Cesàro Statistical Core of Complex Number Sequences

The concept of statistical convergence was first introduced by Fast [1] and further studied by Šalát [2], Fridy [3], and many others, and for double sequences it was introduced and studied by Mursaleen and Edely [4] and Móricz [5] separately in the same year. Many concepts related to the statistical convergence have been introduced and studied so far, for example, statistical limit point, statistical cluster point, statistical limit superior, statistical limit inferior, and statistical core. Recently, Móricz [6] defined the concept of statistical (C,1) summability and studied some tauberian theorems. In this paper, we introduce (C,1)-analogues of the above-mentioned concepts and mainly study C1-statistical core of complex sequences and establish some results on C1-statistical core. Let N be the set of positive integers and K ⊆ N. Let Kn := {k ∈ K : k ≤ n}. Then the natural density of K is defined by δ(K) := limn(1/n)|Kn|, where the vertical bars denote cardinality of the enclosed set. A sequence x = (xk) is said to be statistically convergent to L if for every ε > 0 the set K(ε) := {k ∈ N : |xk − L| ≥ ε} has natural density zero; in this case, we write stlimx = L. By the symbol st, we denote the set of all statistically convergent sequences. Define the (first) arithmetic means σn of a sequence (xk) by setting