A General Theorem on Sharp Evaluation of Polynomial Coefficients

The evaluation of polynomial coefficients is an important problem of algebra, complex analysis and related fields of mathematics. This subject has a long history but is still actively investigated. The paper provides sharp estimates of polynomial coefficients by applying the intrinsic features of univalent functions and of Teichmüller spaces and solves the problem for all n≥5, giving the related extremal polynomial. Also, the case of nonvanishing polynomials in the disk, having an independent interest, is considered.