On q-analogue of a complex summation-integral type operators in compact disks
1 Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA
2 School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi-110078, India
Journal of Inequalities and Applications 2012, 2012:111 doi:10.1186/1029-242X-2012-111Published: 22 May 2012
Recently, Gupta and Wang introduced certain q-Durrmeyer type operators of real variable x ∈ [0, 1] and studied some approximation results in the case of real variables. Here we extend this study to the complex variable for analytic functions in compact disks. We establish the quantitative Voronovskaja type estimate. In this way, we put in evidence the over convergence phenomenon for these q-Durrmeyer polynomials; namely, the extensions of approximation properties (with quantitative estimates) from the real interval [0,1] to compact disks in the complex plane. Some of these results for q = 1 were recently established in Gupta-Yadav.
Mathematical subject classification (2000): 30E10; 41A25.