A generalization of a theorem of Brass and Schmeisser

Let n be an odd positive integer. It was proved by Brass and Schmeisser that for every quadrature \(\mathcal {Q}=\alpha _1f(x_1)+\dots +\alpha _mf(x_m)\) $$\mathcal{Q}={\alpha }_{1}f(x_1)+\cdots +{\alpha }_{m}f(x_m)$$ (with positive weights) of order at least \(n+1\) $n+1$ and for every \(n-\) $n-$ convex function f,  the value of Q on f lies between the values of Gauss-Legendre and Gauss-Lobatto quadratures of order \(n+1\) $n+1$ 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 Gauss-Radau quadratures replace Gauss-Legendre and Gauss-Lobatto ones).