Lower Bounds for Derivatives of Polynomials and Remez Type Inequalities

P. Turán [!Tu] proved that if all the zeros of a polyniomial p lie in the unit interval I def = [−1, 1], then ‖p‖L∞(I) ≥ √ deg(p)/6 ‖p‖L∞(I) . Our goal is to study the feasibility of limn→∞ ‖pn‖X/‖pn‖Y = ∞ for sequences of polynomials {pn}n∈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.