Estimates for the uniform norm of complex polynomials in the unit disk

2010 ◽  
Vol 283 (2) ◽  
pp. 193-199 ◽  
Author(s):  
Richard Fournier ◽  
Gérard Letac ◽  
Stephan Ruscheweyh
Keyword(s):  
Unit Disk ◽  
Uniform Norm ◽  
Complex Polynomials

scholarly journals A relative Grace Theorem for complex polynomials

2016 ◽  
Vol 161 (1) ◽  
pp. 17-30 ◽  
Author(s):  
DANIEL PLAUMANN ◽  
MIHAI PUTINAR
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Circular Domain ◽  
Complex Polynomials ◽  
Polynomial Map ◽  
The Given

AbstractWe study the pullback of the apolarity invariant of complex polynomials in one variable under a polynomial map on the complex plane. As a consequence, we obtain variations of the classical results of Grace and Walsh in which the unit disk, or a circular domain, is replaced by its image under the given polynomial map.

scholarly journals Maximal and linearly inextensible polynomials

2006 ◽  
Vol 99 (1) ◽  
pp. 53 ◽  
Author(s):  
Julius Borcea
Keyword(s):  
Variational Method ◽  
Unit Disk ◽  
Second Order ◽  
Zero Set ◽  
Complex Polynomials ◽  
Closed Unit Disk ◽  
Locally Maximal

Let $S(n,0)$ be the set of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ denote by $|p|_{0}$ the distance from the origin to the zero set of $p'$. We determine all $0$-maximal polynomials of degree $n$, that is, all polynomials $p\in S(n,0)$ such that $|p|_{0}\ge |q|_{0}$ for any $q\in S(n,0)$. Using a second order variational method we then show that although some of these polynomials are linearly inextensible, they are not locally maximal for Sendov's conjecture.

scholarly journals Integral norm estimates for the polar derivative of Lacunary-type complex polynomials

2020 ◽  
pp. 36-36
Author(s):  
Abdullah Mir
Keyword(s):  
Uniform Norm ◽  
Complex Polynomials ◽  
Polar Derivative ◽  
Norm Inequalities ◽  
Norm Estimates ◽  
Integral Norm ◽  
Type Complex

In this paper, we prove some integral-norm inequalities for the polar derivative of Lacunary-type complex polynomials having zeros in closed exterior or closed interior of a circle. The results obtained besides derive polar derivative analogues of some classical Bernstein and Tur?n-type inequalities for the uniform-norm also include several interesting generalizations and refinements of some integral-norm inequalities for polynomials as well.

scholarly journals Certain estimates of Turán’s-type for the maximum modulus of the polar derivative of a polynomial

2020 ◽  
Vol 108 (122) ◽  
pp. 121-130
Author(s):  
Gradimir Milovanovic ◽  
Abdullah Mir ◽  
Abrar Ahmad
Keyword(s):  
Lower Bound ◽  
Unit Disk ◽  
Maximum Modulus ◽  
Uniform Norm ◽  
Polar Derivative ◽  
Derivative Of A Polynomial

We establish some lower bound estimates for the maximum modulus of the polar derivative of a polynomial on the unit disk under the assumption that the polynomial has all zeros in another disk. The obtained results sharpen as well as generalize some estimates of Turan?s-type that relate the uniform-norm of the polar derivative and the polynomial.

scholarly journals On the Study of Trigonometric Polynomials Using Strum Sequence

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Saiful R. Mondal ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad
Keyword(s):  
Unit Disk ◽  
Trigonometric Polynomials ◽  
Complex Polynomials ◽  
Trigonometric Sums ◽  
Geometric Properties

This article constructs trigonometric polynomials of the sine and cosine whose sums are nonnegative. As an application, those nonnegative trigonometric sums are used to study the geometric properties of complex polynomials in the unit disk. The Strum sequences are used to prove the main outcome.

The Yukawa Potential Function and Reciprocal Univalent Function

2013 ◽  
Vol 3 (2) ◽  
pp. 197-202
Author(s):  
Amir Pishkoo ◽  
Maslina Darus
Keyword(s):  
Mathematical Model ◽  
Unit Disk ◽  
Potential Function ◽  
Univalent Function ◽  
Open Problem ◽  
Yukawa Potential ◽  
Reciprocal Theorem ◽  
Problem Statement ◽  
Fundamental Forces

This paper presents a mathematical model that provides analytic connection between four fundamental forces (interactions), by using modified reciprocal theorem,derived in the paper, as a convenient template. The essential premise of this work is to demonstrate that if we obtain with a form of the Yukawa potential function [as a meromorphic univalent function], we may eventually obtain the Coloumb Potential as a univalent function outside of the unit disk. Finally, we introduce the new problem statement about assigning Meijer's G-functions to Yukawa and Coloumb potentials as an open problem.

On a Subclass of Univalent Harmonic Mappings Convex in the Imaginary Direction

2020 ◽  
pp. 445-454
Author(s):  
Deepali Khurana ◽  
Sushma Gupta ◽  
Sukhjit Singh
Keyword(s):  
Present Article ◽  
Unit Disk ◽  
Imaginary Axis ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Distortion Bounds ◽  
Univalent Harmonic Mappings

In the present article, we consider a class of univalent harmonic mappings, $\mathcal{C}_{T} = \left\{ T_{c}[f] =\frac{f+czf'}{1+c}+\overline{\frac{f-czf'}{1+c}}; \; c>0\;\right\}$ and $f$ is convex univalent in $\mathbb{D}$, whose functions map the open unit disk $\mathbb{D}$ onto a domain convex in the direction of the imaginary axis. We estimate coefficient, growth and distortion bounds for the functions of the same class.

Coefficient inequalities related with typically real functions

2020 ◽  
Vol 70 (4) ◽  
pp. 829-838
Author(s):  
Saqib Hussain ◽  
Shahid Khan ◽  
Khalida Inayat Noor ◽  
Mohsan Raza
Keyword(s):  
Unit Disk ◽  
Real Functions ◽  
Coefficient Inequalities ◽  
Typically Real ◽  
Class Of Functions ◽  
Typically Real Functions

AbstractIn this paper, we are mainly interested to study the generalization of typically real functions in the unit disk. We study some coefficient inequalities concerning this class of functions. In particular, we find the Zalcman conjecture for generalized typically real functions.

scholarly journals Some subordination involving polynomials induced by lower triangular matrices

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Saiful R. Mondal ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad
Keyword(s):  
Unit Disk ◽  
Triangular Matrices ◽  
Cesàro Mean ◽  
Cesaro Mean ◽  
Lower Triangular ◽  
Lower Triangular Matrices

Abstract The article considers several polynomials induced by admissible lower triangular matrices and studies their subordination properties. The concept generalizes the notion of stable functions in the unit disk. Several illustrative examples, including those related to the Cesàro mean, are discussed, and connections are made with earlier works.

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