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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
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References:
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