Approximation by complex Durrmeyer-Stancu type operators in compact disks
© Ren et al.; licensee Springer. 2013
Received: 29 April 2013
Accepted: 23 August 2013
Published: 23 September 2013
In this paper we introduce a class of complex Stancu-type Durrmeyer operators and study the approximation properties of these operators. We obtain a Voronovskaja-type result with quantitative estimate for these operators attached to analytic functions on compact disks. We also study the exact order of approximation. More important, our results show the overconvergence phenomenon for these complex operators.
MSC:30E10, 41A25, 41A36.
In 1986, some approximation properties of complex Bernstein polynomials in compact disks were initially studied by Lorentz . Very recently, the problem of the approximation of complex operators has been causing great concern, which has become a hot topic of research. A Voronovskaja-type result with quantitative estimate for complex Bernstein polynomials in compact disks was obtained by Gal  Also, in [3–18] similar results for complex Bernstein-Kantorovich polynomials, Bernstein-Stancu polynomials, Kantorovich-Schurer polynomials, Kantorovich-Stancu polynomials, complex Favard-Szász-Mirakjan operators, complex Beta operators of first kind, complex Baskajov-Stancu operators, complex Bernstein-Durrmeyer polynomials, complex genuine Durrmeyer-Stancu polynomials and complex Bernstein-Durrmeyer operators based on Jacobi weights were obtained.
where α, β are two given real parameters satisfying the condition , , , and .
Note that, for , these operators become the complex Durrmeyer-type operators , this case has been investigated in .
2 Auxiliary results
In the sequel, we shall need the following auxiliary results.
Proof By the definition given by (1), the proof is easy, here the proof is omitted.
Lemma 2 Let , , , , , for all , , we have .
Proof The proof follows directly Lemma 1 and [, Lemma 2]. □
which implies the recurrence in the statement. □
Proof Using the recurrence formula (2), by a simple calculation, we can easily get the recurrence (3), the proof is omitted. □
3 Main results
The first main result is expressed by the following upper estimates.
- (i)For all and , we have
- (ii)(Simultaneous approximation) If are arbitrarily fixed, then for all and , we have
where is defined as in the above point (i).
- (i)For , taking into account that is a polynomial of degree , by the well-known Bernstein inequality and Lemma 2, we get
- (ii)For the simultaneous approximation, denoting by Γ the circle of radius and center 0, since for any and , we have . By Cauchy’s formula, it follows that for all and , we have
which proves the theorem. □
where with , , .
By [, Theorem 1], we have , where with .
Next, let us estimate .
On the other hand, when , using Lemma 1 and (see ), by a simple calculation, we can get .
where , .
In the following theorem, we obtain the exact order of approximation.
where and the constant depends on f, r and α, β, but it is independent of n.
Defining and looking for the analytic function under the form , after replacement in the differential equation, the coefficients identification method immediately leads to for all . This implies that for all and therefore f is constant on , a contradiction with the hypothesis.
this completes the proof. □
Combining Theorem 3 with Theorem 1, we get the following result.
where and the constants in the equivalence depend on f, r and α, β, but they are independent of n.
where and the constants in the equivalence depend on f, r, , p, α and β, but they are independent of n.
In continuation, reasoning exactly as in the proof of Theorem 3, we can get the desired conclusion. □
The authors are most grateful to the editor and anonymous referee for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper. This work is supported by the National Natural Science Foundation of China (Grant no. 61170324), the Class A Science and Technology Project of Education Department of Fujian Province of China (Grant no. JA12324), and the Natural Science Foundation of Fujian Province of China (Grant no. 2013J01017).
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