A multivariate Remez-type inequality with $\varphi$-concave weights
Volume 147 / 2017
Abstract
Let $\varphi:[0,\infty)\to[0,\infty)$ be an increasing twice continuously differentiable function with a positive power index $\beta(\varphi):=\inf_{t \gt 0}(\varphi(t)/\varphi^\prime(t))^\prime$ and let $f:V\to[0,\infty)$ be concave on a convex body $V\subset\mathbb R^m$. In this paper we discuss the following Remez-type inequality for multivariate polynomials $P$ of degree $n$ on measurable sets $E\subseteq V$ equipped with a $\varphi$-concave measure $\mu(E):=\int_E\varphi(f(x))\,dx$: $$ \|P\|_{C(V)}\le T_n\biggl(\frac{2} {1-(1-\mu(E)/\mu(V))^{\beta(\varphi)/(1+m\beta(\varphi))}}-1\bigg)\|P\|_{C(E)}, $$ where $T_n$ is the Chebyshev polynomial of degree $n$. In addition, we describe the classes of all extremal measures $\mu$, bodies $V$, sets $E$, and polynomials $P$ for this inequality.
