Let \(p(z):=\sum_{k=0}^na_kz^k\) be a complex polynomial of degree at most \(n\). According to the celebrated inequality of Bernstein (*) \(| | p^{\prime}| | \leq n| | p| | \), where \(| p| :=\max_{| z| =1}| p(z)| \) and \(n\geq 1\). For various extensions of inequality (*), see, for example, \textit{Q. I. Rahman} and \textit{G. Schmeisser} [Analytic theory of polynomials, (Oxford University Press, Oxford), 2002; Zbl 1072.30006)]. In the paper under review, the authors are concerned with the case when equality holds in a more general version of Bernstein's inequality. This problem has been open since 1982. The authors also prove an inequality of Bernstein type which, under some additional condition, improves the upper bound for \(| | p^{\prime}| | \).