The principle of statistical convergence arose from the first version of the monograph of Zygmund [29] in 1935, and its definition was given in a short note by Fast [12] and later independently by Schoenberg [25] where some specific characteristics of statistical convergence were identified. In recent decades, statistical convergence has arisen in several fields under different names. It appears in such fields as measure theory, approximation theory, Banach spaces, hopfield neural networks, locally convex spaces, summability theory, ergodic theory, number theory, turnpike theory, trigonometric series, Fourier analysis, and optimization. Such authors as Connor [6], Fridy [13], Šalát [27], Rath and Tripathy [22], Et [10], Duman [9], León-Saavedra et al. [18], Weisz [28] have explored statistical convergence from the perspective of spaces of sequences; this is referred to as the theory of summability.

In 1953, Nakano [21] presented the idea of a modulus function for the first time. In 2014, with the benefit of an unbounded modulus function, Aizpuru et al. [1] characterized the notion of f-density, and so introduced a new nonmatrix convergence principle. Using this notion, Bhardwaj et al. [4] have recently extended statistical convergence to the notion of f-statistical boundedness. It has been demonstrated that bounded sequences are definitely those sequences which are f-statistically bounded for every unbounded modulus.

By using a modulus function, Maddox [20], Connor [7], Ruckle [23], Gosh and Srivastava [14], Altin and Et [2], Sarma [24], Kamber [17] and others have constructed various sequence spaces. Further details and applications of the principles of statistical convergence and strong Cesàro summability are available in [5, 8, 11, 15, 16, 26].

In this study, the symbols c and ℓ∞ denote the spaces of convergent and bounded sequences, respectively. The symbols C, R and N denote the sets of all complex, real and natural numbers, respectively.

Definition 2.1. [27] Let U⊂N. The natural density of U and is defined as

in the case the limit exists, where Un=u≤n:u∈U is the cardinality of the indicated set.

It is obvious that δN=1 and δU=δN-δN\U=1-δN\U and also δU=0 if U is a finite subset of N.

Definition 2.2. [27] A sequence xk of complex numbers is statistically convergent (or S-convergent) to l∈C if

for every ε>0. We write S-limxk=l or xk→lS in this particular case. Throughout the paper, the class of all S-convergent sequences will be symbolized by S. That is, we set

Definition 2.3. [21] A function f:R+∪0→R+∪0 is a modulus function (or simply a modulus) if

• fh=0⇔h=0,

• fh1+h2≤fh1+fh2 for all h1,h2∈R+∪0,

• f is increasing,

• f is continuous at 0 from the right.

From the above characteristics we clearly get that a modulus f is continuous on R+∪0. There are bounded and unbounded modulus functions. As an example, fh=hh+1 is a bounded modulus, but fh=logh+1 is an unbounded modulus. Furthermore, for every modulus f and each positive integer n, we have fnh≤nfh from condition 2.

Lemma 2.4. [19] For any modulus f, limh→∞fhh exists and limh→∞fhh=infh∈0,∞fhh.

Definition 2.5. [1] Suppose f is an unbounded modulus. The f-density of a subset U of N is defined by

if the limit exists.

The f-density becomes the natural density if we take fh=h. In the case of the natural density, it is obvious that for any U⊂N, we have δU+δN\U=1. But this conclusion is different for f-density, i.e., δfU+δfN\U=1 does not have to be true, in general. To verify this situation, we may take U=2,4,6,... and the modulus fh=logh+1, then we have δfN\U=1=δfU. But this situation happens for any unbounded modulus function when δfU=0 (for details see Remark 1.2 of [3]). For any finite U⊂N, f-density and natural density have similar concepts, that is, δfU=0 and so that δfU+δfN\U=1.

We know that if U⊂N, δfU=0 implies δU=0 for any unbounded modulus function f (see [1]). The converse need not hold. Indeed, take fh=logh+1 and set U=u2:u∈N. One gets that δU=0 but δfU=12. Moreover, if U⊂N is finite and δU=0, then δfU=0.

Definition 2.6. Suppose f is an unbounded modulus. Then, the sequence xk of complex numbers is f-statistically convergent (or Sf-convergent) to l∈C if

for every ε>0. We write this as Sf-limxk=l or xk→lSf. The class of all Sf-convergent sequences will be symbolized by Sf throughout the paper, that is,

Note that Sf-convergence reduces to S-convergence in the case fh=h.

Lemma 2.7. Suppose xk is any sequence of complex numbers. If xk∈Sf, then its Sf-limit is unique.

Definition 2.8. Suppose f is a modulus. Then, the sequence xk of complex numbers is f-strongly Cesàro summable to l∈C if

The symbol wf denotes the class of all f-strongly Cesàro summable sequences, that is,

Note that this definition does not require the modulus function f to be unbounded.

The concepts of f-strong Cesàro summability and strong Cesàro summability are the same in the case fh=h and the set of all strongly Cesàro summable sequences will be denoted by w, that is, wf will reduce to w if fh=h.

In this section, we give the main results of the study.

3.1. Modulus functions and strong Cesàro summability

Theorem 3.1. Suppose f and g are any modulus functions. If suph∈0,∞fhgh<∞, then wg⊂wf.

Proof. Assume that p=suph∈0,∞fhgh<∞. Then, we have fhgh≤p and so that fh≤pgh for every h∈0,∞. Now, it is apparent that p>0 and if x=xk is g-strongly Cesàro summable to l, we may write

Taking the limits on both sides as n→∞, we obtain that x∈wg implies x∈wf.

