Inequalities involving a Ramanujan Integral

Abstract

In this manuscript, various properties of the Ramanujan integral $I_R(x)$, defined as \begin{align*} I_R(x) = \int_0^\infty e^{-xt} \dfrac{dt}{t(\pi^2 + \log^2 t)}, \quad x>0, \end{align*} are investigated, including its monotonicity, subadditivity, as well as convexity. Furthermore, it is shown that the Ramanujan integral admits an antiderivative that belongs to the class of Bernstein functions. Subsequently, we examine a Turan-type function involving the Ramanujan integral given by \begin{align*} H_n(x;\alpha) = \left(I_R^{(n)}(x)\right)^2 - \alpha I_R^{(n-1)}(x) I_R^{(n+1)}(x), \quad x>0, \end{align*} and establish its complete monotonicity under certain conditions on $\alpha$. Graphical evidences are given for the results where few ranges are yet to be established, providing scope for future research.