A generalization of a theorem of Brass and Schmeisser

Abstract

Let $n$ be an odd positive integer. It was proved by Brass and Schmeisser that for every quadrature $$\mathcal{Q}={\alpha }_{1}f(x_1)+\cdots +{\alpha }_{m}f(x_m),$$ (with positive weights) of order at least $n+1$ and for every $n-$convex function $f,$ the value of $Q$ on $f$ lies between the values of Gauss and Lobatto quadratures of order $n+1$ calculated for the same function $f$. We generalize this result in two directions, replacing $Q$ by an integral with respect to a given measure and allowing the number $n$ to any positive integer (for even $n$ Radau quadratures replace Gauss and Lobatto ones