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
Author(s):  
DANIEL PLAUMANN ◽  
MIHAI PUTINAR

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.


2006 ◽  
Vol 99 (1) ◽  
pp. 53 ◽  
Author(s):  
Julius Borcea

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.


Author(s):  
Abdullah Mir

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.


2020 ◽  
Vol 108 (122) ◽  
pp. 121-130
Author(s):  
Gradimir Milovanovic ◽  
Abdullah Mir ◽  
Abrar Ahmad

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.


2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Saiful R. Mondal ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad

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.


2013 ◽  
Vol 3 (2) ◽  
pp. 197-202
Author(s):  
Amir Pishkoo ◽  
Maslina Darus

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.


Author(s):  
Deepali Khurana ◽  
Sushma Gupta ◽  
Sukhjit Singh

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.


2020 ◽  
Vol 70 (4) ◽  
pp. 829-838
Author(s):  
Saqib Hussain ◽  
Shahid Khan ◽  
Khalida Inayat Noor ◽  
Mohsan Raza

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.


2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Saiful R. Mondal ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad

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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