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An equivalent condition to the Jensen inequality for the generalized Sugeno integral
Journal of Inequalities and Applications volume 2017, Article number: 285 (2017)
Abstract
For the classical Jensen inequality of convex functions, i.e.,
an equivalent condition is proved in the framework of the generalized Sugeno integral. Also, the necessary and sufficient conditions for the validity of the discrete form of the Jensen inequality for the generalized Sugeno integral are given.
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:
Let \((X,\Sigma,\mu)\) be a measure space, f be a realvalued μmeasurable and μintegrable function on a set \(D\in\Sigma\) with \(\mu(D)\in{]}0,\infty[ \). If φ is a convex function on an open interval I in \(\mathbb{R}\) and if \(f(D)\subset I\), then
In recent years, there have been many extensions, refinements and similar results of the classical Jensen inequality, see [1–5].
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
Let Σ be a σalgebra of subsets of X, and let \(\mu: \Sigma\rightarrow[0,\infty[\) be a nonnegative extended realvalued set function. We say that μ is a fuzzy measure iff:
 (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
Let \((X,\Sigma,\mu)\) be a fuzzy measure space. If \(f \in\mathcal {F}_{+}\) and \(A\in\Sigma\), then

(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
(2)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
Let \((X, \Sigma, \mu)\) be a fuzzy measure space with \(A, B \in\Sigma\) and \(f, g \in\mathcal{F}_{+}\). Then

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
Let \(F(\alpha)=\mu(A \cap L_{\alpha}f)\), from parts (5) and (6) of the above theorem, it is very important to note that
Thus, from a numerical point of view, the Sugeno integral can be calculated by solving the equation \(F(\alpha)=\alpha\).
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])
For a μmeasurable function \(f:X\rightarrow Y\), we define the generalized Sugeno integral of f on a set \(A\in\Sigma\) with respect to μ and an operator \(\Delta:Y\times\mu(\Sigma)\rightarrow Y \) by
Let I be a real interval and \(f : I\rightarrow\mathbb{R} \) be a function. Then f is said to be convex (on I) provided
Also,
is said to be the slope of f at \(x_{0}\).
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])
Suppose that \(f : I\rightarrow\mathbb{R}\) is differentiable on I. Then f is convex if and only if
for any \(x, x _{0} \in I \).
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
A triangular norm is a function \(T: [0, 1] \times[0, 1] \longrightarrow[0, 1]\) satisfying the following conditions:
 (\(\mathrm{T}_{1}\)):

\(T(x, 1)= T(1, x)= x\) 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
The following operators are tnorm:

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
For the classical measure μ, the classical Jensen inequality is the following strong property of convex functions (see [26]):
where f is μmeasurable and \(H: [0, \infty[ \longrightarrow[0, \infty[\) is a convex function. The following inequality is known as the discrete Jensen inequality:
where \(H: [a, b] \longrightarrow[0, \infty[\) is a convex function, \(x_{i} \in[a, b]\) and \(\lambda_{i}>0\) for all \(i \in\{1, \ldots, n\}\) and \(\sum_{i=1}^{n}\lambda_{i}= 1\).
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
Assume that \(H : Y\rightarrow Y \) is a differentiable convex function and \(\circ,\star: Y \times\mu(\Sigma)\rightarrow Y \) are nondecreasing operators satisfying the following conditions:

