The q-derivative and applications to q-Szasz Mirakyan operators

Abstract

By using the properties of the q-derivative, we show that q-Szasz Mirakyan operators are convex, if the function involved is convex, generalizing well-known results for q = 1. We also show that q-derivatives of these operators converge to q-derivatives of approximated functions. Futhermore, we give a Voronovskaya-type theorem for monomials and provide a Stancu-type form for the remainder of the q-Szasz Mirakyan operator. Lastly, we give an inequality for a convex function f, involving a connection between two nonconsecutive terms of a sequence of q-Szasz Mirakyan operators.