Some remarks on \(s\)-convex functions
type: Item
instance of: scholarly article
title: Some remarks on \(s\)-convex functions
zbMATH Open document ID: 0823.26004
DOI: 10.1007/BF01837981
published in: Aequationes Mathematicae
publication date: 1994-09-08T00:00:00Z
full work available at URL: https://eudml.org/doc/137594
review text: 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 \textit{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.
Mathematics Subject Classification ID: 26A51
zbMATH DE Number: 636045
zbMATH Keywords: \(s\)-convex functions
zbMATH Keywords: superposition
zbMATH Keywords: composition
zbMATH Keywords: product
author: Henryk Hudzik
author: Lech Maligranda
reviewed by: Wolfgang W. Breckner
MaRDI profile type: Publication
cites work: Q3205805
cites work: A note on the theory of s-normed spaces of φ-integrable functions
cites work: Orlicz spaces and modular spaces
cites work: Q3847669
cites work: Sufficient condition for Pareto optimization in Banach spaces
OpenAlex ID: W2322767775
Recommended article: Remarks on some inequalities for \(s\)-convex functions and applications
Recommended article: Q4697388
Recommended article: Remarks on \((F, t)\)-convex functions
Recommended article: On some inequalities for s-convex functions and applications
Recommended article: Q4295486
Recommended article: $s$-Convex functions in the third sense
Recommended article: Q3774099
Recommended article: Q3222425
Recommended article: Q3629117
Recommended article: A Note on roughly convex functions
description: scientific article; zbMATH DE number 636045