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Journal of Inequalities and Applications volume 2013, Article number: 444 (2013)
In the present paper, we introduce the q-Bernstein-Schurer-Kantorovich operators. We give the Korovkin-type approximation theorem and obtain the rate of convergence of this approximation by means of the first and the second modulus of continuity. Moreover, we compute the order of convergence of the operators in terms of the elements of Lipschitz class functions and the modulus of continuity of the derivative of the function.
MSC: 41A10, 41A25, 41A36.
Some authors have defined general sequences of linear positive operators where the classical sequences can be achieved as particular cases. For instance, Schurer  proposed the following generalization of Bernstein operators in 1962. Let denote the space of a continuous function on . For all , and fixed , the Bernstein-Schurer operators are defined by (see also, )
In 1987, q-based Bernstein operators were defined and studied by Lupaş . In 1996, another q-based Bernstein operator was proposed by Phillips . Then the q-based operators have become an active research area (see [5–9] and ).
Muraru  introduced and investigated the q-Bernstein-Schurer operators. She obtained the Korovkin-type approximation theorem and the rate of convergence of the operators in terms of the first modulus of continuity. These operators were defined, for fixed and for all , by
where . If we choose in (1.1), we get the classical q-Bernstein operators .
Recall that for each nonnegative integer r, is defined as
and the q-factorial of the integer r is defined by
For integers n and r, with , q-binomial coefficients are defined by 
Afterwards, several properties and results of the operators defined by (1.1), such as the order of convergence of these operators by means of Lipschitz class functions, the first and the second modulus of continuity and the rate of convergence of the approximation process in terms of the first modulus of continuity of the derivative of the function, were given by the authors . On the other hand, q-Szasz-Schurer operators were discussed in .
Kantorovich considered the linear positive operators which are defined for as follows:
In 2007, Dalmanoğlu defined Kantorovich-type q-Bernstein operators by 
Notice that, the q-Jackson integral is defined on the interval as follows:
Then she obtained the first three moments and gave the rate of convergence of the approximation process in terms of the first modulus of continuity .
In our definition, the integral that we consider in the q-Schurer-Bernstein-Kantorovich operator is
So, throughout this paper, we will use the following results, which are computed directly by the tools of q-calculus.
Using (1.2), we can find the following results:
where . On the other hand, by (1.2) and (1.3) we get
Recall that the first three moments of the q-Bernstein-Schurer operators were given by Muraru in  as follows.
Lemma 1.1 For the first three moments of we have:
We organize the paper as follows.
Firstly, in section two, we define the q-Bernstein-Schurer-Kantorovich operators and obtain the moments of them. In section three, we obtain the rate of convergence of the q-Bernstein-Schurer-Kantorovich operators in terms of the first modulus of continuity. Also we give the order of approximation by means of Lipschitz class functions and the first and the second modulus of continuity. Furthermore, we compute the degree of convergence of the approximation process in terms of the first modulus of continuity of the derivative of the function.
2 Construction of the operators
For fixed , we introduce the q-Bernstein-Schurer-Kantorovich operators :
for any real number , and . It is clear that is a linear and positive operator for .
For the first three moments and the first and the second central moment, we state the following lemma.
Lemma 2.1 For the q-Bernstein-Schurer-Kantorovich operators we have
Proof (i) From (1.3), we get
Using (1.1), (1.4) and Lemma 1.1, we have
From (1.1), (1.4), (1.5) and then Lemma 1.1, we can calculate the as follows:
Finally, we get
where and are the corresponding moments of the q-Bernstein-Schurer operators.
It is obvious that
Direct calculations yield,(2.2)
By Korovkin’s theorem, we can state the following theorem. □
Theorem 2.2 For all , we have
provided that with and that .
3 Rate of convergence
In this section, we compute the rate of convergence of the operators in terms of the modulus of continuity, elements of Lipschitz classes and the first and the second modulus of continuity of the function. Furthermore, we calculate the rate of convergence in terms of the first modulus of continuity of the derivative of the function.
Now, we give the rate of convergence of the operators by means of the first modulus of continuity. Recall that the first modulus of continuity of f on the interval for is given by
It is known that for all , we have
and for any ,
Theorem 3.1 Let . If , we have
where is the modulus of continuity of f and , which is given as Lemma 2.1.
Proof Using the linearity and positivity of the operator, we get
By the Cauchy-Schwarz inequality,
Now we have
where . Again applying the Cauchy-Schwarz inequality, we get
So, we have
Choosing , we obtain
The proof is concluded. □
Now we give the rate of convergence of the operators in terms of the Lipschitz class , for . Note that a function belongs to if
Theorem 3.2 Let , then
where is the same as in Theorem 3.1.
Proof By the linearity and positivity, we have
Considering (3.2) and then applying the Hölder’s inequality with and , we get
So, we have
where . Again applying Hölder’s inequality with and , we get
Hence, the desired result is obtained. □
Now let us denote by the space of all functions such that . Let denote the usual supremum norm of f. The classical Peetre’s K-functional and the second modulus of smoothness of the function are defined, respectively, by
where . It is known that [, p.177] there exists a constant such that
Theorem 3.3 Let , and . Then, for fixed , we have
for some positive constant C, where
Proof Define an auxiliary operator by
Then, by Lemma 2.1, we get
Then, for a given , it follows by the Taylor formula that
Taking into account (3.7) and using (3.7), we get, for every , that
Then by (3.6),
Hence Lemma 2.1 implies that
Since , considering (3.4) and (3.5), for all and , we may write from (3.8) that
which yields that
Hence we get the result. □
Now, we compute the rate of convergence of the operators in terms of the modulus of continuity of the derivative of the function.
Theorem 3.4 Let and be fixed. If has a continuous derivative and is the modulus of continuity of on , then
where M is a positive constant such that (),
Proof Using the mean value theorem, we have
where . Hence, we have
where is given in (3.9). Hence,
From the Cauchy-Schwarz inequality, for the first term, we get
Finally, we have
This completes the proof. □
4 Concluding remarks
In this paper, we obtain many results in the pointwise sense. On the other hand, we see that the interval is bounded and closed, and also f is continuous on it, so these results can be given in the uniform sense.
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The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.
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Özarslan, M.A., Vedi, T. q-Bernstein-Schurer-Kantorovich Operators. J Inequal Appl 2013, 444 (2013). https://doi.org/10.1186/1029-242X-2013-444
- q-integral operator
- positive linear operators
- q-Bernstein operators
- modulus of continuity