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An equivalent condition to the Jensen inequality for the generalized Sugeno integral
 Mohsen Jaddi^{1},
 Ali Ebadian^{1},
 Manuel de la Sen^{2} and
 Sadegh Abbaszadeh^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s1366001715612
© The Author(s) 2017
 Received: 28 August 2017
 Accepted: 9 November 2017
 Published: 14 November 2017
Abstract
Keywords
 generalized Sugeno integral
 convex function
 the Jensen inequality
MSC
 26D15
 28A25
1 Introduction
The classical Jensen inequality is one of the interesting inequalities in the theory of differential and difference equations, as well as other areas of mathematics. The wellknown Jensen inequality for a convex function is given as follows:
The concepts of fuzzy measures and the Sugeno integral were introduced and initially examined by Sugeno [6]. Further theoretical investigations of these concepts and their generalizations have been pursued by many researchers. Among them, Ralescu and Adams [7] provided several equivalent definitions of the Sugeno integral and proved a monotone convergence theorem for the Sugeno integral; RománFlores et al. [8, 9] discussed levelcontinuity of the Sugeno integral and Hcontinuity of fuzzy measures, while Wang and Klir [10] presented an excellent general overview on fuzzy measurement and fuzzy integration theory. The Sugeno integral has also been successfully applied to various fields (see, e.g., [11–14]).
The study of inequalities for the Sugeno integral was initiated by RománFlores et al. [15]. Since then, fuzzy integral counterparts of several classical inequalities, including the Chebyshev, Jensen, Minkowski, Hadamard and Hölder inequalities, have been presented (see [1–3, 16–18]).
Kaluszka et al. [2] studied the Jensen inequality (1) for the generalized Sugeno integral by using the condition of monotonicity instead of the condition of convexity. The aim of this paper is to study the Jensen inequality for the generalized Sugeno integral without losing the condition of convexity.
The paper is organized as follows. Some basic definitions and summarizations of previous results are given in Section 2. In Section 3, the Jensen inequality for the generalized Sugeno integral is studied. In Section 4, the necessary and sufficient conditions for the validity of the discrete form of the Jensen inequality is presented. A conclusion is given in Section 5.
2 Preliminaries
In this section, some definitions and basic properties of the Sugeno integral which will be used in the next section are presented.
Definition 2.1
 (FM1):

\(\mu(\varnothing )=0 \);
 (FM2):

\(E,F\in\sum\) and \(E \subseteq F\) imply \(\mu(E)\leq\mu (F)\) (monotonicity);
 (FM3):

\(E_{n} \in\Sigma\) (\(n \in\mathbb{N}\)), \(E_{1} \subset E_{2} \subset\cdots\) , imply \(\lim_{n \rightarrow\infty} \mu(E_{n})= \mu(\bigcup_{n= 1}^{\infty} E_{n})\) (continuity from below);
 (FM4):

\(E_{n} \in\Sigma\) (\(n \in\mathbb{N}\)), \(E_{1} \supset E_{2} \supset\cdots\) , \(\mu(E_{1}) < \infty\), imply \(\lim_{n \rightarrow\infty} \mu(E_{n})= \mu(\bigcap_{n= 1}^{\infty} E_{n})\) (continuity from above).
Let \((X,\Sigma,\mu)\) be a fuzzy measure space and f be a nonnegative realvalued function on X. We denote by \(\mathcal{F}_{+}\) the set of all nonnegative measurable functions and by \(L_{\alpha}f\) the set \(\{x\in X \mid f(x)\geq\alpha\}\), the αlevel of f for \(\alpha\geq0\).
Definition 2.2
 (i)The Shilkret integral [20] of f on A with respect to the fuzzy measure μ is given by$$(Sh) \int_{A} f \,d\mu:=\sup_{\alpha\geq0} \bigl\{ \alpha \cdot\mu (A \cap L_{\alpha}f ) \bigr\} ; $$
 (ii)The Sugeno integral [6] of f on A with respect to the fuzzy measure μ is defined by where ∨ and ∧ denote the operations sup and inf on \([0,\infty[\), respectively.
The following theorem gives most elementary properties of the Sugeno integral and can be found in [19, 21].
Theorem 2.3
 1.
;
 2.
for a nonnegative constant k;
 3.
If \(f \leq g\) on A, then ;
 4.
If \(A \subset B\), then ;
 5.
;
 6.
If \(\mu(A) < \infty\), then .
Remark 2.4
Remark 2.5
Let \(Y\subset\mathbb{R}\) be an arbitrary nonempty interval (bounded or unbounded). Throughout this paper, \(Y=[0,1]\) or \(Y=[0,\infty[\). Also, we denote the range of μ by \(\mu(\Sigma)\).
Definition 2.6
(Generalized Sugeno integral [2])
Theorem 2.7
(Muresan [22])
Suppose that \(f : I\rightarrow\mathbb{R} \) is a convex function. Then \(S_{f},{} _{x_{0}} (x)\) is increasing on \(I\setminus\{x _{0}\} \).
Theorem 2.8
(Muresan [22])
Theorem 2.9
(Mitrinović [23])
Suppose that \(f : [0 , +\infty[\rightarrow\mathbb{R} \) is a convex function. If f is a nondecreasing and continuous function on \([0 , +\infty[\) with \(f(0)=0\) and \(\lim_{x\rightarrow+\infty} f(x)= +\infty\), then \(f^{1}\) exists and has the same characteristics as f.
We say that the operator \(\circ: Y \times Y \rightarrow Y \) is nondecreasing if \(a \circ c \geq b \circ d \) for \(a \geq b \) and \(c \geq d\).
Definition 2.10
 (\(\mathrm{T}_{1}\)):

