Estimates for the uniform norm of complex polynomials in the unit disk

Let ‖ · ‖ denote the uniform norm in the unit disk of the complex plane ℂ. The main result in this note is as follows: For any complex polynomial P of degree at most n and any α ∈ ℂ the inequality‖ P ‖ ⩽ ( n + 1)(‖ z P ( z ) + α ‖ ‐ | α |)holds. For any α ≠ 0 the factor n + 1 is best possible, and we determine the cases of equality (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)