Cases of equality for a class of bound-preserving operators over \(P_n\).
A new proof of the well-known inequality \[ | zp'(z) - np(z)| + | zp'(z)| \leq n | p| _D, \quad | z| \leq 1, \tag \(*\) \] is given, where \(p\) is an algebraic polynomial of degree at most \(n\) with complex coefficients and \(| p| _D = \sup_{| z| \leq 1}{| p(z)| }\). All possible cases of the equality in (\(*\)) are described. Also a new proof and all possible cases of equality are given for the inequality \[ | q*p(z)| + | \widetilde q*p(z)| \leq | p| _D, \quad | z| \leq 1, \] where \(q\) is a polynomial of degree at most \(n-1\) with \(q(0)=1\) and \(\inf_{| u| \leq 1} \operatorname{Re} q(u) > \frac{1}{2}\), \(\widetilde q(z) = z^n \overline{q(1/\bar z)}\) and \(*\) denotes the Hadamard product of two polynomials.
scientific article; zbMATH DE number 2104123