Remark 3.2. The converse of Theorem 3.1 does not have to be correct for every modulus functions f and g such that suph∈0,∞fhgh<∞, in general. The example below demonstrates that at least for certain specific modulus functions, the inclusion wg⊂wf can be strict.

Example 3.3. Define the sequence x=xk as

and take the modulus functions fh=hh+1 and gh=h. Then, suph∈0,∞fhgh=1<∞ and so that wg⊂wf. By using the equality f0=0, we have

Since [3]nn→0 as n→∞, we get x∈wf. However,

Since 1n[3]n-1[3]n22→∞ as n→∞ so that x∉wg, where r denotes an integral part of the real number r. Hence, x∈wf-wg and the inclusion wg⊂wf is strict.

Theorem 3.4. Suppose f and g are any modulus functions. If infh∈0,∞fhgh>0, then wf⊂wg.

Proof. Suppose that q=infh∈0,∞fhgh>0. Then, we have fhgh≥q and so that qgh≤fh for every h∈0,∞. Now, if x=xk is f-strongly Cesàro summable to l, we may write

Taking the limits on both sides as n→∞, we obtain that x∈wf implies x∈wg and this fulfills the proof.

Remark 3.5. The converse of Theorem 3.4 does not have to be correct for every modulus functions f and g if infh∈0,∞fhgh>0, in general. For this, recall the sequence x=xk in Example 3.3 and take the modulus functions f(h)=h and g(h)=hh+1. Then, infh∈0,∞fhgh>0 and x∈wg but x∉wf. This shows that at least for certain specific modulus functions, the inclusion wf⊂wg can be strict.

The outcome below is a result of Theorem 3.1 and Theorem 3.4.

Corollary 3.6. Suppose f and g are any modulus functions. If

then wf=wg.

Corollary 3.7. Suppose f is a modulus function. If infh∈0,∞fhh>0, then wf=w.

Proof. Since w⊂wf for any modulus function by the first part of Theorem 3.4 of [3] for the case α=1, taking gh=h in Theorem 3.4, we obtain wf⊂w if infh∈0,∞fhh>0. Therefore, wf=w if infh∈0,∞fhh>0.

3.2. Relations between statistical convergence and strong Cesàro summability according to modulus functions

Theorem 3.8. suppose f and g are unbounded modulus functions. If infh∈0,∞fhgh>0 and limh→∞ghh>0, then every f-strongly Cesàro summable sequence is g-statistically convergent, that is, wf⊂Sg.

Proof. Suppose that β=infh∈0,∞fhgh>0. Then, we have fhgh≥β and so that βgh≤fh for every h∈0,∞. Now, if x=xk is f-strongly Cesàro summable to l, we may write

Taking the limits on both sides as n→∞, we obtain that x∈wf implies x∈Sg since limh→∞ghh>0.

Remark 3.9. The converse of Theorem 3.8 does not have to be correct for every unbounded modulus functions f and g if infh∈0,∞fhgh>0 and limh→∞ghh>0, in general. The following illustration can demonstrate that at least for certain specific unbounded modulus functions, the inclusion wf⊂Sg can be strict.

Example 3.10. Recall the sequence x=xk in Example 3.3 and take the modulus functions f(h)=g(h)=h. Then, we have infh∈0,∞fhgh>0 and limh→∞ghh>0 and also

By taking the limits on both sides as n→∞, we get that x∈Sg. However, x∉wf as shown in Example 3.3.

The outcome below is acquired by taking gh=fh in Theorem 3.8.

Corollary 3.11. Suppose f is an unbounded modulus. If limh→∞fhh>0, then every f-strongly Cesàro summable sequence is f-statistically convergent.

Remark 3.12. Corollary 3.11 was given with the extra condition “fxy≥cfxfy for all x≥0, y≥0 and some positive number c” in [3]. It seems that this extra condition is not necessary and it should be neglected.

The outcome below is acquired by taking gh=h in Theorem 3.8 (see also in [3]).

Corollary 3.13. Suppose f is an unbounded modulus. If infh∈0,∞fhh>0, then every f-strongly Cesàro summable sequence is statistically convergent.

The outcome below is acquired by taking fh=h in Corollary 3.13, which is the first part of Theorem 2.1 of [6], for the case q=1.

Corollary 3.14. A strongly Cesàro summable sequence is statistically convergent.

Theorem 3.15. Suppose f and g are unbounded modulus functions. Then, every bounded and f-statistically convergent sequence is g-strongly Cesàro summable sequence, i.e., ℓ∞∩Sf⊂wg.

Proof. Assuming that f and g are unbounded modulus functions. Since Sf⊂S by the first part of Corollary 2.2 of [1], and since ℓ∞∩S⊂w by the second part of Theorem 2.1 of [6], then we have ℓ∞∩Sf⊂ℓ∞∩S⊂w, that is, ℓ∞∩Sf⊂w. On the other hand, since w⊂wg for any modulus g by the first part of Theorem 3.4 of [3] for the case α=1, it follows that ℓ∞∩Sf⊂wg.

Remark 3.16. The converse of Theorem 3.15 does not have to be correct for every unbounded modulus functions f and g, in general. The following example demonstrates this situation.

Example 3.17. Let us consider the sequence x=xk as

and take the modulus functions gh=fh=logh+1. Then, by using the equality g0=0, we have

So that x∈wg. Although,

That is, x∉Sf. This means that the inclusion ℓ∞∩Sf⊂wg is strict.

The following inclusions are a result of Theorem 3.15.

Corollary 3.18. If f is any unbounded modulus, then we have

• ℓ∞∩Sf⊂wf,

• ℓ∞∩Sf⊂w, and

• ℓ∞∩S⊂wf.