Convergence rate of one class of differentiating sums

Abstract

We consider a differentiation formula for functions analytic in the circle $|z|<1$: $azf'(z)=nf(0)-\sum_{k=1}^n f(\lambda_k z)+R_n(z)$. Here $a\ne 0$ is a real constant, $n=1,2,\dots$, while complex parameters $\lambda_k=\lambda_{n,k}(a)$, $k=1,\dots,n$, are defined as the unique solution of a discrete moment system for Newtonian power sums $\lambda_1^m+\dots+\lambda_n^m=-ma$, $m=1,\dots,n$. Under such choice of the parameters, the function $R_n(z)=R_n(a,f;z)$, which is the remainder in the formula, is of order $O(z^{n+1})$ as $z\to 0$. In this work we show that for each fixed $a>0$ and each $n\geqslant 3\alpha$ ($\alpha:=\max\{a;1\}$) the domain of the applicability of the formula contains the circle $|z|<\exp(-3\sqrt{v}-2v)$, $v:=\alpha/(n+1)$, the radius of which tends to one as $n\to \infty$. We establish an exponential convergence rate of differentiating sums to $nf(0)-a zf'(z)$ in the same circle. This result completes and extends essentially previous results by V.I. Danchenko (2008) and P.V. Chunaev (2020), which, respectively for the cases $a=-1$ and $-n\le a<0$ established the convergence of the differentiating formula but only in the domains contained in fixed compact subsets of the unit circle. The proof of the main results of the paper is based essentially on an approach for constructing a solution for the mentioned moment system; this approach differs essentially from that by Danchenko and Chunaev.