Some properties of the quadrinomials $p(z)=1+\kappa(z+z^{N-1})+z^N$ and $q(z)=1+\kappa(z-z^{N-1})-z^N$

TL;DR

This paper characterizes when the zeros of specific quadrinomials lie on the unit circle, providing inequalities for the parameter , factorization formulas at boundary cases, and applications to derivatives of Feje9r polynomials and univalent polynomials.

Contribution

It offers new inequalities for the parameter  ensuring zeros on the unit circle, explicit factorization formulas at boundary values, and applications to polynomial derivatives and univalent functions.

Findings

Zeros of p(z) lie on the unit circle if and only if inequalities on  are satisfied.

Explicit factorization formulas are provided at limiting values of .

Applications include factorization of Feje9r polynomial derivatives and construction of univalent polynomials.

Abstract

We show that all the zeros of the quadrinomial $p(z)=1+\kappa(z+z^{N-1})+z^N$ lie on the unit circle if and only if the inequalities \[ -1\le\kappa\le 1\; (\mbox{ if $N$ is even}),\;\; -1\le\kappa\le N/(N-2)\; (\mbox{ if $N$ is odd}) \] hold. For the quadrinomial $q(z)=1+\kappa(z-z^{N-1})-z^N$, the corresponding inequalities are \[ -N/(N-2)\le\kappa\le 1\; (\text{ if $N$ is odd}),\;\; -N/(N-2)\le\kappa\le N/(N-2)\; (\text{ if $N$ is even}). \] In the cases of limiting values of the parameter $\kappa$, we provide factorization formulas for the corresponding quadrinomials. For example, when $N$ is odd and $\kappa=N/(N-2)$, the following representation is valid: \[ p(z)=(1+z)^3\prod_{j=1}^{(N-3)/2}[1+z^2-2z\gamma_j], \] where $\gamma_j=1-2\nu_j^2$ with $\{\nu_j\}_{j=1}^{(N-3)/2}$ being the collection of positive roots of the equation $U'_{N-2}(x)=0$; here \[ U_j(x)=U_j(\cos t)=\frac{\sin(j+1)t}{\sin t}=2^j x^j+\ldots \] are Chebyshev polynomials of the second kind and $U'_j(x)$ are their derivatives. Similar factorization formulas are also provided for $q(z)$. As an application of the obtained results, we give the factorization formulas for the derivative of the Fej\'er polynomial, as well as construct certain univalent polynomials related to the polynomials $p(z)$ and $q(z)$.