcal{I}_\phi(t)=\sum_{n=0}^{\infty}\frac{1}{W_\phi(n+1)}\cdot\frac{t^n}{n!},\] where \(W_\phi(1)=1\), \(W_\phi(n+1)=\prod_{k=1}^{n}\phi(k)\) and \(\phi\) is a Bernstein function. Function \(\mathcal{I}_\phi\) with a certain \(\phi\) appeared in a paper of the second author [Ann. Inst. Henri Poincaré, Probab. Stat. 45, No. 3, 667--684 (2009; Zbl 1180.31010)].\par It was shown there that the function \(\mathcal{I}_\phi\) itself, or some of its transformations enjoy interesting analytical properties, such as complete monotonicity. \par In the present paper, the authors prove the properties mentioned above in a more general setting and derive additional ones. Their approach provides also new results for a wide range of special functions since the class of functions \(\mathcal{I}_\phi\) encompasses, as special instances, the modified Bessel functions, hypergeometric functions, the Mittag-Leffler functions, the Wright functions and \(q\)-series.