On Visser's inequality concerning coefficient estimates for a polynomial

TL;DR

This paper generalizes Visser's inequality for polynomial coefficient estimates to polynomials with no zeros inside larger disks, extending previous bounds to broader classes of polynomials.

Contribution

It extends Visser's inequality to polynomials with zeros outside larger disks, broadening the scope of coefficient estimate bounds.

Findings

Generalized inequality for polynomials with zeros outside |z|<ρ, ρ≥1

Derived bounds for coefficients based on polynomial norms

Extended applicability of coefficient estimates to wider polynomial classes

Abstract

If $P(z)=\sum_{j=0}^{n}a_jz^j$ is a polynomial of degree $n$ having no zero in $|z|<1,$ then it was recently proved that for every $p\in[0,+\infty]$ and $s=0,1,\ldots,n-1,$ \begin{align*} \left\|a_nz+\frac{a_s}{\binom{n}{s}}\right\|_{p}\leq \frac{\left\|z+\delta_{0s}\right\|_p}{\left\|1+z\right\|_p}\left\|P\right\|_{p}, \end{align*} where $\delta_{0s}$ is the Kronecker delta. In this paper, we consider the class of polynomials having no zero in $|z|<\rho,$ $\rho\geq 1$ and obtain some generalizations of above inequality.