Positivity https://doi.org/10.1007/s11117-018-0608-z Positivity New quantitative results for the convergence of the iterates of some positive linear operators Marius Mihai Birou 1 Received: 18 September 2017 / Accepted: 24 August 2018 © Springer Nature Switzerland AG 2018 Abstract In this paper we obtain quantitative results for the convergence of the iterates of some positive linear operators which preserve certain functions. Some examples involving q-operators are given. We show that the considered q-operators, 0 < q < 1, provide faster convergence for the iterates than those from the classical case (q = 1). Keywords Positive linear operators · q-operators · Convergence of the iterates · The second order modulus of smoothness Mathematics Subject Classification 41A17 · 41A25 · 41A36 1 Introduction The study of the iterates of the positive linear operators has started by Sikkema [22], Kelisky and Rivlin [15] and Karlin and Ziegler [14]. Later, a lot of researchers had contributions in this field (see [1,2,4,5,7,9,10,12,16,17,19,20]). In this article we obtain estimates for the convergence of the iterates of some positive linear operators which preserve some functions. We use the first and the second order modulus of smoothness. For f ∈ C [0, 1] and δ ≥ 0, we have ω 1 ( f ,δ) = sup {| f (x + h ) − f (x )|: x , x + h ∈[0, 1], 0 ≤ h ≤ δ} , ω 2 ( f ,δ) = sup {| f (x + h ) − 2 f (x ) + f (x − h )|: x , x ± h ∈[0, 1], 0 ≤ h ≤ δ} and ‖ f ‖= max{| f (x )|: x ∈[0, 1]}. B Marius Mihai Birou Marius.Birou@math.utcluj.ro 1 Technical University of Cluj-Napoca, Str. Memorandumului No. 28, 400114 Cluj Napoca, Romania 123