arXiv subjects

A. Swaminathan

Publications and source records attributed to A. Swaminathan.

34 records · Page 2Linked to original sources

arXiv2019

Sufficient Conditions and Radius Problems for a starlike Class Involving a Differential Inequality

Let $\mathcal{A}_n$ be the class of analytic functions $f(z)$ of the form $f(z)=z+\sum_{k=n+1}^\infty a_kz^k,n\in\mathbb{N}$ and let \begin{align*} Ω_n:=\left\{f\in\mathcal{A}_n:\left|zf'(z)-f(z)\right|<\frac{1}{2},\; z\in\mathbb{D}\right\}. \end{align*} We make use of differential subordination technique to obtain sufficient conditions for the class $Ω_n$, and then employ these conditions to construct functions which involve double integrals and members of $Ω_n$. We also consider a subclass $\widehatΩ_n\subsetΩ_n$ and obtain subordination results for members of $\widehatΩ_n$ besides a necessary and sufficient condition. Writing $Ω_1=Ω$, we obtain inclusion properties of $Ω$ with respect to functions defined on certain parabolic regions and as a consequence, establish a relation connecting the parabolic starlike class $\mathcal{S}_p$ and the uniformly starlike $UST$. Various radius problems for the class $Ω$ are considered and the sharpness of the radii estimates is obtained analytically besides graphical illustrations.

math.CV↗

arXiv2018

On a conjecture for trigonometric sums by S. Koumandos and S. Ruscheweyh

S. Koumandos and S. Ruscheweyh posed the following conjecture: For $ρ\in(0,1]$ and $0<μ\leqμ^{\ast}(ρ)$, the partial sum $s_n^μ(z)=\displaystyle\sum_{k=0}^n \frac{(μ)_k}{k!}z^k$, $0<μ\leq1$, $|z|<1$, satisfies % \begin{align*} (1-z)^ρs_n^μ(z) \prec \left(\frac{1+z}{1-z}\right)^ρ, \qquad n\in \mathbb{N}, \end{align*} where $μ^{\ast}(ρ)$ is the unique solution of \begin{align*} \int_0^{(ρ+1)π} \sin(t-ρπ)t^{μ-1}dt=0. \end{align*} This conjecture is already settled for $ρ=\frac{1}{2}$, $\frac{1}{4}$, $\frac{3}{4}$ and $ρ=1$. In this work, we validate this conjecture for an open neighbourhood of $ρ=\frac{1}{3}$ and in a weaker form for $ρ=\frac{2}{3}$. The particular value of the conjecture leads to several consequences related to starlike functions.

math.CV↗

arXiv2018

Biorthogonal rational functions of $R_{II}$ type

In this work, a sequence of orthonormal rational functions that is also biorthogonal to another sequence of rational functions arising from recurrence relations of $R_{II}$ type is constructed. The biorthogonality is proved by a procedure which we call Zhedanov method. A particular case of a sequence of orthonormal rational functions having denominators of special form is considered to motivate the general case. The particular case provides a Christoffel type transformation of the generalized eigenvalue problem with a reformulation different from the existing literature.

math.CA↗

arXiv2018

Biorthogonality and para-orthogonality of $R_I$ polynomials

In this paper, a sequence of linear combination of $R_{I}$ type polynomials such that the terms in this sequence have a common zero is constructed. A biorthogonality relation arising from such a sequence is discussed. Besides a sequence of para-orthogonal polynomials by removing the common zero using suitable conditions is obtained. Finally, a case of hypergeometric functions is studied to illustrate the results obtained.

math.CA↗

arXiv2017

Geometric properties of Cesaro averaging operators

In this paper, using positivity of trigonometric cosine and sine sums whose coefficients are generalization of Vietoris numbers, we find the conditions on the coefficient $\{a_k\}$ to characterize the geometric properties of the corresponding analytic function $f(z)=z+\displaystyle\sum_{k=2}^{\infty} a_kz^k$ in the unit disc $\mathbb{D}$. As an application we also find geometric properties of a generalized Cesàro type polynomials.

math.CV↗

arXiv2017

On generalized Cesàro stable functions

The notion of Cesàro stable function is generalized by introducing Cesàro mean of type $(b-1;c)$ which give rise to a new concept of generalized Cesàro stable function. As an application of generalized Cesàro stable functions we also prove for a convex function of order $λ\in[1/2,1)$, its Cesàro mean of type $(b-1;c)$ is close-to-convex of order $λ$. Further two conjectures are also posed in the direction of generalized Cesàro stable function. Some particular cases of these conjectures are also discussed.

math.CA↗

arXiv2017

Orthogonal polynomials on the real line corresponding to a perturbed chain sequence

In recent years, chain sequences and their perturbations have played a significant role in characterising the orthogonal polynomials both on the real line as well as on the unit circle. In this note, a particular disturbance of the chain sequence related to orthogonal polynomials having their true interval of orthogonality as a subset of $[0,\infty)$ is studied leading to an important consequence related to the kernel polynomials. Such perturbations are shown to be related to transformations of symmetric measures. An illustration using the generalized Laguerre polynomials is also provided.

