Some remarks on \(s\)-convex functions. (English) Zbl 0823.26004

The authors deal with two classes \(K^ 1_ s\) and \(K^ 2_ s\) of \(s\)- convex functions on \(\mathbb{R}_ +\). These classes have been introduced by W. Orlicz [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 9, 157-162 (1961; Zbl 0109.334)] and the reviewer [Publ. Inst. Math., Nouv. Sér. 23(37), 13-20 (1978; Zbl 0416.46029)], respectively. In the first part of the paper they state necessary conditions for functions to be in the classes \(K^ 1_ s\) or \(K^ 2_ s\). Besides, here they prove a theorem about the superposition of functions belonging to the class \(K^ 1_ s\). From this theorem it follows that if \(f, g\in K^ 1_ s\), then \(f+ g\) and \(\max(f, g)\) are also in \(K^ 1_ s\). Moreover, it is shown that any \(f\in K^ 2_ s\) satisfying \(f(0)= 0\) lies in \(K^ 1_ s\), and that both classes \(K^ 1_ s\) and \(K^ 2_ s\) increase if \(s\) decreases. In the second part of the paper, non- negative \(s\)-convex functions are considered. The main result given here refers to the composition and the product of two functions \(f\in K^ 1_{s_ 1}\) and \(g\in K^ 1_{s_ 2}\). Both parts of the paper are completed by various examples and counterexamples.


26A51 Convexity of real functions in one variable, generalizations
Full Text: DOI EuDML


[1] Breckner, W. W.,Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen. Publ. Inst. Math.23 (1978), 13–20. · Zbl 0416.46029
[2] Matuszewska, W. andOrlicz, W.,A note on the theory of s-normed spaces of -integrable functions. Studia Math.21 (1961), 107–115. · Zbl 0202.39903
[3] Musielak, J.,Orlicz spaces and modular spaces. [Lecture Notes in Math. Vol. 1034]. Springer-Verlag, Berlin, 1983. · Zbl 0557.46020
[4] Orlicz, W.,A note on modular spaces. I. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys.9 (1961), 157–162. · Zbl 0109.33404
[5] Rolewicz, S.,Metric linear spaces, 2nd Ed. PWN, Warszawa, 1984. · Zbl 0526.49018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.