arXiv subjects

A. Swaminathan

Publications and source records attributed to A. Swaminathan.

At least 19 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↗

arXiv2024

Orthogonality of quasi-nature spectral polynomials of Jacobi and Laguerre type

In this work, the explicit expressions of coefficients involved in quasi Christoffel polynomials of order one and quasi-Geronimus polynomials of order one are determined for Jacobi polynomials. These coefficients are responsible for establishing the orthogonality of quasi-spectral polynomials of Jacobi polynomials. Additionally, the orthogonality of quasi-Christoffel Laguerre polynomials of order one is derived. In the process of achieving orthogonality, in both cases, one zero is located on the boundary of the support of the measure. This allows us to derive the chain sequence and minimal parameter sequence at the point lying at the end point of the support of the measure. Furthermore, the interlacing properties among the zeros of quasi-spectral orthogonal Jacobi polynomials and Jacobi polynomials are illustrated. Finally, we define the quasi-Christoffel polynomials of order one on the unit circle and analyze the location of their zeros for specific examples, as well as propose the problem in the general setup.

math.CA↗

arXiv2024

Stability of the Toda equations related to a perturbed $R_i$ type recurrence relation

In this manuscript, a modified $R_I$ type recurrence relation is considered whose recurrence coefficients are perturbed by addition or multiplication of a constant. The perturbed system of recurrence coefficients is represented by Toda lattice equations, which are derived. These equations are then represented in a matrix form. With the help of this matrix representation, a known Lax pair is recovered. Inferences about the stability of resulting perturbed system of Toda equations are drawn based on numerical experiments.

math.CA↗

arXiv2024

Inequalities involving a measure of Marcellán class and zeros of corresponding orthogonal polynomials

Let $\tildeΦ_n$ be a quasi-orthogonal polynomial of order 1 on the unit circle, obtained from an orthogonal polynomial $Φ_n$ with measure $μ$, which is in the Marcellán class, if there exist another measure $\tildeμ$ such that $\tildeΦ_n$ is a monic orthogonal polynomial. This article aims to investigate various properties related to the Marcellán class. At first, we study the behaviour of the zeros between $Φ_n$ and $\tildeΦ_n$. Along with numerical examples, we analyze the zeros of $Φ_n$, its POPUC and the linear combination of the POPUC. Further, comparison of the norm inequalities among $Φ_n$ and $\tildeΦ_n$ are obtained by involving their measures. This leads to the study of the Lubinsky type inequality between the measures $μ$ and $\tildeμ$, without using the ordering relation between $μ$ and $\tildeμ$. Additionally, similar type of inequalities for the kernel type polynomials related to $μ$ and $\tildeμ$ are obtained.

math.CA↗

arXiv2024

Spectral transformation associated with a perturbed $R_I$ type recurrence relation

In this work, orthogonal polynomials satisfying $R_I$ type recurrence relation %$\mathcal{P}_{n+1}(z) = (z-c_n)\mathcal{P}_n(z)-λ_n (z-a_n)\mathcal{P}_{n-1}(z),$ with $\mathcal{P}_{-1}(z) = 0$ and $\mathcal{P}_0(z) = 1$ are analyzed when the recurrence coefficients are modified. The structural relationship between the perturbed and the unperturbed polynomials along with the spectral properties and spectral transformation of continued fraction are investigated. It is demonstrated that the transfer matrix method is computationally more efficient than the classical method for obtaining perturbed $R_I$ polynomials. Further, an interesting consequence of co-dilation on the Carathéodary function is presented. Finally, the study of co-recursion and co-dilation in connection to the unit circle is carried out with the help of an illustration. The interlacing and monotonicity of zeros between L-Jacobi polynomials and their perturbed forms are demonstrated.

math.CA↗

arXiv2024

Recovering orthogonality from quasi-nature of Spectral transformations

In this contribution, quasi-orthogonality of polynomials generated by Geronimus and Uvarov transformations is analyzed. An attempt is made to discuss the recovery of the source orthogonal polynomial from the quasi-Geronimus and quasi-Uvarov polynomials of order one. Moreover, the discussion on the difference equation satisfied by quasi-Geronimus and quasi-Uvarov polynomials is presented. Furthermore, the orthogonality of quasi-Geronimus and quasi-Uvarov polynomials is achieved through the reduction of the degree of coefficients in the difference equation. During this procedure, alternative representations of the parameters responsible for achieving orthogonality are derived. One of these representations involves the Stieltjes transform of the measure. Finally, the recurrence coefficients ensuring the existence of a measure that makes the quasi-Geronimus Laguerre polynomial of order one an orthogonal polynomial are calculated.

math.CA↗

arXiv2024

Generalized co-polynomials of $R_{II}$ type and associated quadrature rules

When the co-recursion and co-dilation in the recurrence relation of certain sequences of orthogonal polynomials are not at the same level, the behaviour of the modified orthogonal polynomials is expected to have different properties compared to the situation of the same level of perturbation. This manuscript attempts to derive structural relations between the perturbed and original $R_{II}$ type orthogonal polynomials. The classical result is improved using a transfer matrix approach. It turns out that the $R_{II}$ fraction with perturbation is the rational spectral transformation of the unperturbed one. The derived notions are used to deduce some consequences for the polynomials orthogonal on the real line. A natural question that arises while dealing with perturbations at different levels, i.e., which perturbation, co-recursion or co-dilation, needs to be performed first, is answered.

