Pointwise approximation by a Durrmeyer variant of Bernstein-Stancu operators

In the present paper, we introduce a kind of Durrmeyer variant of Bernstein-Stancu operators, and we obtain the direct and converse results of approximation by the operators.

Motivated by (.), we introduce the following generalization of the operators (.): By Lemma  in Section , we observe that S n,α,β (f , x) can be rewritten as follows: Especially, when α  = α  = β  = β  = , S n,α,β (f , x) reduces to the classical Bernstein-Durrmeyer operators in (.). Many authors have studied some special cases of the operators S n,α,β (f , x). For example, the case α  = α  = β  =  in [] by Jung, Deo, and Dhamija, the case α  = β  =  in [] by Acar, Aral, and Gupta. The main purpose of the present paper is to establish pointwise direct and converse approximation theorems of approximation by S n,α,β (f , x). To state our result, we need some notations: and where x ∼ y means that there exists a positive constant c such that c - y ≤ x ≤ cy.
Our first result can be read as follows.
Theorem  Let f be a continuous function on A n , λ ∈ [, ] be a fixed positive number. Then there exists a positive constant C only depending on λ, α  , α  , β  , and β  such that and ω(f , t) is the usual modulus of continuity of f on A n .
Throughout the paper, C denotes either a positive absolute constant or a positive constant that may depend on some parameters but not on f , x, and n. Their values may be different at different locations.
For the converse result, we have the following.

Auxiliary lemmas
Then which implies that Therefore, By the facts that Lemma  For any given γ ≥ , we have Thus, and thus (.) also holds.

Now, by (.) and (.), we have
Proof By a similar calculation to that of Lemma , we have On the other hand, we have which proves (.).
By Lemma , we have Hence, (.) is proved.

Lemma  If f is r times differentiable on [, ], then
Proof By using Leibniz's theorem, we have Since d r dt r q n+r,k+r (t) = r i= r i (-) i (n + r)! n! q n,k+i (t), we have We obtain the required result by integrating by parts r times. Set Proof Firstly, we prove (.) by considering the following two cases.

Lemma  For
It has been shown in [] that Lemma  and Lemma  are valid when δ n (t) is replaced by δ * n (t), which combining with (.) proves Lemma  and Lemma .

Proof of Theorem 1
Define the auxiliary operators S n,α,β (f , x) as follows: