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


Introduction and preliminaries
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, M óricz [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 C 1 -statistical core of complex sequences and establish some results on C 1 -statistical core.
Let N be the set of positive integers and K ⊆ N. Let K n := {k ∈ K : k ≤ n}.Then the natural density of K is defined by δ(K) := lim n (1/n)|K n |, where the vertical bars denote cardinality of the enclosed set.A sequence x = (x k ) is said to be statistically convergent to L if for every ε > 0 the set K(ε) := {k ∈ N : |x k − L| ≥ ε} has natural density zero; in this case, we write st-limx = L.By the symbol st, we denote the set of all statistically convergent sequences.
Define the (first) arithmetic means σ n of a sequence (x k ) by setting We say that x = (x k ) is statistically summable (C,1) to L if the sequence σ = (σ n ) is statistically convergent to L, that is, st-limσ = L.We denote by C 1 (st) the set of all sequences which are statistically summable (C,1).Let C be the set of all complex numbers.In [7], it was shown that for every bounded complex sequence x, where Note that Knopp's core (or K-core) of a real-bounded sequence x is defined to be the closed interval [lim inf x, lim sup x], and the statistical core (or st-core) as [st-lim inf x, st-lim sup x].
Recently, Fridy and Orhan [8] proved that where if x is a statistically bounded sequence.

Lemmas
In this section, we quote some results on matrix classes which are already known in the literature and these results will be used frequently in the text of the paper.
Let X and Y be any two sequence spaces and A = (a nk ) ∞ n,k=1 an infinite matrix.If for each x ∈ X the sequence Ax = (A n (x)) ∈ Y , we say that the matrix A maps X into Y , where provided that the series on the right converges for each n.We denote by (X,Y ) the class of all matrices A which map X into Y , and by (X,Y ) reg we mean A ∈ (X,Y ) such that the limit is preserved.Let c and l ∞ denote the spaces of convergent and bounded sequences The following is the well-known characterization of regular matrices due to Silverman and Toplitz.
The following result will also be very useful (see Simons [21]).
Analogously, we can easily prove the following.

C 1 -statistical core
In this section, we define (C,1) analogous of some notions related to statistical convergence (see Fridy [3], Fridy and Orhan [23]).Definition 3.1.(i) A sequence x = (x k ) is said to be lower C 1 -statistically bounded if there exists a constant M such that δ({k : σ k < M}) = 0, or equivalently we write δ c1 ({k : x k < M}) = 0. (ii) A sequence x = (x k ) is said to be upper C 1 -statistically bounded if there exists a constant N such that δ({k : σ k > N}) = 0, or equivalently we write δ c1 ({k : We denote the set of all C 1 (st)-bdd sequences by C 1 (st ∞ ).Definition 3.2.For any M,N ∈ R, let then It is clear that every bounded sequence is statistically bounded and every statistically bounded sequence is C 1 (st)-bdd, but not conversely in general.For example, define x = (x k ) by ifk is an even square, 1 ifk is an odd non square, 0 ifk is an even non square. (3.3) Now let us introduce the following notation: a subsequence k=1 .Definition 3.4.The number λ is a C 1 -statistical limit point of the number sequence x = (x k ) provided that there is a nonthin subsequence of x that is (C,1)-summable to λ. Definition 3.5.The number β is a C 1 -statistical cluster point of the number sequence x = (x k ) provided that for every ε > 0 the set {k ∈ N : |σ k − β| < ε} does not have density zero.Now, let us introduce the following notation: if x is a sequence such that x k satisfies property P for all k except a set of natural density zero, then we say that x k satisfies P for "almost all k," and we abbreviate this by "a.a.k." We define the C 1 (st)-core of a sequence x = (x k ) of complex numbers as follows.
Definition 3.6.For any complex sequence x, let C(x) denote the collection of all closed half-planes that contain σ k for almost all k.Then the C 1 -statistical core of x is defined by Note that in defining C 1 (st)-core (x), we have simply replaced x k by its (C,1)-mean in the definition of st-core (x) in the same manner as Moricz has defined statistical summability (C,1) (or C 1 -statistically convergence).Hence, it follows that It is easy to see that for a statistically bounded real sequence x, where σ = (σ k ).
Lemma 3.7.Let x be statistically bounded sequence; for each z ∈ C let The following lemma is an analog to the previous lemma.
Lemma 3.8.Let x be C 1 -statistically bounded sequence; for each z ∈ C let Proof.Let ω be a C 1 -statistical limit point of x, then Thus, ∩ z∈C M x (z) contains all of C 1 -statistical limit points of x.Since C 1 (st)-core (x) is the smallest closed half-planes that contains C 1 -statistical limit points of x, we have Converse set inclusion follows analogously as in the lemma of Fridy and Orhan [8].
Note that the present form of C 1 (st)-core (x) given in the above lemma is not valid if x is not C 1 -statistically bounded.For example, consider x k := 2k − 1 for all k, then C 1 (st)core (x) = ∅ and st − limsup|σ k − z| = ∞ for any z ∈ C. Thus we have
Sufficiency.Assume that (i) and (ii) hold and ω ∈ K-core (Ax).Then for z ∈ C, we have Proof.Necessity.Suppose that and x is convergent to l.Then