Inequalities involving a Ramanujan Integral

TL;DR

This paper explores properties of the Ramanujan integral, including monotonicity and convexity, and introduces a Turan-type function whose complete monotonicity is analyzed, with graphical evidence supporting the theoretical findings.

Contribution

It provides new insights into the properties of the Ramanujan integral and introduces a Turan-type function, establishing conditions for its complete monotonicity.

Findings

Ramanujan integral is monotonic, subadditive, and convex.

The integral admits a Bernstein function antiderivative.

The Turan-type function is completely monotonic under certain conditions.

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.