On maximal ranges of polynomial spaces in the unit disk (Q1122024)

Let \(\Omega\) be a domain in the complex plane with \(0\in \Omega\) and let D be the unit disk. The authors investigate the class \(P^ 0_ n\) of all polynomials p of degree n satisfying \(p(0)=0\) and p(D)\(\subset \Omega\). Let \(\Omega_ n:=\cup_{p\in P^ 0_ n}p(D)\). A polynomial \(q\in P^ 0_ n\) is called extremal if \[ q(\bar D)\cap (\partial \Omega_ n\setminus \partial \Omega)\neq \emptyset. \] The authors give a necessary condition for this extremal property. This condition turns out to be also sufficient if \(\Omega\) is convex. In this case all essential extremal polynomials known till now are univalent in D. The basic theorems are derived in such a manner that they can be extended to other than polynomial classes. As a consequence of these results the authors establish some new inequalities for polynomials and finally give a numerical approach for a related extremal problem.