math.CA↗

arXiv2023

Orthogonality of a new family of $q$-Sobolev type polynomials

In this work, we introduce and construct specific $q$-polynomials that are desired from the well-established families of $q$-orthogonal polynomials, namely little $q$-Jacobi polynomials and $q$-Laguerre polynomials, respectively. We examine these newly constructed $q$-polynomials and observe that they possess integral representations of little $q$-Jacobi polynomials and $q$-Laguerre polynomials. These polynomials solve a third-order $q$-difference equation and display an unconventional four-term recurrence relation. This unique recurrence relation makes us categorize them as $q$-Sobolev-type orthogonal polynomials. This motivation leads to defining the general Sobolev-type orthogonality for $q$-polynomials. Special cases of these polynomials are also explored and discussed. Furthermore, we delve into the behavior of these $q$-orthogonal polynomials of Sobolev type as the parameters approach $1$. We also examine their zeros and interlacing properties.

math.CA↗

arXiv2023

Recovering orthogonality from Quasi-type Kernel Polynomials using specific spectral transformations

In this work, the concept of quasi-type Kernel polynomials with respect to a moment functional is introduced. Difference equation satisfied by these polynomials along with the criterion for orthogonality conditions are discussed. The process of recovering orthogonality for the linear combination of a quasi-type kernel polynomial with another orthogonal polynomial, which is identified by involving linear spectral transformation, is provided. This process involves an expression of ratio of iterated kernel polynomials. This lead to considering the limiting case of ratio of kernel polynomials involving continued fractions. Special cases of such ratios in terms of certain continued fractions are exhibited.

math.SP↗

arXiv2022

Spectral properties related to generalized complementary Romanovski-Routh polynomials

Complementary Romanovski-Routh polynomials play an important role in extracting specific properties of orthogonal polynomials. In this work, a generalized form of the Complementary Romanovski-Routh polynomials (GCRR) that has the Gaussian hypergeometric representation and satisfies a particular type of recurrence called $R_{II}$ type three term recurrence relation involving two arbitrary parameters is considered. Self perturbation of GCRR polynomials leading to extracting two different types of $R_{II}$ type orthogonal polynomials are identified. Spectral properties of these resultant polynomials in terms of tri-diagonal linear pencil were analyzed. The LU decomposition of these pencil matrices provided interesting properties involving biorthogonality. Interlacing properties between the zeros of the polynomials in the discussion are established.

math.CA↗

arXiv2022

Chain sequences and Zeros of a perturbed $R_{II}$ type recurrence relation

In this manuscript, new algebraic and analytic aspects of the orthogonal polynomials satisfying $R_{II}$ type recurrence relation given by \begin{align*} \mathcal{P}_{n+1}(x) = (x-c_n)\mathcal{P}_n(x)-λ_n (x-a_n)(x-b_n)\mathcal{P}_{n-1}(x), \quad n \geq 0, \end{align*} where $λ_n$ is a positive chain sequence and $a_n$, $b_n$, $c_n$ are sequences of real or complex numbers with $\mathcal{P}_{-1}(x) = 0$ and $\mathcal{P}_0(x) = 1$ are investigated when the recurrence coefficients are perturbed. Specifically, representation of new perturbed polynomials (co-polynomials of $R_{II}$ type) in terms of original ones with the interlacing and monotonicity properties of zeros are given. For finite perturbations, a transfer matrix approach is used to obtain new structural relations. Effect of co-dilation in the corresponding chain sequences and their consequences onto the unit circle are analysed. A particular perturbation in the corresponding chain sequence called complementary chain sequences and its effect on the corresponding Verblunsky coefficients is also studied.

math.CA↗

arXiv2021

Sufficiency for Nephroid Starlikeness using Hypergeometric Functions

Let $\mathcal{A}$ consists of analytic functions $f:\mathbb{D}\to\mathbb{C}$ satisfying $f(0)=f'(0)-1=0$. Let $\mathcal{S}^*_{Ne}$ be the recently introduced Ma-Minda type functions family associated with the $2$-cusped kidney-shaped {\it nephroid} curve $\left((u-1)^2+v^2-\frac{4}{9}\right)^3-\frac{4 v^2}{3}=0$ given by \begin{align*} \mathcal{S}^*_{Ne}:= \left\{f\in\mathcal{A}:\frac{zf'(z)}{f(z)}\precφ_{\scriptscriptstyle {Ne}}(z)=1+z-z^3/3\right\}. \end{align*} In this paper, we adopt a novel technique that uses the geometric properties of {\it hypergeometric functions} to determine sharp estimates on $β$ so that each of the differential subordinations \begin{align*} p(z)+βzp'(z)\prec \begin{cases} \sqrt{1+z}; 1+z; e^z; \end{cases} \end{align*} imply $p(z)\precφ_{\scriptscriptstyle{Ne}}(z)$, where $p(z)$ is analytic satisfying $p(0)=1$. As applications, we establish conditions that are sufficient to deduce that $f\in\mathcal{A}$ is a member of $\mathcal{S}^*_{Ne}$.

