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 $$\|f\|_A := \sup_{x \in A}{|f(x)|}$$ for real-valued functions $f$ defined on a set $A \subset {\Bbb R}$. 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 $$c_1 \frac nk\leq \min_{P \in {\mathcal P}_{n,k}}{\frac{\|P^{\prime}\|_{[0,1]}}{V_0^1(P)}} \leq \min_{P \in {\mathcal P}_{n,k}}{\frac{\|P^{\prime}\|_{[0,1]}}{|P(1)|}} \leq c_2 \left( \frac nk + 1 \right)$$ for all integers $n \geq 1$ and $k \geq 1$. We also prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 \left(\frac nk\right)^{1/2} \leq \min_{P \in {\mathcal P}_{n,k}}{\frac{\|P^{\prime}(x)\sqrt{1-x^2}\|_{[0,1]}}{V_0^1(P)}} \leq \min_{P \in {\mathcal P}_{n,k}}{\frac{\|P^{\prime}(x)\sqrt{1-x^2}\|_{[0,1]}}{|P(1)|}} \leq c_2 \left(\frac nk + 1\right)^{1/2}$$ for all integers $n \geq 1$ and $k \geq 1$.