math.CA↗

arXiv2017

Orthogonal Polynomials related to g-fractions with missing terms

The purpose of the present paper is to investigate some structural and qualitative aspects of two different perturbations of the parameters of $g$-fractions. In this context the concept of \emph{gap} $g$-fractions is introduced. While tail sequences of a continued fraction play a significant role in the first perturbation, Schur fractions are used in the second perturbation of the $g$-parameters that are considered. Illustrations are provided using Gaussian hypergeometric functions. Using a particular gap $g$-fraction, some members of the class of Pick functions are also identified.

math.CA↗

arXiv2016

Orthogonal Polynomials Associated with Complementary Chain Sequences

Using the minimal parameter sequence of a given chain sequence, we introduce the concept of complementary chain sequences, which we view as perturbations of chain sequences. Using the relation between these complementary chain sequences and the corresponding Verblunsky coefficients, the para-orthogonal polynomials and the associated Szegö polynomials are analyzed. Two illustrations, one involving Gaussian hypergeometric functions and the other involving Carathéodory functions are also provided. A connection between these two illustrations by means of complementary chain sequences is also observed.

math.CA↗

arXiv2016

Pick Functions Related to the Multiple Gamma Functions of order $n$

Let $G_n$ be the Barnes multiple Gamma function of order $n$ and the function $f_n(z)$ be defined as \begin{align*} f_n(z)=\dfrac{\log G_n(z+1)}{z^n\Log z},\quad z\in \mathbb{C}\setminus (-\infty,0]. \end{align*} In this work, a conjecture to find the Stieltjes representation is proposed such that $f_n(z)$ is a Pick function. The conjecture is established for the particular case $n=3$ by examining the properties of $f_3(z)$.

math.CA↗

arXiv2014

Inclusion properties of Generalized Integral Transform using Duality Techniques

Let $\mathcal{W}_β^δ(α,γ)$ be the class of normalized analytic functions $f$ defined in the region $|z|<1$ and satisfying \begin{align*} {\rm Re\,} e^{iϕ}\left(\dfrac{}{}(1\!-\!α\!+\!2γ)\!\left({f}/{z}\right)^δ+\left(α\!-\!3γ+γ\left[\dfrac{}{}\left(1-{1}/δ\right)\left({zf'}/{f}\right)+ {1}/δ\left(1+{zf"}/{f'}\right)\right]\right)\right.\\ \left.\dfrac{}{}\left({f}/{z}\right)^δ\!\left({zf'}/{f}\right)-β\right)>0, \end{align*} with the conditions $α\geq 0$, $β<1$, $γ\geq 0$, $δ>0$ and $ϕ\in\mathbb{R}$. For a non-negative and real-valued integrable function $λ(t)$ with $\int_0^1λ(t) dt=1$, the generalized non-linear integral transform is defined as \begin{align*} V_λ^δ(f)(z)= \left(\int_0^1 λ(t) \left({f(tz)}/{t}\right)^δdt\right)^{1/δ}. \end{align*} The main aim of the present work is to find conditions on the related parameters such that $V_λ^δ(f)(z)\in\mathcal{W}_{β_1}^{δ_1}(α_1,γ_1)$, whenever $f\in\mathcal{W}_{β_2}^{δ_2}(α_2,γ_2)$. Further, several interesting applications for specific choices of $λ(t)$ are discussed.

math.CV↗

arXiv2014

Convexity of the Generalized Integral Transform and Duality Techniques

Let $\mathcal{W}_β^δ(α,γ)$ be the class of normalized analytic functions $f$ defined in the domain $|z|<1$ satisfying \begin{align*} {\rm Re\,} e^{iϕ}\left(\dfrac{}{}(1\!-\!α\!+\!2γ)\!\left({f}/{z}\right)^δ+\left(α\!-\!3γ+γ\left[\dfrac{}{}\left(1-{1}/δ\right)\left({zf'}/{f}\right)+ {1}/δ\left(1+{zf''}/{f'}\right)\right]\right)\right.\\ \left.\dfrac{}{}\left({f}/{z}\right)^δ\!\left({zf'}/{f}\right)-β\right)>0, \end{align*} with the conditions $α\geq 0$, $β<1$, $γ\geq 0$, $δ>0$ and $ϕ\in\mathbb{R}$. Moreover, for $0<δ\leq\frac{1}{(1-ζ)}$, $0\leqζ<1$, the class $\mathcal{C}_δ(ζ)$ be the subclass of normalized analytic functions such that \begin{align*} {\rm Re}{\,}\left(1/δ\left(1+zf''/f'\right)+(1-1/δ)\left({zf'}/{f}\right)\right)>ζ,\quad |z|<1. \end{align*} In the present work, the sufficient conditions on $λ(t)$ are investigated, so that the generalized integral transform \begin{align*} V_λ^δ(f)(z)= \left(\int_0^1 λ(t) \left({f(tz)}/{t}\right)^δdt\right)^{1/δ},\quad |z|<1, \end{align*} carries the functions from $\mathcal{W}_β^δ(α,γ)$ into $\mathcal{C}_δ(ζ)$. Several interesting applications are provided for special choices of $λ(t)$.

