Convergence of the q-Stancu-Szász-Beta type operators

In this paper, we study on q-Stancu-Szász-Beta type operators. We give these operators convergence properties and obtain a weighted approximation theorem in the interval [0,∞). MSC:41A25, 41A36.


Introduction
In [], Mahmudov constructed q-Szász operators and obtained rate of global convergence in the frame of weighted spaces and a Voronovskaja type theorem for these operators. In [], Gupta and Mahmudov studied on the q-analog of the Szász-Beta type operators. In [], Yüksel and Dinlemez gave a Voronovskaja type theorem for q-analog of a certain family Szász-Beta type operators. In [], Govil and Gupta introduced the q-analog of certain Beta-Szász-Stancu operators. They estimated the moments and established direct results in terms of modulus of continuity and an asymptotic formula for the q-operators. In [-], interesting generalization about q-calculus were given. Our aims are to give approximation properties and a weighted approximation theorem for q-Stancu-Szász-Beta type operators. We use without further explanation the basic notations and formulas, from the theory of q-calculus as set out in [-]. Let A >  and f be a real valued continuous function defined on the interval [, ∞). For  < q ≤ , q-Stancu-Szász-Beta type operators are defined as If we write q =  and α = β =  in (.), then the operators B (α,β) n,q (f , x) are reduced to Szász-Beta type operators studied in [-].

Auxiliary results
For the sake of brevity, the notation F q s (n) = s i= [ni] q and G q β (n) = ([n] q + β) will be used throughout the article. Now we are ready to give the following lemma for the Korovkin test functions.
Proof Using the q-Gamma and q-Beta functions in [, ], we obtain the following equality: Then, using (.), for m = , we get and the proof of (i) is finished. With a direct computation, we obtain (ii) as follows: Using the equality we get and so we have the proof of (iii).
To obtain our main results we need to compute the second moment.
Lemma  Let q ∈ (, ) and n > . Then we have the following inequality: .
Proof From the linearity of the B (α,β) n,q operators and Lemma , we write the second moment as .
And the proof of Lemma  is now finished.

Direct estimates
The Peetre K -functional is defined by where Thus we are ready to give direct results. The following lemma is routine and its proof is omitted.
And the proof of the Lemma  is now completed.
Theorem  Let (q n ) ⊂ (, ) a sequence such that q n →  as n → ∞. Then for every n > , Proof Using (.) for any g ∈ W  ∞ , we obtain the following inequality: From Lemma , we get By using equality (.) we have Taking the infimum over g ∈ W  ∞ on the right-hand side of the above inequality and using the inequality (.), we get the desired result.
Theorem  Let (q n ) ⊂ (, ) a sequence such that q n →  as n → ∞.

Competing interests
The author declares to have no competing interests.