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Inequalities involving a Ramanujan Integral

Deepshikha Mishra, A. Swaminathan

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(π^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;α) = \left(I_R^{(n)}(x)\right)^2 - αI_R^{(n-1)}(x) I_R^{(n+1)}(x), \quad x>0, \end{align*} 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.

Inequalities involving a Ramanujan Integral

Abstract

In this manuscript, various properties of the Ramanujan integral , defined as \begin{align*} I_R(x) = \int_0^\infty e^{-xt} \dfrac{dt}{t(π^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;α) = \left(I_R^{(n)}(x)\right)^2 - αI_R^{(n-1)}(x) I_R^{(n+1)}(x), \quad x>0, \end{align*} and establish its complete monotonicity under certain conditions on . Graphical evidences are given for the results where few ranges are yet to be established, providing scope for future research.

Paper Structure

This paper contains 12 theorems, 62 equations, 4 figures.

Key Result

Theorem 1

Bouwkamp_1971_rama int asymp Let $n \geq 0$ and $a > 0$. Then, as $x \to \infty$, the following holds where the coefficients $\Phi_k(a, n)$ are determined by the generating function

Figures (4)

  • Figure 1: Graph of the derivative of $\phi_1(t)$
  • Figure 2: Graph of $\phi_0(t)$ upto its fifth derivatives
  • Figure 3: Graph of $g(x)$
  • Figure 4: Graph of $H_3(x;0.55)$ upto its fifth derivative

Theorems & Definitions (27)

  • Theorem 1
  • Remark 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Theorem 4
  • proof
  • Remark 3
  • Definition 2
  • ...and 17 more