arXiv subjects

Priyanka Sangal

Publications and source records attributed to Priyanka Sangal.

5 recordsLinked to original sources

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

Stable Functions of Janowski Type

A function $f\in \mathcal{A}_1$ is said to be stable with respect to $g\in \mathcal{A}_1 $ if \begin{align*} \frac{s_n(f(z))}{f(z)} \prec \frac{1}{g(z)}, \qquad z\in\mathbb{D}, \end{align*} holds for all $n \in \mathbb{N}$ where $\mathcal{A}_1$ denote the class of analytic functions $f$ in the unit disk $\mathbb{D} =\{z\in \mathbb{C}: |z|<1 \}$ normalized by $f(0)=1$. Here $s_n(f(z))$, the $n^{th}$ partial sum of $f(z)=\displaystyle\sum_{k=0}^{\infty} a_kz^k$ is given by $s_n(f(z)) = \displaystyle\sum_{k=0}^{n} a_kz^k, \ n\in \mathbb{N} \cup \{0\}$. In this work, we consider the following function \begin{align*} v_λ(A,B,z)=\left(\frac{1+Az}{1+Bz}\right)^λ \end{align*} for $-1\leq B < A \leq 1$ and $0\leq λ\leq 1 $ for our investigation. The main purpose of this paper is to prove that $v_λ(A,B,z)$ is stable with respect to $\displaystyle v_λ(0,B,z)= \frac{1}{(1+Bz)^λ}$ for $0 < λ\leq 1 $ and $-1\leq B < A \leq 0$. Further, we prove that $v_λ(A,B,z)$ is not stable with respect to itself, when $0 < λ\leq 1 $ and $-1\leq B < A <0$. \end{abstract}

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↗

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↗