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We introduce a q-generalization of Szász-Mirakjan operators Sn,q and discuss their properties for fixed q∈(0,1). We show that the q-Szász-Mirakjan operators Sn,q have good shape-preserving properties. For example, Sn,q are variation-diminishing, and preserve monotonicity, convexity, and concave modulus of continuity. For fixed q∈(0,1), we prove that the sequence {Sn,qf} converges to B∞,q(f) uniformly on [0,1] for each f∈C[0, 1/(1-q)], where B∞,q is the limit q-Bernstein operator. We obtain the estimates for the rate of convergence for {Sn,qf} by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions

abstract and applied analysis

Shape-Preserving and Convergence Properties for the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math>-Szász-Mirakjan Operators for Fixed<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>q</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>0,1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>

Shape-Preserving and Convergence Properties for theq-Szász-Mirakjan Operators for Fixedq∈(0,1)