Shape-preserving and convergence properties for the \(q\)-Szász-Mirakjan operators for fixed \(q \in(0,1)\) (Q1724389)

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