Open Access

Rate of Convergence of a New Type Kantorovich Variant of Bleimann-Butzer-Hahn Operators

Journal of Inequalities and Applications20092009:852897

DOI: 10.1155/2009/852897

Received: 28 September 2009

Accepted: 16 November 2009

Published: 24 November 2009


A new type Kantorovich variant of Bleimann-Butzer-Hahn operator is introduced. Furthermore, the approximation properties of the operators are studied. An estimate on the rate of convergence of the operators for functions of bounded variation is obtained.

1. Introduction

In 1980, Bleimann et al. [1] introduced a sequence of positive linear Bernstein-type operators (abbreviated in the following by BBH operators) defined on the infinite interval by


where denotes the set of natural numbers.

Bleimann et al. [1] proved that as for (the space of all bounded continuous functions on ) and give an estimate on the rate of convergence of measured with the second modulus of continuity of .

In the present paper, we introduce a new type of Kantorovich variant of BBH operator , also defined on by

where , , and is Lebesgue measure.

The operator (1.2) is different from another type of Kantorovich variant of BBH operator :

which was first considered by Abel and Ivan in [2]. The integrand function in the operator (1.3) has been replaced with new integrand function in the operator (1.2). In this paper we will study the approximation properties of for the functions of bounded variation. The rate of convergence for functions of bounded variation was investigated by many authors such as Bojanić and Vuilleumier [3], Chêng [4], Guo and Khan [5], Zeng and Piriou [6], Gupta et al. [7], involving several different operators.

Throughout this paper the class of function is defined as follows:

Our main result can be stated as follows.

Theorem 1.1.

Let and let be the total variation of on interval . Then, for sufficiently large, one has

2. Some Lemmas

In order to prove Theorem 1.1, we need the following lemmas for preparation. Lemma 2.1 is the well-known Berry-Esséen bound for the classical central limit theorem of probability theory. Its proof and further discussion can be founded in Feller [8, page 515].

Lemma 2.1.

Let be a sequence of independent and identically distributed random variables. And , , then, there holds

where , , .

In addition, let be the random variables with two-point distribution
where . Then we can easily obtain that
Let , then we also have
On the other hand, can be written by following integral form:
. It is easy to verify that .

Lemma 2.2.

If is fixed and is sufficiently large, then

(a)for , there holds
(b)for , there holds


We first prove (a). Since , , then . Hence, we have
Direct calculation gives

Hence , for sufficiently large.

The proof of is similar.

Lemma 2.3 (see [9, Theorem  1] or, cf. [10]).

For every , there holds

3. Proof of Theorem 1.1

Let , and , Bojanic-Cheng decomposition yields

where is defined as in (1.6) and
Obviously, . Thus it follows from (3.1) that
First of all, we estimate

where .

Assuming that , for some ( ), then we have

By Lemma 2.3 combining some direct computations, we can easily obtain

Set , then by (2.4) and using Lemma 2.1, we have

Thus, by (3.7), (3.8) we have

Finally, we estimate .

First, interval can be decomposed into four parts as
So can be divided into four parts

where .

Noticing and for , we have .


Next, let .

Now, we recall the Lebesgue-Stieltjes integral representation, and by using partial Lebesgue-Stieltjes integration, we get
An application of (a) in Lemma 2.2 yields
Furthermore, since
we have
Putting in the last integral, we have
It follows from (3.16) and (3.17) that
By a similar method and using Lemma 2.2(b), we obtain

Now, the remainder of our work is to estimate .

For satisfying the growth condition for some positive integer as , we obviously have
Thus, for sufficiently large, there exists a , such that the following inequalities hold:
where . By the definition of the Stirling numbers of the second kind, we readily have
where the Stirling numbers satisfy
Thus we can write
From , , we can easily find . For a fixed , when , we have . Thus there holds
Now using the similar method as that in the proof of Lemma  4 of [11], we deduce that
From (3.21), (3.24), and (3.27), we obtain
Finally, by combining (3.12), (3.18), (3.19), and (3.28), we deduce that

Theorem 1.1 now follows from (3.3), (3.9), and (3.29).



This work is supported by the National Natural Science Foundation of China and the Fujian Provincial Science Foundation of China. The authors thank the associate editor and the referee(s) for their several important comments and suggestions which improve the quality of the paper.

Authors’ Affiliations

Department of Mathematics, Minjiang University
Department of Mathematics, Xiamen University


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© L. Chen and X.-M. Zeng. 2009

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