<p>In this manuscript, various properties of the Ramanujan integral <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$I_R(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, defined as <Equation ID="Equ21"> <EquationSource Format="TEX">$\begin{aligned} I_R(x) = \int _0^\infty e^{-xt} \dfrac{dt}{t(\pi ^2 + (\log t)^2 )}, \quad x>0, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>I</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>∞</mi> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>x</mi> <mi>t</mi> </mrow> </msup> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="italic">dt</mi> </mrow> <mrow> <mi>t</mi> <mo stretchy="false">(</mo> <msup> <mi>π</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>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 Turán-type function involving the Ramanujan integral given by <Equation ID="Equ22"> <EquationSource Format="TEX">$\begin{aligned} H_n(x;\alpha ) = \left( I_R^{(n)}(x)\right) ^2 - \alpha I_R^{(n-1)}(x) I_R^{(n+1)}(x), \quad x>0, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>H</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mfenced close=")" open="("> <msubsup> <mi>I</mi> <mi>R</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mn>2</mn> </msup> <mo>-</mo> <mi>α</mi> <msubsup> <mi>I</mi> <mi>R</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msubsup> <mi>I</mi> <mi>R</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>and establish its complete monotonicity under certain conditions on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. Graphical evidences are given for the results where few ranges are yet to be established, providing scope for future research.</p>

Inequalities involving a Ramanujan integral

期刊论文
发表日期：

2026年5月18日

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Deepshikha Mishra,
A. Swaminathan

摘要

In this manuscript, various properties of the Ramanujan integral $I_R(x)$ $I_{R} (x)$ , defined as $\begin{aligned} I_R(x) = \int _0^\infty e^{-xt} \dfrac{dt}{t(\pi ^2 + (\log t)^2 )}, \quad x>0, \end{aligned}$ $\begin{matrix} I_{R} (x) = \int_{0}^{\infty} e^{- x t} \frac{dt}{t (π^{2} + {(log t)}^{2})}, x > 0, \end{matrix}$ 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 Turán-type function involving the Ramanujan integral given by $\begin{aligned} H_n(x;\alpha ) = \left( I_R^{(n)}(x)\right) ^2 - \alpha I_R^{(n-1)}(x) I_R^{(n+1)}(x), \quad x>0, \end{aligned}$ $\begin{matrix} H_{n} (x; α) = {(I_{R}^{(n)}, (x))}^{2} - α I_{R}^{(n - 1)} (x) I_{R}^{(n + 1)} (x), x > 0, \end{matrix}$ and establish its complete monotonicity under certain conditions on $\alpha $ $α$ . Graphical evidences are given for the results where few ranges are yet to be established, providing scope for future research.