The Impact of the Limit q-Durrmeyer Operator on Continuous Functions

Abstract

The limit q-Durrmeyer operator, \(D_{\infty ,q}\), was introduced and its approximation properties were investigated by Gupta (Appl. Math. Comput. 197(1):172–178, 2008) during a study of q-analogues for the Bernstein–Durrmeyer operator. In the present work, this operator is investigated from a different perspective. More precisely, the growth estimates are derived for the entire functions comprising the range of \(D_{\infty ,q}\). The interrelation between the analytic properties of a function f and the rate of growth for \(D_{\infty ,q}f\) are established, and the sharpness of the obtained results are demonstrated.