Inequalities for the derivative of a polynomial (Q5904094)

In this paper some results concerning polynomials and their derivatives are derived. In particular it is proved: Theorem. Let \[ P(z)= \prod^n_{j=1} (z-z_j) \] be a polynomial of degree \(n\) which does not vanish in the disk \(D_k- \{z:| z|<k\}\) where \(k\leq 1\), and let \[ Q(z)= z^n\overline {P(1/ \overline z)}. \] If \(| P'(z)|\) and \(| Q'(z)|\) become maximum at the same point on the circle \(C_1= \{z:| z|=1\}\), then \[ \max_{| z |=1} \leq {1\over 1+k^n} \left\{n-k^n \sum^n_{j=1} {| z_j |-k \over| z_j|+k} \right\} \max_{| z|=1} \bigl| P(z) \bigr|. \] The result is best possible and equality holds for the polynomial \(P(z)= z^n+k^n\), where \(k\leq 1\).