On a function studied by Ramanujan and connected with discrete mixtures of gamma densities (Q1047811)

The study of the function \[ \phi(t)=\sum_{k=1}^{\infty} \frac{k^{k-1}}{k!}e^{-kt} \;(t\geq 0) \] goes back to the work of Lambert, Lagrange, Euler and Ramanujan. The authors of the present paper extend certain results for functions of the form \[ \psi(x)=\sum_{k=1}^{\infty} p_k\frac{t_k^{\alpha_k}}{\Gamma(\alpha_k)}x^{\alpha_k-1}e^{-t_kx} \;(x>0), \] where \((p_k)\) is a probability law, \((t_k)\) is a strictly increasing sequence of positive real numbers and \((\alpha_k)\) is a sequence of positive real numbers such that \(\alpha_k \geq 1\) (\(k\geq 1\)) and \(\sup_{k\geq 1} \alpha_k/t_k< \infty\). Convexity and subadditivity properties of the function \(\phi^{\lambda}\) (\(\lambda \in {\mathbb R}\)) and functional inequalities involving \(\phi\) are also presented.