# American Institute of Mathematical Sciences

doi: 10.3934/mfc.2021043
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## On complex modified Bernstein-Stancu operators

 Ankara Hacı Bayram Veli University, Polatlı Faculty of Science and Letters, Department of Mathematics, 06900, Ankara, Turkey

Received  August 2021 Revised  November 2021 Early access January 2022

The present paper deals with complex form of a generalization of perturbed Bernstein-type operators. Quantitative upper estimates for simultaneous approximation, a qualitative Voronovskaja type result and the exact order of approximation by these operators attached to functions analytic in a disk centered at the origin with radius greater than 1 are obtained in this study.

Citation: Nursel Çetin. On complex modified Bernstein-Stancu operators. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021043
##### References:
 [1] A. M. Acu and P. N. Agrawal, Better approximation of functions by genuine Bernstein-Durrmeyer type operators, Carpathian J. Math., 35 (2019), 125-136.  doi: 10.37193/CJM.2019.02.01. [2] A. M. Acu and G. Başcanbaz-Tunca, Approximation by complex perturbed Bernstein-type operators, Results Math., 75 (2020), Paper No. 120, 16 pp. doi: 10.1007/s00025-020-01244-x. [3] A. M. Acu, G. Başcanbaz-Tunca and N. Çetin, Approximation by certain linking operators, Ann. Funct. Anal., 11 (2020), 1184-1202.  doi: 10.1007/s43034-020-00081-x. [4] A. M. Acu, V. Gupta and G. Tachev, Better numerical approximation by Durrmeyer type operators, Results Math., 74 (2019), Paper No. 90, 24 pp. doi: 10.1007/s00025-019-1019-6. [5] S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Commun. Kharkov Math. Soc., 13 (1912/1913), 1-2. [6] N. Çetin, Approximation and geometric properties of complex $\alpha -$Bernstein operator, Results Math., 74 (2019), Paper No. 40, 16 pp. doi: 10.1007/s00025-018-0953-z. [7] N. Çetin, A new generalization of complex Stancu operators, Math. Methods Appl. Sci., 42 (2019), 5582-5594.  doi: 10.1002/mma.5622. [8] N. Çetin and G. Başcanbaz-Tunca, Approximation by a new complex generalized Bernstein operators, An. Univ. Oradea Fasc. Mat., 26 (2019), 127-139. [9] R. DeVore and G. G. Lorentz, Constructive Approximation, Springer, Berlin, 1993. [10] S. G. Gal, Approximation by Complex Bernstein and Convolution Type Operators, Series on Concrete and Applicable Mathematics, vol. 8, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009. doi: 10.1142/9789814282437. [11] V. Gupta, G. Tachev and A. M. Acu, Modified Kantorovich operators with better approximation properties, Numer. Algorithms, 81 (2019), 125-149.  doi: 10.1007/s11075-018-0538-7. [12] H. Khosravian-Arab, M. Dehghan and M. R. Eslahchi, A new approach to improve the order of approximation of the Bernstein operators: Theory and applications, Numer. Algorithms, 77 (2018), 111-150.  doi: 10.1007/s11075-017-0307-z. [13] G. G. Lorentz, Bernstein Polynomials, 2$^{nd}$ edition, Chelsea Publ, New York, 1986. [14] D. D. Stancu, Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20 (1983), 211-229.  doi: 10.1007/BF02575593.

show all references

##### References:
 [1] A. M. Acu and P. N. Agrawal, Better approximation of functions by genuine Bernstein-Durrmeyer type operators, Carpathian J. Math., 35 (2019), 125-136.  doi: 10.37193/CJM.2019.02.01. [2] A. M. Acu and G. Başcanbaz-Tunca, Approximation by complex perturbed Bernstein-type operators, Results Math., 75 (2020), Paper No. 120, 16 pp. doi: 10.1007/s00025-020-01244-x. [3] A. M. Acu, G. Başcanbaz-Tunca and N. Çetin, Approximation by certain linking operators, Ann. Funct. Anal., 11 (2020), 1184-1202.  doi: 10.1007/s43034-020-00081-x. [4] A. M. Acu, V. Gupta and G. Tachev, Better numerical approximation by Durrmeyer type operators, Results Math., 74 (2019), Paper No. 90, 24 pp. doi: 10.1007/s00025-019-1019-6. [5] S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Commun. Kharkov Math. Soc., 13 (1912/1913), 1-2. [6] N. Çetin, Approximation and geometric properties of complex $\alpha -$Bernstein operator, Results Math., 74 (2019), Paper No. 40, 16 pp. doi: 10.1007/s00025-018-0953-z. [7] N. Çetin, A new generalization of complex Stancu operators, Math. Methods Appl. Sci., 42 (2019), 5582-5594.  doi: 10.1002/mma.5622. [8] N. Çetin and G. Başcanbaz-Tunca, Approximation by a new complex generalized Bernstein operators, An. Univ. Oradea Fasc. Mat., 26 (2019), 127-139. [9] R. DeVore and G. G. Lorentz, Constructive Approximation, Springer, Berlin, 1993. [10] S. G. Gal, Approximation by Complex Bernstein and Convolution Type Operators, Series on Concrete and Applicable Mathematics, vol. 8, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009. doi: 10.1142/9789814282437. [11] V. Gupta, G. Tachev and A. M. Acu, Modified Kantorovich operators with better approximation properties, Numer. Algorithms, 81 (2019), 125-149.  doi: 10.1007/s11075-018-0538-7. [12] H. Khosravian-Arab, M. Dehghan and M. R. Eslahchi, A new approach to improve the order of approximation of the Bernstein operators: Theory and applications, Numer. Algorithms, 77 (2018), 111-150.  doi: 10.1007/s11075-017-0307-z. [13] G. G. Lorentz, Bernstein Polynomials, 2$^{nd}$ edition, Chelsea Publ, New York, 1986. [14] D. D. Stancu, Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20 (1983), 211-229.  doi: 10.1007/BF02575593.
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