On the minimum moduli of normalized polynomials with two prescribed values
Let \(P_ n(\mu)\) denote the set of complex polynomials of degree at most n which satisfy \(p(0)=1\) and \(p(1)=\mu\). Let \(S_ n(\mu)\) denote the supremum over P(\(\mu)\) of the quantity min\(\{| p(z)|:| z| \leq 1\}\). The authors determine the exact value of \(S_ n(\mu)\) for all \(n\geq 1\) and all complex \(\mu\). For \(\mu >0\), Corollary 1 states that \(S_ n(\mu)=\mu\) if \(0<\mu \leq 1\); \(S_ n(\mu)=0\) if \(\mu \geq 2^ n\); and if \(1<\mu <2^ n\) then \(S_ n(\mu)=\sigma\), the unique solution with \(0<\sigma <1\) of \(\sigma =T_{n+1}(\sigma^{-1/(n+1)})\), where \(T_{n+1}\) is the standard Chebyshev polynomial. For \(1<\mu <2^ n\), there is a unique extremal polynomial and it has positive coefficients. This latter result, for \(\mu =2\), has an application to the analysis of global descent methods for polynomials. The proof of the main theorem is not easy but is easy to follow since the paper is well-written and almost self-contained.
scientific article; zbMATH DE number 3971123
scientific article