Reverse Markov- and Bernstein-type inequalities for incomplete polynomials

Abstract

Let Pn,k be the set of all algebraic polynomials, with real coefficients, of degree at most n+k having at least n+1 zeros at 0. Let ‖f‖A≔supx∈A|f(x)|for real-valued functions f defined on a set A⊂R. Let Vab(f)≔∫ab|f′(x)|dxdenote the total variation of a continuously differentiable function f on an interval [a,b]. We prove that there are absolute constants c1>0 and c2>0 such that c1nk≤minP∈Pn,k‖P′‖[0,1]V01(P)≤minP∈Pn,k‖P′‖[0,1]|P(1)|≤c2nk+1for all integers n≥1 and k≥1. We also prove that there are absolute constants c1>0 and c2>0 such that c1nk1∕2≤minP∈Pn,k‖P′(x)1−x2‖[0,1]V01(P)≤minP∈Pn,k‖P′(x)1−x2‖[0,1]|P(1)|≤c2nk+11∕2for all integers n≥1 and k≥1.