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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 563613, 8 pages
http://dx.doi.org/10.1155/2014/563613
Research Article

Shape-Preserving and Convergence Properties for the -Szász-Mirakjan Operators for Fixed

1School of Mathematical Sciences, BCMIIS, Capital Normal University, Beijing 100048, China
2Jia Huiming Educational Technology Co. (Ltd.), Beijing 100068, China
3School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received 22 February 2014; Accepted 14 April 2014; Published 6 May 2014

Academic Editor: Sofiya Ostrovska

Copyright © 2014 Heping Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a -generalization of Szász-Mirakjan operators and discuss their properties for fixed . We show that the -Szász-Mirakjan operators have good shape-preserving properties. For example, are variation-diminishing, and preserve monotonicity, convexity, and concave modulus of continuity. For fixed , we prove that the sequence converges to uniformly on for each , where is the limit -Bernstein operator. We obtain the estimates for the rate of convergence for by the modulus of continuity of , and the estimates are sharp in the sense of order for Lipschitz continuous functions.