On oscillating polynomials

Abstract

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 ] . We prove that if a symmetric polynomial f , with all its zeros in [ − 1 , 1 ] , attains its maximal absolute value at the end-points, then | 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.