Some New Difference Sequence Spaces of Invariant Means Defined by Ideal and Modulus Function

where t k m (x) = (x m + x σ(m) + ⋅ ⋅ ⋅ + x σ k (m) )/(k + 1). In case σ is the translation mapping n → n + 1, σ-mean is often called a Banach limit and V σ , the set of bounded sequences all whose invariant means are equal, is the set of almost convergent sequences (see Lorentz [2]). Using the concept of invariant means Mursaleen et al. [3] introduced the following sequence spaces as a generalization of Das and Sahoo [4]:


Introduction and Background
Let ℓ ∞ and be the Banach spaces of real bounded and convergent sequences with the usual supremum norm.
Let be the mapping of the set of all positive integers into itself. A continuous linear functional on ℓ ∞ is said to be an invariant mean or -mean if and only if where ( ) = ( + ( ) + ⋅ ⋅ ⋅ + ( ) )/( + 1).
In case is the translation mapping → + 1, -mean is often called a Banach limit and , the set of bounded sequences all whose invariant means are equal, is the set of almost convergent sequences (see Lorentz [2]). Using the concept of invariant means Mursaleen et al. [3] introduced the following sequence spaces as a generalization of Das and investigated some of its properties. The notion of statistical convergence for number sequences was studied at the initial stage by Fast [5] and later investigated by Connor [6], Fridy [7], Maddox [8],Šalát [9], and many others. International Journal of Analysis Definition 1 (see [5]). A number sequence = ( ) is said to be statistically convergent to a number (denoted by − lim → ∞ = ) provided that, for every > 0, where the vertical bars denote the cardinality of the enclosed set. By a lacunary sequence, we mean an increasing sequence = ( ) of positive integers such that 0 = 0 and ℎ = − −1 → ∞ as → ∞. The intervals determined by will be denoted by = ( −1 , ], where the ratio / −1 is denoted by . The space of lacunary strongly convergent sequence was defined by Freedman et al. [10] as follows: Fridy and Orhan [11] generalized the concept of statistical convergence by using lacunary sequence which is called lacunary statistical convergence. Further, lacunary sequences have been studied by Fridy and Orhan [12], Pehlivan and Fisher [13], Et and Gökhan [14], and Tripathy and Dutta [15]. Quite recently, Karakaya [16] combined the approach of lacunary sequence with invariant means and introduced the notion of strong -lacunary statistically convergence as follows. Definition 2 (see [16]). Let = ( ) be a lacunary sequence. A sequence = ( ) is said to be lacunary strong -lacunary statistically convergent if, for every > 0, where denotes the set of all lacunary strong -lacunary statistically convergent sequences.
Another interesting generalization of statistical convergence was introduced in ∈ N}. Throughout the present work, I denotes a nontrivial admissible ideal.
Definition 3 (see [17]). Let I ⊂ P(N) be a nontrivial ideal in N and let ( , ) be a metric space. A sequence = ( ) in is said to be I-convergent to if, for each > 0, the set ( ) = { ∈ N : ( , ) ≥ } ∈ I. In this case, we write Recently, Das et al. [18] unified the idea of lacunary statistical convergence with ideal convergence and presented the following interesting generalization of statistical convergence.
Definition 4 (see [18]). Let = ( ) be a lacunary sequence. A sequence = ( ) of numbers is said to be I-lacunay statistical convergent or (I)-convergent to , if, for every > 0 and > 0, In this case, we write → ( (I)) or (I)−lim → ∞ = . The set of all I-lacunary statistically convergent sequences will be denoted by (I).
The notion of modulus function was introduced by Nakano [28] as follows: by a modulus function, we mean a function from is increasing, and (iv) is continuous from right at 0. It follows that must be continuous everywhere on [0, ∞). A modulus function may be bounded or unbounded. In the recent past the notion of modulus function was investigated from different aspects and sequence spaces have been studied by Ruckle [29], Maddox [30], Et [31], Pehlivan and Fisher [13], Savas [32], Et and Gökhan [14], Kumar et al. [33], and many others.
The following well-known lemma is required for establishing a very important result in our paper. Lemma 6. Let be a modulus function and let 0 < < 1. Then, for each > , we have ( ) ≤ (2 ⋅ (1) )/ .
International Journal of Analysis 3 The following inequality will be used throughout the paper. Let = ( ) be a positive sequence of real numbers with 0 < ≤ sup = and = max(1, 2 −1 ). Then, for all , ∈ C, for all ∈ N, we have Inspired by the above works, we presently introduce some new kind of sequence spaces by using ideal convergence, modulus function, and invariant mean. Further, we also obtain some relevant connections of these spaces with Δ (I)-convergence.

Main Results
Throughout the paper, I ⊂ P(N) is considered a nontrivial admissible ideal and ( ) denotes the space of all sequences = ( ) ∈ .
Definition 7. Let I be an admissible ideal, let be a modulus function, and let = ( ) be any sequence of strictly positive real numbers. Then, for each > 0, we define the following sequence spaces: uniformly in .
Remark 8. By taking some particular cases, we obtain the following. Proof. We will prove the assertion only for [Δ , I, ] 0 and others can be treated similarly. Suppose that = ( ), = ( ) ∈ [Δ , I, ] 0 . Then, for every > 0 and uniformly in , the sets belong to I. Let , ∈ C and Δ is linear; then, where , are two positive numbers such that | | ≤ and | | ≤ .

4
International Journal of Analysis Then, for given > 0, we have the following containment: uniformly in .
Since , ∈ [Δ , I, ] 0 , it follows that the later sets belong to I. By using the property of an ideal the set on the left hand side in the above expression also belongs to I. This completes the proof.

Theorem 10. For ≥ 1, then the inclusion
is strict.
Proof. We will prove the result for Since is a modulus function, therefore we have the following inequality: uniformly in . Now, for given > 0, we have the following containment: uniformly in . Both the sets on the right hand side in the above containment belong to I by (16). It follows that ∈ [Δ , I, ] 0 .
(ii) This result can be proved by the following inequality: uniformly in , where sup = and = max(1, 2 −1 ). Proof. This can be proved similarly as in Theorem 12(i).

Δ (I)-Convergence
In this section, we define the notion of Δ (I)-convergence with the help of ideal and invariant means and difference sequences. Further, we also establish some relations between Δ (I)-convergence and [Δ , I, ] 0 .  Proof. Assume that ∈ [Δ , I, ] 0 and > 0 is given. Then, we have uniformly in . Then, for every > 0, we have the following containment: Proof. This part is the direct consequence of Theorems 17 and 18.