math.CV↗

arXiv2020

Differential Subordinations for Starlike Functions Associated With A Nephroid Domain

Let $\mathcal{A}$ be the set of all analytic functions $f$ defined in the open unit disk $\mathbb{D}$ and satisfying $f(0)=f'(0)-1=0$. In this paper, we consider the function $φ_{\scriptscriptstyle {Ne}}(z):=1+z-z^3/3$, which maps the unit circle $\{z:|z|=1\}$ onto a $2$-cusped curve called nephroid given by $\left((u-1)^2+v^2-\frac{4}{9}\right)^3-\frac{4 v^2}{3}=0$, and the function class $\mathcal{S}^*_{Ne}$ defined as \begin{align*} \mathcal{S}^*_{Ne}:=\left\{f\in\mathcal{A}:\frac{zf'(z)}{f(z)}\precφ_{\scriptscriptstyle {Ne}}(z)\right\}, \end{align*} where $\prec$ denotes subordination. We obtain sharp estimates on $β\in\mathbb{R}$ so that the first-order differential subordination \begin{align*} 1+β\frac{zp'(z)}{p^j(z)}\prec\mathcal{P}(z), \quad j=0,1,2 \end{align*} implies $p\precφ_{\scriptscriptstyle{Ne}}$, where $\mathcal{P}(z)$ is certain Carathéodory function with nice geometrical properties and $p(z)$ is analytic satisfying $p(0)=1$. Moreover, we use properties of Gaussian hypergeometric function in order to get the subordination $p\precφ_{\scriptscriptstyle{Ne}}$ whenever $p(z)+βzp'(z)\prec\sqrt{1+z}$ or $1+z$. As applications, we establish sufficient conditions for $f\in\mathcal{A}$ to be in the class $\mathcal{S}^*_{Ne}$.

math.CV↗

arXiv2020

Radius Problems For Functions Associated with a Nephroid Domai

Let $\mathcal{S}^*_{Ne}$ be the collection of all analytic functions $f(z)$ defined on the open unit disk $\mathbb{D}$ and satisfying the normalizations $f(0)=f'(0)-1=0$ such that the quantity $zf'(z)/f(z)$ assumes values from the range of the function $φ_{\scriptscriptstyle{Ne}}(z):=1+z-z^3/3\,,z\in\mathbb{D}$, which is the interior of the nephroid given by \begin{align*} \left((u-1)^2+v^2-\frac{4}{9}\right)^3-\frac{4 v^2}{3}=0. \end{align*} In this work, we find sharp $\mathcal{S}^*_{Ne}$-radii for several geometrically defined function classes introduced in the recent past. In particular, $\mathcal{S}^*_{Ne}$-radius for the starlike class $\mathcal{S}^*$ is found to be $1/4$. Moreover, radii problems related to the families defined in terms of ratio of functions are also discussed. Sharpness of certain radii estimates are illustrated graphically.

math.CV↗

arXiv2020

Vietoris type theorem related to positivity of trigonometric polynomials

In this work, a Vietoris type theorem for the positivity of sine and cosine sum for a particular sequence of real numbers is provided. In this connection, the positivity of a particular type of sine sum involving ratio of some parameters is given, which is new in the literature. Various new results that follow from the Vietoris type theorem include improved estimates for the location of the zeros of a class of trigonometric polynomials and new positive sums for orthogonal polynomials. An open problem is also provided for the partial sums of the generalized polylogarithm.

math.CA↗

arXiv2019

Starlike And Convex Functions Associated with A Nephroid domain having Cusps On The Real Axis

In this paper, we show that the Carathéodory function $φ_{\scriptscriptstyle {Ne}}(z)=1+z-z^3/3$ maps the open unit disk $\mathbb{D}$ onto the interior of the nephroid, a $2$-cusped kidney-shaped curve, \begin{align*} \left((u-1)^2+v^2-\frac{4}{9}\right)^3-\frac{4 v^2}{3}=0, \end{align*} and introduce new Ma-Minda type function classes $\mathcal{S}^*_{Ne}$ and $\mathcal{C}_{Ne}$ associated with it. Apart from studying the characteristic properties of the region bounded by this nephroid, the structural formulas, extremal functions, growth and distortion results, inclusion results, coefficient bounds and Fekete-Szegö problems are discussed for the classes $\mathcal{S}^*_{Ne}$ and $\mathcal{C}_{Ne}$. Moreover, for $β\in\mathbb{R}$ and some analytic function $p(z)$ satisfying $p(0)=1$, we prove certain subordination implications of the first order differential subordination $1+β\frac{zp'(z)}{p^j(z)}\precφ_{\scriptscriptstyle {Ne}}(z),\,j=0,1,2,$ and obtain sufficient conditions for some geometrically defined function classes available in the literature.

math.CV↗