Reverse Markov- and Bernstein-type inequalities for incomplete polynomials

TL;DR

This paper establishes reverse inequalities of Markov- and Bernstein-type for incomplete polynomials with specific zero constraints, providing bounds involving derivatives, polynomial values, and total variation.

Contribution

It introduces new reverse inequalities for incomplete polynomials, linking derivatives, polynomial values, and total variation with explicit bounds.

Findings

Derived bounds for derivatives of incomplete polynomials

Established inequalities involving polynomial values at 1

Connected total variation with polynomial derivative norms

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 $k \geq 1$.