Error bound for an osculatory quadrature formula (Q1098031)

Rabinoitz derived the error term for an integration rule given by \[ \int^{1}_{-1}f(x)dx=2f(-1)+B_{on}f'(- 1)+\sum^{n}_{k=1}B_{kn}f'(x_{kn})+R(f), \] where \[ R(f)=\frac{2^{2n+4}[(n+1)!]^ 4(n+2)}{(2n+3)[(rn+2)!]^ 3}f^{(2n+2)}(\eta),\quad -1<\eta <1, \] provided \(f\in C^{2n+2}[- 1,1]\). The n abscissas \(x_{kn}\) in the above formula are the zeros of the polynomial \(P(t)=(t+1)^ 2P_ n^{(1,1)}(t)\), where \(P_ n^{(1,1)}(t)=2^{-n}\left( \begin{matrix} 2n+2\\ n\end{matrix} \right)t^ n+terms\) of degree n-2. In this note some asymptotic estimate from above is obtained for functions analytic in a region D. The following theorem is typical. Theorem: Let D be a region, and \(C: | Z| =R\) be a circle contained in D, and f is analytic in D. Then \[ | R(f)| \leq K_ n\frac{M(R)}{R^{2n+2}}(1+O(R^{-1})),\quad R\to +\infty, \] where \[ K_ n=\frac{2^{2n+4}[(n+1)!]^ 4(3n+4)}{(2n+3)(2n+2)[(2n+2)!]^ 2} \] and \(M(R)=Max| f(z)|\) on \(| z| =R\).