Polynomials with zeros and small norm on curves (Q2845869)

Let \(P_n\) be a polynomial of degree at most \(n\) such that \(P_n(z_0)=1\), where \(z_0\) is a point inside a Jordan curve \(\Gamma\). This paper studies how the zeros lying on \(\Gamma\) influence the norm of \(P_n\), and it is done through three theorems and two corollaries. If \(P_n\) has \(k_n\) zeros on \(\Gamma\), then it is proved that (Theorem 1) NEWLINE\[NEWLINE\|P_n\|_\Gamma\geq 1+c k_n/n.NEWLINE\]NEWLINE This estimate was formerly obtained by \textit{V. V. Andrievskii} and \textit{H.-P. Blatt} [Discrepancy of signed measures and polynomial approximation. New York, NY: Springer (2002; Zbl 0995.30001)], but in the present version the analyticity of \(\Gamma\) is replaced by \(\Gamma\in C^{1+\alpha}\) (\(\Gamma \in C^1\) and the derivative of its arc length parametrization is in \(\operatorname{Lip}\alpha\)). Besides, it is not assumed that the zeros on \(\Gamma\) are separated.NEWLINENEWLINETheorem 2 is an extension of a result by \textit{P. P. Varjú} and the author in [Acta Sci. Math. 73, No. 3--4, 593--611 (2007; Zbl 1174.26013)]. This theorem establishes that NEWLINE\[NEWLINE\|P_n\|_\Gamma\geq \exp(c\,k_n^2/n),NEWLINE\]NEWLINE where \(c>0\) only depends on \(\Gamma\) and \(z_0\). Here, it is assumed that \(P_n\) has at least \(k_n+n \mu_\Gamma(J)\) zeros on a subarc \(J\) of \(\Gamma\), where \(\mu_\Gamma\) is the equilibrium measure of \(\Gamma\).NEWLINENEWLINENEWLINEA consequence of Theorem 3 is that the estimate in Theorem 2 cannot be improved when \(\Gamma\in C^2\).