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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.

##### MSC:
 26A51 Convexity of real functions in one variable, generalizations
##### Keywords:
$$s$$-convex functions; superposition; composition; product
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##### References:
 [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
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