Distribution of the zeros of polynomials near the unit circle (Q6632947)

Let \(P_n=c_n \prod_{j=1}^n (z- \alpha_j)=\sum_{j=0}^n c_j z^j\) be a polynomial in \(\mathbb{C}[z]\) of degree \(n\), with \(c_0\neq 0\) and roots \(\alpha_j=\rho_j e(\theta_j), j=1,2,\dots,n,\) where \(e^{2\pi i \theta}\). For \(0<p<\infty\), a \(p\)-norm of \(P_n\) is defined on unit circle by\N\[\N\|P_n\|_p=\Bigg(\frac{1}{2\pi} \int_{\mathbb{T}} |P_n(e^{i\theta})|^p \mathrm{d} \theta \Bigg)^{\frac{1}{p}}= \Bigg(\int_{\mathbb{R}/\mathbb{Z}} |P_n(e(\theta)|^p \mathrm{d} \theta \Bigg)^{\frac{1}{p}}\N\]\NThe logarithmic \(p\)-norm is defined by\N\[\NB_p(P_n)=\begin{cases} \log\Big(\frac{\|P_n\|_\infty}{\sqrt{|c_0 c_n|}}\Big) & \text{if } p=\infty, \\\N(1-|E|)\log \Big(\frac{\|P_n\|_p}{\sqrt{|c_0c_n|}}\Big)+\frac{1}{ep} & \text{if } 0<p<\infty. \end{cases}\N\]\N\NTheorem 1. Let \(P_n(z)=\sum_{k=0}^n c_k z^k\) be a polynomial of degree \(n\) such that \(P_n(0)\neq 0\) and \(\theta\) be a fixed real number in \([\tfrac{1}{2},1]\). Then any disk of radius \(\gamma_n=7 \frac{(2 B_\infty(P_n))^\theta}{\sqrt{n}} \leq 1\) with ceter on the unit circle \(|z| = 1\), contains at least \(\sqrt{n}(2B_\infty(P_n))^\theta\) zeros of \(P_n\).\N\NFurthermore, if \(p \in (0,\infty)\) and \(P_n\) satisfy the conditions \(|c_n c_0|\geq 1\) and \(\|P_n\|_p\geq 1\), then any disk of radius \(\gamma_n =9 \frac{(2B_p(P_n))^\theta}{\sqrt{n}} \leq 1\) with center on the unit circle \(|z|=1\), contains at least \(\sqrt{n}(2B_p(P_n))^\theta\) zeros of \(P_n\).\N\NConsidered class of polynomials \(\mathcal{G}_n=\{P_n : P_n(z)=\sum_{k=0}^n a_k z^k, a_k \in \mathbb{C}, |a_0|=|a_n|=1, |a_k|\leq 1 \}\).\N\NTheorem 2. Let \(P_n \in \mathcal{G}_n\) a polynomial and \(\theta \in [\tfrac{1}{2},1]\) be any fixed real number. Then any open disk of radius \(\gamma_n=9 \frac{(log n)^\theta}{\sqrt{n}}\leq 1\) with center on \(|z|=1\) contains at least \(\sqrt{n}(\log n)^\theta\) zeros of \(P_n\).