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Introduction
The first idea of statistical convergence goes back to the first edition of the famous Zygmund's monograph []. The statistical convergence was introduced for real and complex sequences by Steinhaus []. Fast [] extended the usual concept of sequential limit and called it statistical convergence. Schoenberg [] called it as D-Convergence. The idea depends on a certain density of subsets of N. The natural density (or asymptotic density) of a set A ⊂ N is defined by δ(A) = lim n→∞  n |{k ≤ n : k ∈ A}| if the limit exists, where |A(n)| is cardinality of the set A(n) (see []). A sequence x = (x k ) of complex numbers is said to be statistically convergent to some number if δ({k ∈ N : |x k -| ≥ ε}) has natural density zero for ε > . is necessarily unique, which is statistical limit of (x k ), and written as S-lim x k = . The space of all statistically convergent sequences is denoted by S (see The order of statistical convergence of a sequence of positive linear operators was given by Gadjiev  Let s  denote the space of all double sequences, and let  ∞ , c  and c   be the linear spaces of bounded, convergent and null sequences x = (x jk ) with complex terms, respectively, normed by x (∞,) = sup j,k |x jk |, where j, k ∈ N = {, , . . .}.
A double sequence x = (x j,k ) ∞ j,k= has Pringsheim limit provided that for every ε >  there exists N ∈ N such that |x j,k -| < ε whenever j, k > N . In this case, we write P-lim x = [].
x = (x j,k ) ∞ j,k= is bounded if there exists a positive number M such that |x j,k | < M for all j and k, that is, The double natural density of K is defined by f is continuous from the right at . Hence, f must be continuous everywhere on [, ∞). A modulus function may be bounded or unbounded. For example, Aizpuru et al. [] introduced and discussed the concepts of f -statistical convergence and f -statistically Cauchy sequences, a single sequence of numbers, where f is an unbounded modulus function. Bhardwaj

f λ,μ -double statistical convergence of order α
In this section, we introduce f λ,μ -double statistical convergence of order α for double sequences.
Throughout this paper, we take s, t, u, v ∈ (, ] as otherwise indicated. We will write α instead of (s, t) and β instead of (u, v). Also, we define the following: We begin with the following definitions. Let λ = (λ n ) and μ = (μ m ) be two non-decreasing sequences of positive real numbers tending to ∞ with λ n+ ≤ λ n + , λ  = ; μ n+ ≤ μ n + , μ  =  and α ∈ (, ] be given. Let K ⊆ N × N be a two-dimensional set of positive integers and f be an unbounded modulus function. Then δ f  α (λ, μ)-double density of K is defined as f (j, k) ∈ I n × I m : (i, j) ∈ K if the limit exists.
Definition . Let λ = (λ n ) and μ = (μ m ) be two non-decreasing sequences of positive real numbers as above and α ∈ (, ] be given. (x jk ) is said to be f λ,μ -statistically convergent of order α if there is a complex number such that, for every ε > , In this case we write S  α (f , λ, μ)-lim j,k x jk = , and we denote the set of all f λ,μ -statistically convergent double sequences of order α by S  α (f , λ, μ), where f is an unbounded modulus function.
In the case of f (x) = x, α ∼ =  and λ n = n, μ m = m, f λ,μ -statistical convergence of order α reduces to the statistical convergence of double sequences []. If x = (x jk ) is f λ,μstatistically convergent of order α to the number , then is determined uniquely. f λ,μdouble statistical convergence of order α is well defined for α ∈ (, ] but it is not well defined for α . For this, let us define x = (x jk ) as follows: Since lim t→∞ for α , that is, s >  and t > , so that x = (x jk ) is f λ,μ -statistically convergent of order α both to  and , i.e., S  α (f , λ, μ)-lim x jk =  and S  α (f , λ, μ)-lim x jk = . But this is impossible.
Theorem . Let f be an unbounded modulus function and α ∈ (, ]. Let x = (x jk ), y = (y jk ) be any two sequences of complex numbers. Then Theorem . Let f be an unbounded modulus function andα,β be two real numbers such that  α β . Then S  α (f , λ, μ) ⊆ S β (f , λ, μ) and strict inclusion may occur.
Corollary . If x = (x jk ) is f λ,μ -statistically convergent of order α to , for some α such that α ∈ (, ], then it is f λ,μ -statistically convergent to , and the inclusion is strict.
Definition . Let f be a modulus function andα be a positive real number. We have Proof (i) The proof of (i) is trivial.
Let ε >  and choose δ with  < δ <  such that f (t) < ε for  ≤ t < δ. Now we write  (λ n μ m )α j∈J n k∈I n f |x jk | =  +  , where the first summation is over |x jk | ≤ δ and the second is over |x jk | > δ. Then  ≤