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Article Citation - WoS: 1Citation - Scopus: 1Shape-Preserving Properties of the Limit q-durrmeyer Operator(Academic Press inc Elsevier Science, 2024) Yilmaz, Ovgu Gurel; Ostrovska, Sofiya; Turan, MehmetThe present work aims to establish the shape-preserving properties of the limit q- Durrmeyer operator, D q for 0 < q < 1. It has been proved that the operator is monotonicity- and convexity-preserving. What is more, it maps a function m - convex along {q (j)}(infinity)(j =0) to a function m - convex along any sequence { xq( j )}(infinity)(j =0) , x is an element of (0, 1). (c) 2024 Elsevier Inc. All rights reserved.Article Citation - WoS: 1Citation - Scopus: 1On the Rate of Convergence for the q-durrmeyer Polynomials in Complex Domains(Walter de Gruyter Gmbh, 2024) Gurel, Ovgu; Ostrovska, Sofiya; Turan, MehmetThe q-Durrmeyer polynomials are one of the popular q-versions of the classical operators of approximation theory. They have been studied from different points of view by a number of researchers. The aim of this work is to estimate the rate of convergence for the sequence of the q-Durrmeyer polynomials in the case 0 < q < 1. It is proved that for any compact set D subset of C, the rate of convergence is O(q(n)) as n -> infinity. The sharpness of the obtained result is demonstrated.