Lower bounds for derivatives of polynomials and Remez type inequalities

P. Turan [Uber die Ableitung von. Polynomen, Comositio Math. 7 (1939), 89-95] proved that if all the zeros of a polynomial p lie in the unit interval I <(def)double under bar> [-1, 1], then parallel to p'parallel to(L infinity(I)) greater than or equal to root deg(p)/6 parallel to p parallel to(L infinity(I)). Our goal is to study the feasibility of lim(n-->infinity)parallel to p(n)'parallel to X/parallel to p(n) parallel to Y = infinity for sequences of polynomials {p(n)}(n is an element of N) whose zeros satisfy certain conditions, and to obtain lower bounds for derivatives of (generalized) polynomials and Remez type inequalities for generalized polynomials in various spaces.