Reverse Markov- and Bernstein-type inequalities for incomplete polynomials

Abstract

Let ${\mathcal{P}}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${\mathcal{P}}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let $$\Vert f{\Vert }_A:=\underset{x\in A}{\sup }|f(x)|$$ for real-valued functions $f$ defined on a set $A\subset \mathbb{R}$. Let $$V_a^b(f):={\int }_a^b|f^{\prime }(x)|dx$$ denote the total variation of a continuously differentiable function $f$ on an interval $[a,b]$. We prove that there are absolute constants $c_1>0$ and $c_2>0$ such that $$c_1\frac{n}{k}\le \underset{P\in {\mathcal{P}}_{n,k}}{\min }\frac{\Vert P^{\prime }{\Vert }_{[0,1]}}{V_0^1(P)}\le \underset{P\in {\mathcal{P}}_{n,k}}{\min }\frac{\Vert P^{\prime }{\Vert }_{[0,1]}}{|P(1)|}\le c_2\left(\frac{n}{k}+1\right)$$ for all integers $n\ge 1$ and $k\ge 1$. We also prove that there are absolute constants $c_1>0$ and $c_2>0$ such that $$c_1{\left(\frac{n}{k}\right)}^{1/2}\le \underset{P\in {\mathcal{P}}_{n,k}}{\min }\frac{\Vert P^{\prime }(x)\sqrt{1-x^2}{\Vert }_{[0,1]}}{V_0^1(P)}\le \underset{P\in {\mathcal{P}}_{n,k}}{\min }\frac{\Vert P^{\prime }(x)\sqrt{1-x^2}{\Vert }_{[0,1]}}{|P(1)|}\le c_2{\left(\frac{n}{k}+1\right)}^{1/2}$$ for all integers $n\ge 1$ and $k\ge 1$.

submitted to Journal of Approximation Theory