arXiv subjects

Deepshikha Mishra

Publications and source records attributed to Deepshikha Mishra.

3 recordsLinked to original sources

arXiv2025

Inequalities involving a Ramanujan Integral

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.

math.GM↗

arXiv2025

Complete Monotonicity of the function involving derivatives of Barnes G-function

In this manuscript, we present the complete monotonicity of functions defined in terms of the poly-double gamma function \begin{align*} ψ_2^{(n)}(x) = (-1)^{n+1} n! \sum_{k=0}^{\infty} \dfrac{(1+k)}{(x+k)^{n+1}}, \quad x > 0, \ n\geq 2. \end{align*} Consequently, we derive bounds for the ratio involving $ψ_2^{(n)}(x)$ and apply these bounds to establish the convexity, subadditivity and superadditivity of $ψ_2^{(n)}(x)$. In the process, various fundamental properties of $ψ_2^{(n)}(x)$ are established, including recurrence relations, integral representations, asymptotic expansions, complete monotonicity, and related inequalities. Graphical illustrations are provided to support the theoretical results.

math.GM↗

arXiv2025

Integral representation and functional inequalities involving generalized polylogarithm

The purpose of this manuscript is to derive two distinct integral representations of the generalized polylogarithm using two different techniques. The first approach involves the Dirichlet series and its Laplace representation, which leads to a single integral representation. The second approach utilizes the Hadamard convolution, resulting in a double integral representation. As a consequence, an integral representation of the Lerch transcendent function is obtained. Furthermore, we establish properties such as complete monotonicity, Turan inequality, convexity, and bounds of the generalized polylogarithm. Finally, we provide an alternative proof of an existing integral representation of the generalized polylogarithm using the Hadamard convolution.

math.CV↗