On the core of weighted means of sequences

Abstract

Let $(p_n)$ be a sequence of nonnegative numbers such that $p_0>0$ and $P_n:=\sum _{k=0}^{n}p_k$. The sequence $(t_n)$ of n-th weighted means of a sequence $(u_n)$ is defined by

$$\begin{aligned} t_n:=\frac{1}{P_n}\sum _{k=0}^{n}p_k u_k\quad (n =0,1,2,\ldots ). \end{aligned}$$

It is well-known from the Knopp’s core theorem that $\mathcal {K}-core(t)\subseteq \mathcal {K}-core(u)$ for every real sequence $(u_n)$. But the converse of this inclusion is not true in general. In this paper, we obtain sufficient conditions under which the converse inclusion holds.