On the length of lemniscates
For a monic polynomial \(p\) of degree \(d\), we write \(E(p)=\{z:|p(z)|=1\}\). \textit{P. Erdős, F. Herzog}, and \textit{C. Piranian} [J. Anal. Math. 6, 125-148 (1958; Zbl 0088.25302)] conjectured that the length \(|E(p)|\) is maximal when \(p(z)=z^d+1\). Recently \textit{P. Borwein} [Proc. Am. Math. Soc. 123, No. 3, 797-799 (1995; Zbl 0821.31002)] proved that \(|E(p)|\leq 68.32 d.\) In the paper under review it is shown that \(|E(p)|\leq 9.173 d.\) The authors also prove that extremal polynomials exist, and that there is an extremal polynomial \(p_o\) such that all critical points of \(p_o\) lie in \(E(p_o)\) and \(E(p_o)\) is connected. A similar problem for rational functions is also studied and completely solved. In the proofs the authors make ingenious use of a variety of tools: quasiconformal maps, subharmonic functions, logarithmic capacity, the Riemann-Hurwitz theorem, integral-geometric formulae, and special properties of complex polynomials.
scientific article; zbMATH DE number 1548313