Convex functions of order \(n\) and \(P_n\)-simple functionals


A certain characterization of convex functions of order \(n\) on an interval \(I\) which are \((n+1)\) times differentiable on \(I\), by the use of a \(P_{n}\)-simple functional \(L\), is proved to be in connection with a certain behaviour of the functional \(L\) with respect to the strictly quasiconvex functions of order \(n-1\). In context it is also proved that the necessary and sufficient condition for an \(n\) times continuously differentiable real function \(f\) be strictly quasiconvex of order \(n\ (n\geq 0)\) on \(I\) is that f be convex or concav e or order \(n\) on \(I\), or there exist \(c\in I\) such that \(f\) be concave of order \(n\) on \(I\cap (-\infty,c)\) and convex of order \(n\) on \(I\cap [c,+\infty]\).


Radu Precup
University of Cluj-Napoca, Department of Mathematics, Cluj-Napoca, Romania



Cite this paper as:

R. Precup, Convex functions of order \(n\) and \(P_n\)-simple functionals, Rev. Anal. Numér. Théor. Approx., 18 (1989) no. 2, pp. 161-170.

About this paper


Mathematica-Revue d’analyse numérique et de théorie de l’approximation

Publisher Name

Academia Republicii S.R.

Print ISSN

Not available yet.

Online ISSN

Not available yet.

MR: 92d:41048.

Google Scholar Profile


[1] A. Lupaş, Mean value theorems for the Fourier-Jacobi coefficients. (Romanian) Rev. Anal. Numer. Teoria Aproximaţiei 3 (1974), no. 1, 79-84 (1975), MR0387937.

[2] E. Popoviciu, Teoreme de medie din analiza matematică şi legătura lor cu teoria interpolării, Ed. Dacia, Cluj, 1972.

[3] E. Popoviciu, Sur une allure de quasi-convexite d’ordre supeŕrieur, Math. Rev. Anal. Numér Théor. Approximation, anal. Numér. Théor, Approximation 11 (1982), pp. 129-137.

[4] T. Popoviciu, Deux remarques sur les fonctions convexes. (French) Bull. Sect. Sci. Acad. Roum. 20 (1938), 187-191 (or 45-49) (1939), MR0000418.

[5] T. Popoviciu, Les fonctions convexes, Hermann & Cie, Paris, 1945.

[6] T. Popoviciu, Asupra restului în unele formule liniare de aproximare ale analizei, Stud. Cerc. Mat. (Cluj) 10 (1959), pp. 337-389.

[7] R. Precup, Fonctions convexes et fonctionnelles de forme simple. (French) [Convex functions, and functionals of simple form] Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca, 1988), 269-274, Preprint, 88-6, Univ. “Babeş-Bolyai”, Cluj-Napoca, 1988, MR0993581.

[8] H. T. Wang, Convex functions and Fourier coefficients, Proc. Amer. Math. Soc. 94 (1985), pp. 641-646.

Related Posts