An equivalent condition to the Jensen inequality for the generalized Sugeno integral

For the classical Jensen inequality of convex functions, i.e., H(1μ(D)∫Dfdμ)≤1μ(D)∫DH∘fdμ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} H \biggl(\frac{1}{\mu(D)} \int_{D}f\,d\mu \biggr)\leq\frac{1}{\mu (D)} \int_{D} H\circ f\,d\mu, \end{aligned}$$ \end{document} 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.


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 well-known Jensen inequality for a convex function is given as follows: Let (X, , μ) be a measure space, f be a real-valued μ-measurable and μ-integrable function on a set D ∈ with μ(D  Kaluszka et al. [] studied the Jensen inequality () 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 . In Section , the Jensen inequality for the generalized Sugeno integral is studied. In Section , the necessary and sufficient conditions for the validity of the discrete form of the Jensen inequality is presented. A conclusion is given in Section .

Preliminaries
In this section, some definitions and basic properties of the Sugeno integral which will be used in the next section are presented.
Definition . (Fuzzy measure [, ]) Let be a σ -algebra of subsets of X, and let μ : → [, ∞[ be a non-negative extended real-valued set function. We say that μ is a fuzzy measure iff: Let (X, , μ) be a fuzzy measure space and f be a non-negative real-valued function on X. We denote by F + the set of all non-negative measurable functions and by L α f the set {x ∈ X | f (x) ≥ α}, the α-level of f for α ≥ .
Definition . Let (X, , μ) be a fuzzy measure space. If f ∈ F + and A ∈ , then (i) The Shilkret integral [] of f on A with respect to the fuzzy measure μ is given by (ii) The Sugeno integral [] of f on A with respect to the fuzzy measure μ is defined by where ∨ and ∧ denote the operations sup and inf on [, ∞[, respectively.
The following theorem gives most elementary properties of the Sugeno integral and can be found in [, ].
, from parts () and () 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 we define the generalized Sugeno integral of f on a set A ∈ with respect to μ and an operator : Let I be a real interval and f : I → R be a function. Then f is said to be convex (on I) provided Also, for any x, x  ∈ I. We say that the operator  T(y, z)) for any x, y, z ∈ [, ].
Example . The following operators are t-norm:

Results and discussion
For the classical measure μ, the classical Jensen inequality is the following strong property of convex functions (see []): The following inequality is known as the discrete Jensen inequality: is sharp if and only if, for any y ∈ Y and b ∈ μ( ),

H(y) • b ≥ H(y b). ()
Proof Sufficiency. Let y ∈ Y . Since H is a differentiable convex function, by Theorem . we have and by assumption (), for all x ∈ X. Therefore, by the monotonicity of μ, we deduce for an arbitrary set A ∈ . On the other hand, since H(Y ) ⊆ Y , we have Combining () with () and using the monotonicity of •, we get

So, we conclude that
Necessity. Inequality () is satisfied for any arbitrary set A ∈ and any measurable function f : X → Y ; in particular, for f (x) = yχ A (x) with y ∈ Y , inequality () is true. At first, we define f (x) := yχ A (x) with y ∈ Y and A ∈ . By Theorem . in [], we have because otherwise there exists x ∈ A such that H(f (x)) ≥ a. So yS H ,  (y) = H(y) ≥ a, and hence S H ,  (y) ≥ a y , which is a contradiction (note that the slope function is nondecreasing). Now, by the monotonicity of y − → y • b and the conditions a •  =  and For an arbitrary subset A ∈ and a measurable function f : X → Y such that A f μ ∈ Y , the Jensen inequality for the Sugeno integral Based on Theorem ., the proof is obvious.
Proof Choosing the operators , • =: ·, we get Based on Theorem ., the proof is obvious.
Proof Let Y = [, ] and , • := ⊕. Then () takes the form (). Since H is differentiable, we have and so for  < b, y < . Hence, taking the limit as b approaches , we have H (y) ≥ . Now, by Theorem ., we obtain the assertion of this corollary.
In the next theorems, the sufficient and necessary conditions for the reverse of inequality () are given. The proofs are similar to the proof of Theorem . and are omitted.  H(y b).

Generalized Sugeno integral and discrete Jensen inequality
In this section, we deal with the discrete Jensen inequality for the generalized Sugeno

.
A f μ ∈ Y , then Proof According to Theorem ., For the left-hand side of (), by the hypotheses of the theorem, we have