Journal of Applied Mathematics

Volume 2012 (2012), Article ID 454579, 14 pages

doi:10.1155/2012/454579

Research Article

Approximation Theorems for Generalized Complex Kantorovich-Type Operators

N. I. Mahmudov and M. Kara

Eastern Mediterranean University, Gazimagusa, T.R. North Cyprus, Mersiin 10, Turkey

Received 28 April 2012; Accepted 3 September 2012

Academic Editor: Jinyun Yuan

Copyright © 2012 N. I. Mahmudov and M. Kara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The order of simultaneous approximation and Voronovskaja-type results with quantitative estimate for complex q-Kantorovich polynomials () attached to analytic functions on compact disks are obtained. In particular, it is proved that for functions analytic in , , the rate of approximation by the q-Kantorovich operators () is of order versus for the classical Kantorovich operators.

1. Introduction

For each integer , the -integer and the -factorial are defined by

For integers , the -binomial coefficient is defined by

For fixed , we denote the -derivative of by

Let be a disc in the complex plane . Denote by the space of all analytic functions on . For we assume that .

In several recent papers, convergence properties of complex -Bernstein polynomials, proposed by Phillips [1], defined by

and attached to an analytic function in closed disks, were intensively studied by many authors; see [2] and references their in. It is known that the cases and are not similar to each other. This difference is caused by the fact that, for , are positive linear operators on while for , the positivity fails. The lack of positivity makes the investigation of convergence in the case essentially more difficult than that for . There are few papers [3–6] studying systematically the convergence of the -Berntsein polynomials in the case . If then qualitative Voronovskaja-type and saturation results for complex -Bernstein polynomials were obtained in Wang and Wu [5]. Wu [6] studied saturation of convergence on the interval for the -Bernstein polynomials of a continuous function for arbitrary fixed . On the other hand, Gal [7, 8], Anastassiou and Gal [9, 10], Mahmudov [11–13], and Mahmudov and Gupta [14] obtained quantitative estimates of the convergence and of the Voronovskaja’s theorem in compact disks, for different complex Bernstein-Durrmeyer type operators.

The goal of the present note is to extend these type of results to complex Kantorovich operators based on the -integers, in the case , defined as follows:

Notice that in the case , these operators coincide with the classical Kantorovich operators. For the operator is positive and for , it is not positive. The problems studied in this paper in the case were investigated in [2, 9].

We start with the following quantitative estimates of the convergence for complex -Kantorovich-type operators attached to an analytic function in a disk of radius and center .

Theorem 1.1. Let .

(i)Let and . For all and , one has

(ii)Let and . For all and , one has

Remark 1.2. (i) Since as in the estimate in Theorem 1.1(i) we do not obtain convergence of to . But this situation can be improved by choosing with as . Since in this case as , from Theorem 1.1(i) we get uniform convergence in .

(ii) Theorem 1.1(ii) says that for functions analytic in , , the rate of approximation by the -Kantorovich operators () is of order versus for the classical Kantorovich operators.

Let . Let us define

It is not difficult to show that

Here we used the identity

The next theorem gives Voronovskaja-type result in compact disks, for complex -Kantorovich operators attached to an analytic function in , and center .

Theorem 1.3. Let .

(i)Let and . For all and one has

(ii)Let and . For all and , one has

Remark 1.4. (i) In the hypothesis on in Theorem 1.3(i) choosing with as , it follows that

uniformly in any compact disk included in the open disk .

(ii) Theorem 1.3(ii) gives explicit formulas of Voronovskaja-type for the -Kantorovich polynomials for .

(iii) Obviously the best order of approximation that can be obtained from the estimate Theorem 1.3(i) is and for , while the order given by Theorem 1.3(ii) is , , which is essentially better.

Next theorem shows that , is continuous about the parameter for , .

Theorem 1.5. Let and . Then for any ,  ,

uniformly on .

As an application of Theorem 1.3, we present the order of approximation for complex -Kantorovich operators.

Theorem 1.6. Let (or ) and . If is not a constant function then the estimate

holds, where the constant depends on and but is independent of .

2. Auxiliary Results

Lemma 2.1. Let . For all , one has

where .

Proof. The recurrence formula can be derived by direct computation.

Lemma 2.2. For all one has

Proof. Indeed, using the inequality (see [3]), we get

Lemma 2.3. For all , , , one has

Proof. We know that (see [2])

Taking the derivative of the formula (2.1) and using the above formula we have

It follows that

Here we used the identity

For define

Here it is assumed that .

Lemma 2.4. Let , .

(a)If , one has the following recurrence formula:

(b)If , one has

Proof. We give the proof for the case . The case is similar to that of .

(b) It is immediate that is a polynomial of degree less than or equal to and that .

Using the formula (2.5), we get

A simple calculation leads us to the following relationship:

which is the desired recurrence formula.

Remark 2.5. Lemmas 2.3 and 2.4 are true in the case . In the formulae, we have to replace -derivative by the ordinary derivative.

3. Proofs of the Main Results

We give proofs for the case . The case and are similar to that of .

Proof of Theorem 1.1. The use of the above recurrence we obtain the following relationship:

We can easily estimate the sum in the above formula as follows:

It is known that by a linear transformation, the Bernstein inequality in the closed unit disk becomes

(where ) which combined with the mean value theorem in complex analysis implies

for all , where is a complex polynomial of degree . From the above recurrence formula (3.1), we get

By writing the last inequality for , we easily obtain, step by step, the following:

Since is analytic in , we can write

which together with (3.6) immediately implies for all

Proof of Theorem 1.3. A simple calculation and the use of the recurrence formula (2.5) lead us to the following relationship:

Firstly, we estimate , . It is clear that

Secondly, using the known inequality

to estimate , , .

Finally, we estimate , . We use [2, Theorem 1.1.2]

Using (3.6), (3.10), (3.12), and (3.13) in (3.9) finally we have ()

As a consequence, we get

This inequality combined with

immediately implies the required estimate in statement.

Note that since and the series is absolutely convergent for all , it easily follows the finiteness of the involved constants in the statement.

Proof of Theorem 1.6. For all and , we get

We apply

to get

Because by hypothesis is not a constant in , it follows . Indeed, assuming the contrary, it follows that for all that is

for all . Thus ,  . Thus, is constant, which is contradiction with the hypothesis.

Now, by Theorem 1.3, we have

Consequently, there exists (depending only on and ) such that for all we have

which implies

For , we have

which finally implies that

for all , with .

Proof of Theorem 1.5. Proof is similar to that of Theorem 1.3 [5].

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