math.CV↗

arXiv2014

Starlikeness of the generalized integral transform using duality techniques

For $α\geq 0$, $δ>0$, $β<1$ and $γ\geq 0$, the class $\mathcal{W}_β^δ(α,γ)$ consist of analytic and normalized functions $f$ along with the condition \begin{align*} {\rm Re\,} e^{iϕ}(\dfrac{}{}(1\!-\!α\!+\!2γ)\!({f}/{z})^δ+(α\!-\!3γ\!+\!γ[\dfrac{}{}(1-{1}/δ)({zf'}/{f})+ {1}/δ(1+{zf''}/{f'})]).\\ .\dfrac{}{}({f}/{z})^δ\!({zf'}/{f})-β)>0, \end{align*} where $ϕ\in\mathbb{R}$ and $|z|<1$, is taken into consideration. The class $\mathcal{S}^\ast_s(ζ)$ be the subclass of the univalent functions, defined by the analytic characterization ${\rm Re}{\,}({zf'}/{f})>ζ$, for $0\leq ζ< 1$, $0<δ\leq\frac{1}{(1-ζ)}$ and $|z|<1$. The admissible and sufficient conditions on $λ(t)$ are examined, so that the generalized and non-linear integral transforms \begin{align*} V_λ^δ(f)(z)= (\int_0^1 λ(t) ({f(tz)}/{t})^δdt)^{1/δ}, \end{align*} maps the function from $\mathcal{W}_β^δ(α,γ)$ into $\mathcal{S}^\ast_s(ζ)$. Moreover, several interesting applications for specific choices of $λ(t)$ are discussed, that are related to some well-known integral operators.

math.CV↗

arXiv2014

Order of Starlikeness and Convexity of certain integral transforms using duality techniques

For $α\geq 0$, $β<1$ and $γ\geq 0$, the class $\mathcal{W}_β(α,γ)$ satisfies the condition \begin{align*} {\rm Re\,} \left( e^{iϕ}\left((1-α+2γ)f/z+(α-2γ)f'+ γzf''-β\right)\frac{}{}\right)>0, \quad ϕ\in {\mathbb{R}},{\,}z\in {\mathbb{D}}; \end{align*} is taken into consideration. The Pascu class of $ξ$-convex functions of order $σ$ $(M(σ,{\,}ξ))$, having analytic characterization \begin{align*} {\rm Re\,}\frac{ξz(zf'(z))'+(1-ξ)zf'(z)}{ξzf'(z)+(1-ξ)f(z)}>σ,\quad 0\leq σ< 1,\quad z\in {\mathbb{D}}, \end{align*} unifies starlike and convex functions class of order $σ$.The admissible and sufficient conditions on $λ(t)$ are investigated so that the integral transforms \begin{align*} V_λ(f)(z)= \int_0^1 λ(t) \frac{f(tz)}{t} dt, \end{align*} maps the function from $\mathcal{W}_β(α,γ)$ into $M(σ,{\,}ξ)$. Further several interesting applications, for specific choice of $λ(t)$ are discussed which are related to the classical integral transform.

math.CV↗

arXiv2013

Integral transforms of functions to be in the Pascu class using duality techniques

Let $W_β(α,γ)$, $β<1$, denote the class of all normalized analytic functions $f$ in the unit disc ${\mathbb{D}}=\{z\in {\mathbb{C}}: |z|<1\}$ such that \begin{align*} {\rm Re\,} \left(e^{iϕ}\left((1-α+2γ)\frac{f}{z}+(α-2γ)f'+γzf"-β\right)\frac{}{}\right)>0, \quad z\in {\mathbb{D}}, \end{align*} for some $ϕ\in {\mathbb{R}}$ with $α\geq 0$, $γ\geq 0$ and $β< 1$. Let $M(ξ)$, $0\leq ξ\leq 1$, denote the Pascu class of $ξ$-convex functions given by the analytic condition \begin{align*} {\rm Re\,}\frac{ξz(zf'(z))'+(1-ξ)zf'(z)}{ξzf'(z)+(1-ξ)f(z)}>0 \end{align*} which unifies the class of starlike and convex functions. The aim of this paper is to find conditions on $λ(t)$ so that the integral transforms of the form \begin{align*} V_λ(f)(z)= \int_0^1 λ(t) \frac{f(tz)}{t} dt. \end{align*} carry functions from $W_β(α,γ)$ into $M(ξ)$. As applications, for specific values of $λ(t)$, it is found that several known integral operators carry functions from $W_β(α,γ)$ into $M(ξ)$. Results for a more generalized operator related to $V_λ(f)(z)$ are also given.

math.CV↗

arXiv2011

Inclusion Criteria for Subclasses of Functions and Gronwall's Inequality

A normalized analytic function f is shown to be univalent in the open unit disk D if its second coefficient is sufficiently small and relates to its Schwarzian derivative through a certain inequality. New criteria for analytic functions to be in certain subclasses of functions are established in terms of the Schwarzian derivatives and the second coefficients. These include obtaining a sufficient condition for functions to be strongly $α$-Bazileviĉ of order $β$.

math.CV↗