On oscillating polynomials

Extremal problems of Markov type are studied, concerning maximization of a local extremum of the derivative in the class of real polynomials of bounded uniform norm and with maximal number of zeros in [−1,1][−1,1]. We prove that if a symmetric polynomial ff, with all its zeros in [−1,1][−1,1], attains its maximal absolute value at the end-points, then |f′||f′| attains maximal value at the end-points too. As an application of the method developed here, we show that the classic Zolotarev polynomials have maximal derivative at one of the end-points.