Inequalities for polynomials with two equal coefficients (Q1086448)

Consider a polynomial \(p(z)=\sum^{n}_{k=0}a_ kz^ k\), \(n\geq 2\), such that \(| p(z)| \leq 1\) for \(| z| \leq 1\). It is known that \(| p'(z)| \leq n/2\) for \(| z| =1\) whenever \(a_ k=\bar a_{n-k}\) for \(0\leq k\leq n\) [see the second author and \textit{G. Schmeisser}, Séminaire de Mathématiques Supérieures No.86 (1983; Zbl 0525.30001)]. The corresponding question for the condition (*): \(a_ k=a_{n-k}\), \(0\leq k\leq n\) remains unsolved. It is interesting to observe [see the authors, Trans. Am. Math. Soc. (to appear)] that there exists a polynomial p(z) satisfying (*) such that for \(| z| =1\), max \(| p'(z)| \geq (n-1)\) max \(| p(z)|\). The authors show that for all real \(\theta\), \(| p'(e^{i\theta})| \leq n-1+| a_ 0| | e^{in\theta}-1|\) whenever \(a_ 0=a_ n\).