The idea of statistical convergence which is, in fact, a generalization of the usual notion of convergence was introduced by Fast [17] and Steinhaus [31] independently in the same year 1951 and since then several generalizations and applications of this concept have been investigated by various authors namely Bhardwaj et al. [4, 5, 6, 7], Connor [13, 14], Et and Şengül [16], Fridy [18], Işık [21], Işık and Akbaş [22], Mursaleen [26], Rath and Tripathy [28], Salat [30], Temizsu [32] et al. and many others.

The idea of statistical convergence depends upon the density of subsets of the set ℕ of natural numbers. The natural density δ(K) of a subset K of the set ℕ of natural numbers is defined by

where |{k ≤ n : k ∈ K}| denotes the number of elements of K not exceeding n. Obviously, we have δ(K) = 0 provided that K is a finite set.

A sequence x = (x_k) is said to be statistically convergent to L if for every ɛ > 0,

In this case we write S – lim x_k = L. Since lim x_k = L implies S – lim x_k = L, statistical convergence may be considered as a regular summability method. The set of all statistically convergent sequences is denoted by S.

Connor [14], Ghosh and Srivastava [20], Bhardwaj and Singh [8, 9, 10], Çolak [12], Altin and Et [3] and some others have used a modulus function to extend the theory of statistical convergence and construct some new sequence spaces.

The idea of a modulus function was structured by Nakano [27] in 1953. Following Ruckle [29] and Maddox [25], we recall that a modulus f is a function from [0,∞) to [0,∞) such that (i) f(x) = 0 if and only if x = 0, (ii) f(x + y) ≤ f(x) + f(y) for x ≥ 0, y ≥ 0, (iii) f is increasing, (iv) f is continuous from the right at 0. Hence f must be continuous everywhere on [0,∞). A modulus may be unbounded or bounded. For example, f(x) = x_p where 0 < p ≤ 1, is unbounded, but f(x)=x1+x is bounded.

In the year 2014, Aizpuru et al. [2] have defined a new concept of density with the help of an unbounded modulus function and as a consequence they obtained a new concept of non-matrix convergence which is intermediate between the ordinary convergence and the statistical convergence, and agrees with the statistical convergence when the modulus function is the identity mapping.

Contributing in this direction, Bhardwaj et al. [6] introduced and studied a new concept of f-statistical boundedness by using the approach of Aizpuru et al. [2]. It is shown that the concept of f–statistical boundedness is intermediate between the ordinary boundedness and the statistical boundedness. It is also proved that bounded sequences are precisely those sequences which are f–statistically bounded for every unbounded modulus f.

We now recall some definitions that will be needed in the sequel

Definition 1.1

([2]) Let f–be an unbounded modulus function. The f–density of a set K ⊂ ℕ is defined by

provided the limit exists.

Remark 1.2

The concept of f-density reduces to that of natural density when f(x) = x. The equality δ^f (K) + δ^f (ℕ − K) = 1 does not hold, in general, where f is an unbounded modulus. However we can assert that if δ^f (K) = 0, then δ^f (ℕ − K) = 1.

Remark 1.3

For any unbounded modulus f and K ⊂ ℕ, δ^f (K) = 0 implies that δ(K) = 0. But converse need not be true in the sense that a set having zero natural density may have non-zero f–density with respect to some unbounded modulus f.

Definition 1.4

([2]) Let f be an unbounded modulus function. A number sequence x = (x_k) is said to be f–statistically convergent to ℓ, or S^f–convergent to ℓ, if for each ɛ > 0,

and we write it as S^f – lim x_k = ℓ. The set of all f–statistically convergent sequences is denoted by S^f.

In view of Remark 1.3, it follows that every f–statistically convergent sequence is statistically convergent, but a statistically convergent sequence need not be f–statistically convergent for every unbounded modulus f.

In 1932, R.P. Agnew [1] introduced the concept of deferred Cesàro mean of real (or complex) valued sequences x = (x_k) defined by

where p = {p(n) : n ∈ ℕ} and q = {q(n) : n ∈ ℕ} are the sequences of non-negative integers satisfying

Agnew also showed that the method given by (1) has many more important properties besides being regular.

Motivating from the work of Agnew, the concepts of deferred density and deferred statistical convergence were given by Küçükaslan and Yılmaztürk [24, 33] as follows:

Let K be a subset of ℕ and denote the set {k : p (n) < k ≤ q (n), k ∈ K} by K_p,q (n).

