# On the Markov inequality in the $L_2$-norm with the Gegenbauer weight

## TL;DR

This paper investigates the best constant in the Markov inequality for polynomials under the Gegenbauer weight in the $L_2$-norm, providing explicit bounds valid for all degrees and parameters.

## Contribution

It derives explicit lower and upper bounds for the Markov constant in the Gegenbauer-weighted $L_2$-norm for all polynomial degrees and parameters.

## Key findings

- Established explicit bounds for the Markov constant $c_n(\lambda)$
- Bounds are valid for all polynomial degrees $n$ and parameters $\lambda$
- Results contribute to understanding polynomial inequalities in weighted $L_2$ spaces.

## Abstract

Let $w_{\lambda}(t) := (1-t^2)^{\lambda-1/2}$, where $\lambda > -\frac{1}{2}$, be the Gegenbauer weight function, let $\|\cdot\|_{w_{\lambda}}$ be the associated $L_2$-norm, $$ \|f\|_{w_{\lambda}} = \left\{\int_{-1}^1 |f(x)|^2 w_{\lambda}(x)\,dx\right\}^{1/2}\,, $$ and denote by $\mathcal{P}_n$ the space of algebraic polynomials of degree $\le n$. We study the best constant $c_n(\lambda)$ in the Markov inequality in this norm $$ \|p_n'\|_{w_{\lambda}} \le c_n(\lambda) \|p_n\|_{w_{\lambda}}\,,\qquad p_n \in \mathcal{P}_n\,, $$ namely the constant $$ c_n(\lambda) := \sup_{p_n \in \mathcal{P}_n} \frac{\|p_n'\|_{w_{\lambda}}}{\|p_n\|_{w_{\lambda}}}\,. $$ We derive explicit lower and upper bounds for the Markov constant $c_n(\lambda)$, which are valid for all $n$ and $\lambda$.