Abstract
The main purpose of this paper is to define a new family of Szasz-Mirakyan operators that depends on a non-negative parameter, say alpha. This new family of Szasz-Mirakyan operators is crucial in that it includes both the existing Szasz-Mirakyan operator and allows the construction of new operators for different values of alpha. Then, the convergence properties of the new operators with the aid of the Popoviciu-Bohman-Korovkin theorem-type property are presented. The Voronovskaja-type theorem and rate of convergence are provided in a detailed proof. Furthermore, with the help of the classical modulus of continuity, we deduce an upper bound for the error of the new operator. In addition to these, in order to show that the convex or monotonic functions produced convex or monotonic operators, we obtain shape-preserving properties of the new family of Szasz-Mirakyan operators. The symmetry of the properties of the classical Szasz-Mirakyan operator and of the properties of the new sequence is investigated. Moreover, we compare this operator with its classical correspondence to show that the new one has superior properties. Finally, some numerical illustrative examples are presented to strengthen our theoretical results.