Numer Algor DOI 10.1007/s11075-016-0117-8 ORIGINAL PAPER New estimates for the differences of positive linear operators Ana Maria Acu 1 · Ioan Ras ¸a 2 Received: 12 November 2015 / Accepted: 18 February 2016 © Springer Science+Business Media New York 2016 Abstract Using Taylor’s formula some inequalities for a positive linear functional are considered in this paper. These results lead us to new estimates of the differ- ences of certain positive linear operators. Applications for some known positive linear operators are given. Keywords Positive linear operators · Moduli of continuity · Bernstein operators · Genuine Bernstein-Durrmeyer operators · Beta-type operators Mathematics Subject Classification (2010) 41A10 · 41A15 · 41A25 · 41A36 1 Introduction The present note is motivated by the recent results which give a solution to a problem proposed by A. Lupas ¸ in [16]. One of the questions raised by him was to give an estimate for B n ◦ B n − B n ◦ B n =: U n − S n ,  Ana Maria Acu acuana77@yahoo.com Ioan Ras ¸a ioan.rasa@math.utcluj.ro 1 Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, Str. Dr. I. Ratiu, No.5-7, 550012 Sibiu, Romania 2 Faculty of Automation and Computer Science, Department of Mathematics, Technical University of Cluj-Napoca, Str. Memorandumului nr. 28, Cluj-Napoca, Romania