On an inequality of S. Bernstein and the Gauss-Lucas theorem (Q2708927)

The reviewed paper contains a very nice generalization of the well known Gauss-Lucas Theorem as well as some inequalities for polynomials of the Bernstein type. NEWLINENEWLINENEWLINEAs examples we quote the following two results from the paper:NEWLINENEWLINENEWLINECorollary 1. If all the zeros of the nth degree polynomial \(P(z)\) lie in \(|z|\leq r\), then so do the zeros of \(P(Rz)-\beta P(z)\) for every real or complex number \(\beta\) with \(|\beta|\leq 1\) and \(R>1\).NEWLINENEWLINENEWLINETheorem: If \(P(z)\) is a polynomial of degree n which does not vanish in \(|z|<1\), then, for every real or complex number \(\beta\) with \(|\beta|\leq 1\) and \(R>1\), NEWLINE\[NEWLINE |P(Rz)-\beta P(z)|\leq\left\{\frac{|R^n-\beta||z|^n+|1-\beta|}{2}\right\} \max_{|z|=1}|P(z)|\quad\text{for}\quad|z|\geq 1. NEWLINE\]NEWLINE The result is best possible and equality holds for \(P(z)=z^n+1\). The proofs of the results follow from some lemmas where the maximum modulus principle and self inverse polynomials where used in clever, interesting manner.NEWLINENEWLINEFor the entire collection see [Zbl 0947.00027].