The purpose of this paper is twofold. Firstly, we present an equivalence property involving isotonic linear functionals. Secondly, by using the contraction principle, we give a  method for obtaining the limit of iterates of some classes of linear positive operators.


Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania


Popoviciu-Bohman-Korovkin criterion;Jensen inequality; Jenssen inequality; Banach fixed-point theorem;  Bernstein operators; iterates of an approximation process.

Paper coordinates

O. Agratini, Inequalities and approximation theory, In Inequalities and Applications, eds. Themistocles M. Rassias, Dorin Andrica, pp. 1-12, Cluj University Press, 2008, ISBN 978-973-610-793-1


About this paper

Publisher Name

Cluj University Press

Print ISSN
Online ISSN

google scholar link

[1] Agratini, O, On the iterates of a class of summation-type linear positive operators.  Computers and Mathematics with Applicaitons, 55(2008), no.6, 1178-1180.
[2] Agratini, O., Rus, I.A.,  Iterates of a class of discrete linear operators via contraciton principle,  Comment. Math. Univ. Caroline 44(2003), f.3, 555-563.
[3] Agratini, O., Rus, I.A.,  Iterates of some bivariate approximation processes via weakly Picared operators,  Nonlinear Analysis Forum, 8 (2003), f.2, 159-168.
[4] Andrica, D.,  On a maximum problem,  In: Proc. Colloq. on approx. and Optimiz., Cluj-Napoca, October 25-27, 1984, Univ. of Cluj-Napoca, 1985, 173-177.
[5] Andrieca, D.,  Note on an abstract approximation theorem,  In: Approx. Theory and Appl., M. Rassias Ed., Hadronic Press, Palm Habor, Florida, U.S.A., 1999, 1-10.
[6] Andrica, D., Badea, C.,  Jensen’s and Jessen’s inequality, convesity-preserving and approximating polynomial operators and Korovkin’s theorem,  Babes-Bolyai Univ., Cluj-Napoca, Seminar on Mathematical Analysis, Preprint nr.4(1986), 7-16.
[7] Andrica, D., Rasa, I., The Jensen inequality: refinements and applicaitons,  Anal. Numer.Theor. Approx., 14(1985), 105-108.
[8] Andrica, D., Rasa, I., Toader, Gh., On some inequalities involving convex sequence,  Anal. N umer. Theor. Approx., 13(1984), 5-7.
[9] Beesack, P.R., Pecaric, J.E., On Jenssen’s inequality for convex functions. J. Math. Anal. Appl., 110(1985), 536-552.
[10] Gonska, H.H., Kacso, D., Pitul, P., The degree of convergence of over-iterated positive linear operators, Schriftenreihe des Fachbereichs Mathematik, SM-DU-600, 2005, Universitat Duisburg-Essen, 1-20.
[11] Jenssen, B.,  Bemaerkinger on Konvekse Funcioner og Uligheder imellem Middelvaerdier I, Mat.Tidsskrift B(1931), 17-28.
[12] Karlin, S., Zeigler, Z.,  Iteration of positive approximation operators, J. Approx. Theory, 3(1970), 310=339.
[13] Kelinsky, R.P., Rivlin, T.J.,  Iterates of Bernstein polynomials,  Pacifi J. Math., 21(1967),511-520.
[14] King, J.P., Positive linear operators which preserve x², Acta Mathematica Hungarica, 99(2003), f.3, 203-208.
[15] Lupas, A,  Die Folge der Betaoperatoren, Dissertation, Stuttgart, 1972.
[16] Mitrinovic, D.S.,  Analytic Inequalities,  Berlin-Heidelberg-New York, Springer V erlag, 1970.
[17] Pecare, J. E.,  Note on multidimensional generalization of Slater’s inequality,  J. Approx. Theory, vol. 44(1985), 292-294.
[18] Pecare, J.E., Andrica, D.,  Abstract Jensse’s inequality for convex funcitons and applications,  Mathematica (Cluj), 29(52), 1, 1987, 61-65.
[19] Popoviciu, T.,  Sur l’approximation des fonctions convexes d’ordre superieur,  Mathematica (Cluj), 10 (1935), 49-54.
[20] Popoviciu, T., Asupra demonstrației teoremei lui Weierstrass cu ajutorul polinoamelor de interpolare,  Lucrările Sesiunii Gen. Șt. Acd. Române,2-12 iunie 1950, Editura Academiei Republicii Populare Române, 1951, p. 1664-1667. [Translated in English by Daniela Kacso: On the proof of Weierstrass theorem using interpolation polynomials, East Journal on Approximations, 4(1998), f.1, 107-110].
[21] Roberts, A.W., Varberg, D.E.,  Convex Funcitons, Academic Press., New York and London, 1973.
[22] Rus, I.A., Iterates of Bernstein operators via contraction principle, J. Math. Anal. Appl., 292(2004), 259-261.
[23] Slater, M.I.,  A compasion inequality to Jenssen’s inequality, J. Approx. Theory, 32(1981), 160-166.


Related Posts