Available online at www.sciencedirect.com Fuzzy Sets and Systems 222 (2013) 18 – 32 www.elsevier.com/locate/fss Jensen and Chebyshev inequalities for pseudo-integrals of set-valued functions Mirjana Štrboja a , Tatjana Grbi´ c b , Ivana Štajner-Papuga a , ∗ , Gabrijela Gruji´ c b , Slavica Medi´ c b a Department of Mathematics and Informatics, University of Novi Sad, Serbia b Faculty of Technical Sciences, University of Novi Sad, Serbia Received 13 January 2012; received in revised form 20 July 2012; accepted 22 July 2012 Available online 31 July 2012 Abstract Set-valued functions are an important mathematical notion and play a crucial role in several practical areas. At the same time, pseudo-analysis as a background allows extension of some classical mathematical notions to the forms that are highly applicable in some nonstandard situations. This paper focuses on pseudo-integration of set-valued functions, which is generalization of Aumann’s research, and corresponding extensions of the Jensen and Chebyshev integral inequalities to the set-valued case. Since the integral inequalities in question are widely used in various aspects of mathematics, the main motivation for the presented research lies in the possibility of expanding the applicability of these inequalities by combining the properties of set-valued functions with pseudo-analysis. © 2012 Elsevier B.V. All rights reserved. Keywords: Jensen-type inequality; Chebyshev-type inequality; Pseudo-operations; Pseudo-integral; Set-valued functions 1. Introduction Set-valued functions, besides being an important mathematical notion, have become an essential tool in several practical areas, especially in economic analysis [13]. The integration of set-valued functions has roots in Aumann’s research based on the classical Lebesgue integral [5]. Some generalizations of this approach are based on different types of integrals, such as the ⊥-integral [30], the Choquet integral [12], the Sugeno integral and generalized fuzzy integrals [29] and pseudo-integrals [8,10]. Different integral inequalities, including Hölder, Minkowski, Jensen and Chebyshev inequalities [6], are widely used in various aspects of mathematics, such as in probability theory, differential equations, decision-making under risk and information sciences. The problem of generalization of different classical integral inequalities is a contemporary issue [1–4,7,11,24,27]. The focus of this paper is on extensions of the Jensen and Chebyshev integral inequalities to the set-valued case. The main motivation for this research lies in the possibility of expanding the applicability of these ∗ Corresponding author. Tel.: +381 21 485 2869. E-mail addresses: mirjana.strboja@dmi.uns.ac.rs (M. Štrboja), tatjana@uns.ac.rs (T. Grbi´ c), ivana.stajner-papuga@dmi.uns.ac.rs (I. Štajner-Papuga), gabrijela@neobee.net (G. Gruji´ c), slavicam@uns.ac.rs (S. Medi´ c). 0165-0114/$-see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.07.011