A New Estimate on the Rate of Convergence of Durrmeyer-Bézier Operators
© P.Wang and Y. Zhou. 2009
Received: 20 February 2009
Accepted: 13 April 2009
Published: 5 May 2009
We obtain an estimate on the rate of convergence of Durrmeyer-Bézier operaters for functions of bounded variation by means of some probabilistic methods and inequality techniques. Our estimate improves the result of Zeng and Chen (2000).
In 2000, Zeng and Chen  introduced the Durrmeyer-Bézier operators which are defined as follows:
Concerning the approximation properties of operators and some results on approximation of functions of bounded variation by positive linear operators, one can refer to [2–7]. Authors of  studied the rate of convergence of the operators for functions of bounded variation and presented the following important result.
Since the Durrmeyer-Bézier operators are an important approximation operator of new type, the purpose of this paper is to continue studying the approximation properties of the operators for functions of bounded variation, and give a better estimate than that of Theorem A by means of some probabilistic methods and inequality techniques. The result of this paper is as follows.
It is obvious that the estimate (1.5) is better than the estimate (1.3). More important, the estimate (1.5) is true for all . This is an important improvement comparing with the fact that estimate (1.3) holds only for .
2. Some Lemmas
In order to prove Theorem 1.1, we need the following preliminary results.
The inequality (2.1) is proved.
we get the inequality (2.2). Lemma 2.1 is proved.
Thus Lemma 2.2 is proved.
3. Proof of Theorem 1.1
We first estimate , from [1, page 11] we have the following equation:
Next we estimate . From (15) of , it follows the inequality
From (3.7) and (3.8) we obtain
Thus from (3.9) we obtain
Theorem 1.1 now follows by collecting the estimations (3.3), (3.5), and (3.12).
The present work is supported by Project 2007J0188 of Fujian Provincial Science Foundation of China.
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