\(T(x, 1)= T(1, x)= x\) for any \(x \in[0, 1]\);
 (\(\mathrm{T}_{2}\)):

T is increasing;
 (\(\mathrm{T}_{3}\)):

\(T(x, y)= T(y, x)\) for any \(x, y \in[0, 1]\);
 (\(\mathrm{T}_{4}\)):

\(T (T(x, y), z )= T (x, T(y ,z) )\) for any \(x, y, z \in[0, 1]\).
Example 2.11
 1.
\(M (x, y)= \min\{x, y\}\).
 2.
\(\sqcap(x, y)= x \cdot y\).
 3.
\(O_{L} (x, y)= \max\{x+ y 1, 0\}\).
3 Results and discussion
The aim of this section is to characterize the Jensen inequality for the generalized Sugeno integral when f is a convex function. Throughout this section, let \((X, \Sigma, \mu)\) be a fuzzy measure space.
Theorem 3.1
 1.
\(a \star0 = a \circ0 = 0\);
 2.
\(H (0 ) = 0\) and \(H ^{\prime}(y)\geq1 \) for all \(y \in Y\);
 3.
\(\int_{A} f\star\mu\in Y \) for an arbitrary set \(A \in\Sigma \) and a measurable function \(f:X \rightarrow Y\).
Proof
Corollary 3.2
Proof
Corollary 3.3
Proof
Corollary 3.4
Proof
In the next theorems, the sufficient and necessary conditions for the reverse of inequality (5) are given. The proofs are similar to the proof of Theorem 3.1 and are omitted.
Theorem 3.5

For all \(y \in Y \) and \(b \in\mu(\Sigma)\), \(H(y)\circ b \leq H (y \star b )\).
Theorem 3.6
Let \(H : Y\rightarrow Y \) be a differentiable concave function such that \(H (Y )= Y\), \(H (0 ) = 0\) and \(H ^{\prime}(y)\leq1 \) for all \(y \in Y\). Let \(a \star0 = a \circ0 = 0\) for any \(a \in Y\), and the functions \(y \longmapsto y \circ b\) and \(y \longmapsto y \star b\) are nondecreasing. For an arbitrary set \(A \in\Sigma\) and a measurable function \(f : X \rightarrow Y \) such that \(\int_{A} f \star\mu\in Y\), the Jensen inequality \(\int_{A} H (f)\circ\mu\leq H(\int_{A} f\star\mu) \) is sharp iff \(H(y)\circ b \leq H (y \star b )\) for any \(y \in Y \) and \(b \in\mu(\Sigma)\).
4 Generalized Sugeno integral and discrete Jensen inequality
In this section, we deal with the discrete Jensen inequality for the generalized Sugeno integral of convex functions.
Theorem 4.1
 1.
\(a \star0 = a \circ0 = 0\),
 2.
\(\int_{A} f\star\mu\in Y\),
Proof
5 Conclusion
The classical Jensen inequality is one of the most important results for convex (concave) functions defined on an interval with a natural geometrical interpretation. In order to obtain a characterization for the classical Jensen inequality for the generalized Sugeno integral, it is clear that the classical conditions must be changed. Previously, some equivalent conditions have been obtained by losing the basic condition of convexity and replacing this condition with monotonicity (see [2]). This paper studied the Jensen inequality for the generalized Sugeno integral by maintaining the condition of convexity. Also, the necessary and sufficient conditions for the validity of the discrete form of the Jensen inequality for the generalized Sugeno integral were investigated. For further investigations of integral inequalities in the area of the generalized Sugeno integral and their applications in other sciences, the results of this paper will be useful and effective.
Declarations
Availability of data and materials
Not applicable.
Funding
The third author is grateful to UPV /EHU for Grant PGC 17/33.
The fourth author is very grateful to the Iranian National Science Foundation for its support of this research through Grant No. 95004084.
Authors’ contributions
All the authors conceived of the study, participated in its design and read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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