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 \(f:X \rightarrow Y\).
Then the Jensen inequality
is sharp if and only if, for any \(y \in Y \) and \(b \in\mu(\Sigma)\),
Proof
Sufficiency. Let \(y\in Y\). Since H is a differentiable convex function, by Theorem 2.8 we have
and by assumption (2),
for all \(t \in Y\). So we have
for all \(x \in X\). Therefore, by the monotonicity of μ, we deduce
for an arbitrary set \(A \in\Sigma\). On the other hand, since \(H(Y)\subseteq Y\), we have
Combining (7) with (8) and using the monotonicity of ∘, we get
Now, we define \(h(y ):= y\star\mu (A \cap L_{y}f ) \) for each \(y \in Y\). From (9) we have
By the assumption, \(h (y )\in Y \) and \(\sup_{ y \in Y } h (y ) : = \int_{A} f\star\mu\in Y\).
Since H is continuous and \(H (0 ) = 0\), then the function \(S_{H} ,{} _{0}(t)\) (i.e., the function \(S_{H} ,{} _{0}: H \setminus\{ 0\} \rightarrow\mathbb{R}\); \(t \rightarrow\frac{ H (t)}{t}\)) is increasing. From \(h(y _{ n })\uparrow\sup_{ y\in Y} h(y )\) we get
So,
for all \(y \in Y\). Consequently,
So, we conclude that
Necessity. Inequality (5) is satisfied for any arbitrary set \(A \in\Sigma\) and any measurable function \(f:X \rightarrow Y\); in particular, for \(f(x )= y \chi_{A} (x ) \) with \(y \in Y\), inequality (5) is true. At first, we define \(f (x ):= y \chi_{A} (x ) \) with \(y \in Y \) and \(A \in\Sigma\). By Theorem 2.2 in [2], we have
On the other hand, the assumption \(a \leq H(y)\) shows that if \(x \in A\), then \(H(f(x)) = H(y)\geq a\). So
and hence \(A \cap L_{a} H(f)= A\). Also, the assumption \(a > H(y)\) shows that if \(x \in A \), then \(H(f(x)) =H(y)< a\) and \(y S_{H},{} _{0} (y) =H(y) < a \), and hence \(S_{H},{} _{0} (y) < \frac{a}{y}\). Thus,
because otherwise there exists \(x \in A \) such that \(H (f(x) )\geq a\). So \(y S_{H},{} _{0} (y)= H(y)\geq a\), and hence \(S_{H},{} _{0} (y) \geq\frac{a}{y}\), which is a contradiction (note that the slope function is nondecreasing). Now, by the monotonicity of \(y \longmapsto y \circ b\) and the conditions \(a \circ0 = 0 \) and \(H (0 )= 0\), we have
as \(H(y) \circ\mu(A) \in Y \subseteq[0 , \infty]\). Therefore, combining (5), (11) and (12), we conclude that
□
We now investigate for which functions \(H : Y \rightarrow Y \) the assertion of Theorem 3.1 is valid if either \(Y=[0,1]\) or \(Y=[0,\infty[\), \(\mu(\Sigma)= Y\) and the operators ⋆,∘ are chosen from the following tnorms:
Corollary 3.2
Suppose that \(Y \subseteq[0 , \infty[ \) and \(H : Y\rightarrow Y \) is a differentiable convex function such that \(H (0 )= 0\) and \(H ^{\prime}(y)\geq1 \) for all \(y \in Y\). For an arbitrary subset \(A \in\Sigma\) and a measurable function \(f : X \rightarrow Y \) such that \(\int_{A} f\star\mu\in Y\), the Jensen inequality for the Sugeno integral
is fulfilled iff
Proof
Let ⋆,∘ = :∧. Clearly
and
Based on Theorem 3.1, the proof is obvious. □
Corollary 3.3
Suppose that \(Y=[0,1]\) and \(H : Y\rightarrow Y \) is a differentiable convex function such that \(H (0 )= 0\) and \(H ^{\prime}(y)\geq1 \) for all \(y \in Y\). For any measurable subset \(A \subseteq Y \) and any measurable function \(f : X \rightarrow Y\), the Jensen inequality for the Shilkret integral
is fulfilled iff
Proof
Choosing the operators ⋆,∘ = :⋅, we get
and
Based on Theorem 3.1, the proof is obvious. □
Corollary 3.4
Suppose that \(Y=[0,1]\) and \(H : Y\rightarrow Y \) is a differentiable convex function with \(H (0 )= 0\). For any measurable subset \(A \subseteq Y \) and any measurable function \(f : X \rightarrow Y\), the Jensen inequality
is fulfilled iff
Proof
Let \(Y=[0,1]\) and ⋆,∘: = ⊕. Then (6) takes the form (14). Since H is differentiable, we have
and so
for \(0 < b \), \(y < 1\). Hence, taking the limit as b approaches 1, we have \(H ^{\prime}(y)\geq1 \). Now, by Theorem 3.1, we obtain the assertion of this corollary. □
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
Let \(\circ,\star:H(Y )\times\mu(\Sigma)\rightarrow\mathbb{R}\) be arbitrary operators and \(H : Y \rightarrow\mathbb{R}\) be a differentiable concave function such that \(Y \subseteq H(Y)\), \(H (0 ) = 0\) and \(H ^{\prime}(y)\leq1 \) for all \(y \in Y \) and satisfies the following condition:

For all \(y \in Y \) and \(b \in\mu(\Sigma)\), \(H(y)\circ b \leq H (y \star b )\).
If \(\int_{A} f\star\mu\in Y \) for \(A \in\Sigma\) and a measurable function \(f : X \rightarrow Y\), then
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
Let \((X, \Sigma, \mu)\) be a fuzzy measure space. Let \(f \in\mathcal {F}_{+}\) and \(A= \{x_{1}, \ldots, x_{n}\}\) be a finite subset of X with \(f(x_{1}) \geqslant f(x_{2}) \geqslant\cdots\geqslant f(x_{n})\). If \(H : Y\rightarrow Y \) is a differentiable convex function with \(H (0 ) = 0\) and \(H ^{\prime}(y)\geq1 \) for all \(y \in Y\), \(\lambda_{i}= \mu (\{x_{1}, \ldots, x_{i}\} )\) for all \(i \in\{1, \ldots, n\}\) and \(\circ,\star: Y \times\mu(\Sigma)\rightarrow Y \) are nondecreasing operators satisfying the following conditions:

1.
\(a \star0 = a \circ0 = 0\),

2.
\(\int_{A} f\star\mu\in Y\),
then
Proof
According to Theorem 3.1,
For the lefthand side of (15), by the hypotheses of the theorem, we have
For the righthand side of (15), by using the order \(H(Y) \subseteq Y\), we get
From (15), (16) and (17) we conclude that
□
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.
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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.
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Jaddi, M., Ebadian, A., de la Sen, M. et al. An equivalent condition to the Jensen inequality for the generalized Sugeno integral. J Inequal Appl 2017, 285 (2017). https://doi.org/10.1186/s1366001715612
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DOI: https://doi.org/10.1186/s1366001715612