Positivity DOI 10.1007/s11117-016-0442-0 Positivity On a new sequence of positive linear operators related to squared Bernstein polynomials Ioan Gavrea 1 · Mircea Ivan 1 Received: 11 April 2016 / Accepted: 19 August 2016 © Springer International Publishing 2016 Abstract We define a new sequence of positive linear approximation operators by means of the squared Bernstein polynomials and estimate the rate of approximation. Keywords Bernstein polynomials · Approximation order · Rate of approximation Mathematics Subject Classification 41A36 · 41A17 · 42A16 1 Introduction Let us consider the well-known Bernstein polynomials (see, e.g., [2, Chapter 10], [6]) p n,k (x ) =  n k  x k (1 − x ) n−k , n = 0, 1,... k = 0, 1,..., n. There is currently a growing interest in studying the properties of the sums of squared Bernstein polynomials (see, e.g., [3, 4, 7] and many papers in http://arxiv. org/ ). In [3] we give an affirmative simple solution to a conjecture concerning the inequality B Mircea Ivan mircea.ivan@math.utcluj.ro Ioan Gavrea ioan.gavrea@math.utcluj.ro 1 Department of Mathematics, Technical University of Cluj-Napoca, Str. Memorandumului nr. 28, 400114 Cluj-Napoca, Romania