Jain, P.; Persson, L. E.; Upreti, P. Inequalites and properties of some generalized Orlicz classes and spaces. (English) Zbl 1164.26016 Acta Math. Hung. 117, No. 1-2, 161-174 (2007). The generalized Orlicz classes \(\widetilde{X}_{\Phi}\) and generalized Orlicz spaces \(X_{\Phi}\) are defined by replacing the space \(L^{1}\) in the classical construction of Orlicz classes \(\widetilde{L}_{\Phi }\) and Orlicz spaces \(L_{\Phi }\) by an arbitrary Banach function space \(X.\) The authors prove some results which are important for the study of inequalities in these spaces. In particular, they prove some natural generalizations of the classical Minkowski’s inequality, Hölder’s inequality and Young’s inequality. Reviewer: Gheorghe Toader (Cluj-Napoca) Cited in 6 Documents MSC: 26D10 Inequalities involving derivatives and differential and integral operators 26D15 Inequalities for sums, series and integrals 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:inequalites; function spaces; Orlicz class; Orlicz space; Luxemburg norm; Banach function space; Young function; Young’s inequality; Minkowski’s inequality; Hölder’s inequality PDFBibTeX XMLCite \textit{P. Jain} et al., Acta Math. Hung. 117, No. 1--2, 161--174 (2007; Zbl 1164.26016) Full Text: DOI References: [1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press (London, 1988). · Zbl 0647.46057 [2] M. A. Krasnosel’skii and J. B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff Ltd. (Groningen, 1961). [3] S. G. Krein, Ju. I. Petunin and E. M. Semenov, Interpolation of Linear Operators, A.M.S. (Providence, 1982). [4] S. G. Krein, Ju. I. Petunin and E. M. Semenov, Scales of Banach lattices of measurable functions, Trudy Moskov Mat. Obsc., 17 (1967), 293–322 (in Russian). · Zbl 0193.09302 [5] A. Kufner, J. Oldrich and F. Svatopluk, Function Spaces, Noordhoff International Publishing (Leydon, 1977). [6] G. Ju. Lozanovskii, On reflexive KB-spaces, Dokl. Akad. Nauk SSSR, 158 (1964), 516–519 (in Russian). [7] W. A. J. Luxemburg, Banach Function Spaces, Ph.D. Thesis, Technische Hogeschool te Delft (1955). · Zbl 0068.09204 [8] L. Maligranda and L. E. Persson, Generalized duality of some Banach function spaces, Proc. Konin. Nederlands Akad. Wet., 92 (1989), 323–338. · Zbl 0704.46018 [9] L. E. Persson, Some elementary inequalities in connection with X p-spaces, in: Constructive Theory of Functions (1987), pp. 367–376. [10] L. E. Persson, On some generalized Orlicz classes and spaces, Research Report 1988–3, Department of Mathematics, Lulea University of Technology (1988). [11] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker Inc. (New York, Basel, Hong Kong, 1991). · Zbl 0724.46032 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.