Article

Convergence rate of one class of differentiating sums

Komarov M.A.

10.13108/2021-13-4-41

$|z|<1$

$azf'(z)=nf(0)-\sum_{k=1}^n f(\lambda_k z)+R_n(z)$

$a\ne 0$

$n=1,2,\dots$

$\lambda_k=\lambda_{n,k}(a)$

$k=1,\dots,n$

$\lambda_1^m+\dots+\lambda_n^m=-ma$

$m=1,\dots,n$

$R_n(z)=R_n(a,f;z)$

$O(z^{n+1})$

$z\to 0$

$a>0$

$n\geqslant 3\alpha$

$\alpha:=\max\{a;1\}$

$|z|<\exp(-3\sqrt{v}-2v)$

$v:=\alpha/(n+1)$

$n\to \infty$

$nf(0)-a zf'(z)$

$a=-1$

$-n\le a<0$