- Open Access
Approximation by complex Durrmeyer-Stancu type operators in compact disks
Journal of Inequalities and Applications volume 2013, Article number: 442 (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.
The aim of the present article is to obtain approximation results for complex Durrmeyer-Stancu type operators which are defined for continuous on by
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.
Lemma 1 Let , , , , , then we have that is a polynomial of degree less than or equal to and
Proof By the definition given by (1), the proof is easy, here the proof is omitted.
Let , according to [, Lemma 1], by a simple computation, we have
Lemma 2 Let , , , , , for all , , we have .
Proof The proof follows directly Lemma 1 and [, Lemma 2]. □
Lemma 3 Let , , and , then we have
then we have
By a simple calculation, we obtain
It follows that
Also, using integration by parts, we have
So, in conclusion, we have
which implies the recurrence in the statement. □
Lemma 4 Let , , , , and , we have
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.
Theorem 1 Let , , . Suppose that is analytic in , i.e., for all .
For all and , we have
(Simultaneous approximation) If are arbitrarily fixed, then for all and , we have
where is defined as in the above point (i).
Proof Taking , by the hypothesis that is analytic in , i.e., for all , it is easy for us to obtain
Therefore, we get
For , taking into account that is a polynomial of degree , by the well-known Bernstein inequality and Lemma 2, we get
On the one hand, when , for , by Lemma 1, we have
When , for , , , in view of , using the recurrence formula (3) and the above inequality, we have
By writing the last inequality, for , we easily obtain step by step the following:
In conclusion, for any , , , we have
from which it follows that
By assuming that is analytic in , we have and the series is absolutely convergent in , so we get , which implies .
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. □
Theorem 2 Let , , . Suppose that is analytic in , i.e., for all . For any fixed and all , , we have
where with , , .
Proof For all , we have
By [, Theorem 1], we have , where with .
Next, let us estimate .
By f is analytic in , i.e., for all , we have
On the one hand, when , since , by Lemma 1, we obtain
By the proof of [, Corollary 3], for any , , , we have
Hence, for any , , , we can get
for any , , , we get
On the other hand, when , using Lemma 1 and (see ), by a simple calculation, we can get .
So, for any , , , we have
Hence, we have
where , .
In conclusion, we obtain
In the following theorem, we obtain the exact order of approximation.
Theorem 3 Let , , . Suppose that is analytic in . If f is not a polynomial of degree 0, then for any , we have
where and the constant depends on f, r and α, β, but it is independent of n.
Proof Define and
For all and , we have
In view of the property , it follows
Considering the hypothesis that f is not a polynomial of degree 0 in , we have
Indeed, supposing the contrary, it follows that
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.
Using inequality (4), we get
Therefore, there exists an index , depending only on f, r and α, β, such that for all , we have
For , we have
As a conclusion, we have
this completes the proof. □
Combining Theorem 3 with Theorem 1, we get the following result.
Corollary 1 Let , , . Suppose that is analytic in . If f is not a polynomial of degree 0, then for any , we have
where and the constants in the equivalence depend on f, r and α, β, but they are independent of n.
Theorem 4 Let , , . Suppose that is analytic in . Also, let and be fixed. If f is not a polynomial of degree , then we have
where and the constants in the equivalence depend on f, r, , p, α and β, but they are independent of n.
Proof Taking into account the upper estimate in Theorem 1, it remains to prove the lower estimate only. Denoting by Γ the circle of radius and center 0, by Cauchy’s formula, it follows that for all and , we have
Keeping the notation there for , for all , we have
By using Cauchy’s formula, for all , we get
Passing now to and denoting , it follows
Since for any and we have , so, by inequality (5), we get
In continuation, reasoning exactly as in the proof of Theorem 3, we can get the desired conclusion. □
Lorentz GG (Ed): Bernstein Polynomials. 2nd edition. Chelsea, New York; 1986.
Gal SG: Voronovskaja’s theorem and iterations for complex Bernstein polynomials in compact disks. Mediterr. J. Math. 2008, 5: 253–272.
Anastassiou GA, Gal SG: Approximation by complex Bernstein-Schurer and Kantorovich-Schurer polynomials in compact disks. Comput. Math. Appl. 2009, 58: 734–743.
Gal SG: Approximation by complex Bernstein-Kantorovich and Stancu-Kantorovich polynomials and their iterates in compact disks. Rev. Anal. Numér. Théor. Approx. 2008, 37: 159–168.
Gal SG: Exact orders in simultaneous approximation by complex Bernstein-Stancu polynomials. Rev. Anal. Numér. Théor. Approx. 2008, 37: 47–52.
Gal SG (Ed): Approximation by Complex Bernstein and Convolution Type Operators. World Scientific, Singapore; 2009.
Gal SG: Exact orders in simultaneous approximation by complex Bernstein polynomials. J. Concr. Appl. Math. 2009, 7: 215–220.
Gal SG: Approximation by complex Bernstein-Stancu polynomials in compact disks. Results Math. 2009, 53: 245–256.
Gal SG: Approximation by complex genuine Durrmeyer type polynomials in compact disks. Appl. Math. Comput. 2010, 217: 1913–1920.
Gal SG: Approximation by complex Bernstein-Durrmeyer polynomials with Jacobi weights in compact disks. Math. Balk. 2010, 24: 103–110.
Gal SG, Gupta V: Approximation by a Durrmeyer-type operator in compact disks. Ann. Univ. Ferrara 2011, 57: 261–274.
Gal SG, Gupta V: Approximation by complex beta operators of first kind in strips of compact disks. Mediterr. J. Math. 2013, 10: 31–39.
Gupta V: Approximation properties by Bernstein-Durrmeyer type operators. Complex Anal. Oper. Theory 2013, 7: 363–374.
Gal SG, Gupta V, et al.: Approximation by complex Baskakov-Stancu operators in compact disks. Rend. Circ. Mat. Palermo 2012, 61: 153–165.
Gal SG, Mahmudov NI, Kara M: Approximation by complex q -Szász-Kantorovich operators in compact disks, . Complex Anal. Oper. Theory 2012. 10.1007/s11785-012-0257-3
Mahmudov NI: Approximation properties of complex q -Szász-Mirakjan operators in compact disks. Comput. Math. Appl. 2010, 60: 1784–1791.
Mahmudov NI, Gupta V: Approximation by genuine Durrmeyer-Stancu polynomials in compact disks. Math. Comput. Model. 2012, 55: 278–285.
Ren MY, Zeng XM: Approximation by a kind of complex modified q -Durrmeyer type operators in compact disks. J. Inequal. Appl. 2012., 2012: Article ID 212 10.1186/1029-242X-2012-212
Gupta V, Maheshwari P: Bezier variant of a new Durrmeyer type operators. Riv. Mat. Univ. Parma 2003, 7: 9–21.
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).
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
About this article
Cite this article
Ren, MY., Zeng, XM. & Zeng, L. Approximation by complex Durrmeyer-Stancu type operators in compact disks. J Inequal Appl 2013, 442 (2013). https://doi.org/10.1186/1029-242X-2013-442
- complex Durrmeyer-Stancu type operators
- Voronovskaja-type result
- exact order of approximation
- simultaneous approximation