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

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https://ictp.acad.ro/jnaat/journal/article/view/1989-vol18-no2-art7/Precup

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

MR: 92d:41048.

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