Inequalities involving a Ramanujan Integral

2025/11/03 by Mishra, Deepshikha, Swaminathan, A.
#26A48 #26D07 #33E20 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2511.07443

Abstract

In this manuscript, various properties of the Ramanujan integral I_R(x), defined as I_R(x) = ∫₀^∞ e^-xt \dfracdtt(π² + log² t), xgt;0, 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 H_n(x;α) = (I_R⁽ⁿ⁾(x))² - αI_R^(n-1)(x) I_R⁽ⁿ⁺¹⁾(x), xgt;0, and establish its complete monotonicity under certain conditions on α. Graphical evidences are given for the results where few ranges are yet to be established, providing scope for future research.

Citations

• Evaluation of beta integrals of Ramanujan type and integral representations for bilateral hypergeometric series
• Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function

Related

• Ramanujan's Beautiful Integrals
• A note on the Ramanujan master theorem
• Sharp Approximations for the Ramanujan Constant
• Remarks on Ramanujan Function A_q(z)
• Reflections on Ramanujan's Mathematical Gems