Definition 1.5

The deferred density of K is defined by

The vertical bars indicate the cardinality of the enclosed set K_p,q (n). If q (n) = n, p (n) = 0, then deferred density coincides with natural density of K.

Definition 1.6

A real valued sequence x = (x_k) is said to be deferred statistically convergent to L, if for each ɛ > 0

In this case we write S_p,q–lim x_k = L. The set of all deferred statistically convergent sequences will be denoted by S_p,q. If q (n) = n, p (n) = 0, then deferred statistical convergence coincides with usual statistical convergence.

Quite recently, Et et al. [15] introduced and examined the concept of deferred statistical boundedness of order α and established the realtion between statistical boundedness and deferred statistical boundedness of order α.

In the present paper we extend the notion of deferred-density to that of deferred f–density in the same way as natural density was extended to f–density by Aizpuru et al. [2] and then introduce a new and more general non-matrix summability method, namely deferred f–statistical convergence where f is an unbounded modulus. It is shown that the terms of a deferred f–statistical convergent sequence (x_k) coincide to that of a convergent sequence for almost all k deferred with respect to f. Apart from studying various inclusion relations, a decomposition theorem is also estblished. Finally we conclude the paper by the introduction of the concept of strongly deferred Cesàro summable sequences defined by modulus f and it is shown that a bounded sequence which is deferred f–statistical convergent to ℓ is strongly deferred Cesàro summable with respect to f to ℓ.

Throughout the paper, we consider the sequences of non negative integers p = {p(n) : n ∈ ℕ}and q = {q(n) : n ∈ ℕ} satisfying p(n) < q(n) and limn→∞q(n)=∞. Any other restriction ( if needed) on p(n) and q(n) will be mentioned in the related theorems.

We begin this section by introducing a new concept of deferred f–density of a subset of ℕ.

Definition 2.1

The deferred f–density of a subset K of ℕ is defined as

provided the limit exists. The verticle bars indicate the cardinality of the enclosed set. It is obvious that finite sets have zero deferred f–density for any unbounded modulus f.

Remark 2.2

For q(n) =n p(n) = 0 the deferred f–density reduces to f–density and when f(x) = x, the deferred f–density coincides with deferred density. If q(n) =n p(n) = 0 and f(x) = x, the deferred f–density turns out to be natural density.

The equality δ_p,q(K) + δ_p,q(ℕ − K) = 1 remains no longer true, if deferred density is replaced by deferred f–density, i.e., δp,qf(K)+δp,qf(ℕ-K)=1 does not hold, in general where f is an unbounded modulus. Let us demonstrate it with the help of following example.

Example 2.3

Take K = (2, 4, 6, …), q(n) = 4n, p(n) = 2n and f(x) = log(x+1). Then

Also, δp,qf(ℕ-K)=1.

Proposition 2.4

If for any unbounded modulus f,δp,qf(K)=0then δ_p,q(K) = 0.

Remark 2.5

Converse of above proposition need not be true in the sense that a set having zero deferred density may have non-zero deferred f–density. This is illustrated by the following example.

Example 2.6

Let f(x) = log(x+1) and K = (1, 4, 9, 16, …). Take q(n) = n² and p(n) = n. Then

and so δ_p,q(K) = 0 and

This implies that δp,qf≠0.

Before proceeding further we first introduce the following notation

For an unbounded modulus f, if x = (x_k) is a sequence such that x_k satisfies property P for all k, except a set of deferred f–density zero, then we say x = (x_k) satisfies P for “almost all k deferred with respect to f” and we abbreviate this by “ a.a. k deferred w.r.t. f”.

Definition 2.7

Let f be an unbounded modulus. A sequence x = (x_k) is said to be deferred f–statistically convergent to ℓ if for each ɛ > 0,

In this case we write Sp,qf-lim xk=ℓ.

Proposition 2.8

Let f be an unbounded modulus. Then every deferred f-statistically convergent sequence is deferred statistically convergent but converse need not be true.

Proposition 2.9

Let f be an unbounded modulus. ThenSp,qf-lim xk=ℓif and only if there exists K ⊂ ℕ such thatδp,qf(K)=0andlim xkk∈ℕ-K=ℓ.