ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 26 Hejcman, Jan, University of Education, Hradec Králové, Czech Republic A description of quasiconvex functions A real - valued function f is said to he quasiconvex on an interval J ⊂ ℝ if f(tx + (1 − t)y) ≤ max{f(x), f(y)} for each x, y ∈ J, 0 ≤ t ≤ 1. A function f is quasiconcave if −f is quasiconvex. Every convex function is quasiconvex. A function is both quasiconvex and quasiconcave if and only if it is monotone. Theorem. Let a function f be not monotone on an interval J. Then f is quasiconvex on J if and only if there exist disjoint intervals J_{1}, J_{2} such that J_{1} ∩ J_{2} = J, (x ∈ J_{1} ∧ y ∈ J_{2}) ⇒ x < y and f is decreasing on J_{1} and increasing on J_{2}.