The Sugeno fuzzy integral of logconvex functions
 Sadegh Abbaszadeh^{1}Email author,
 Madjid Eshaghi^{1, 2} and
 Manuel de la Sen^{3}
DOI: 10.1186/s1366001508626
© Abbaszadeh et al. 2015
Received: 15 May 2015
Accepted: 5 October 2015
Published: 14 November 2015
Abstract
In this paper, we give an upper bound for the Sugeno fuzzy integral of logconvex functions using the classical Hadamard integral inequality. We present a geometric interpretation and some examples in the framework of the Lebesgue measure to illustrate the results.
Keywords
Sugeno fuzzy integral the Hadamard inequality logconvex function seminormed Sugeno fuzzy integralMSC
26A51 28E10 39B621 Introduction
Aggregation is a process of combining several numerical values into a single one which exists in many disciplines, such as image processing [1], pattern recognition [2] and decision making [3, 4]. To obtain a consensus quantifiable judgments, some synthesizing functions have been proposed. For example, arithmetic mean, geometric mean and median can be regarded as a basic class, because they are often used and very classic. However, these operators are not able to model an interaction between criteria. For having a representation of interaction phenomena between criteria, fuzzy measures have been proposed [5].
The properties and applications of the fuzzy measures and fuzzy integrals have been studied by many authors. Ralescu and Adams [6] studied several equivalent definitions of fuzzy integrals. RománFlores et al. [7–11] studied the levelcontinuity of fuzzy integrals, Hcontinuity of fuzzy measures and geometric inequalities for fuzzy measures and integrals, respectively. Wang and Klir [12] had a general overview on fuzzy measurement and fuzzy integration theory.
Two main classes of the fuzzy integrals are Choquet and Sugeno integrals. Choquet and Sugeno integrals are idempotent, continuous and monotone operators. Recently, many authors have studied the most wellknown integral inequalities for fuzzy integral. Agahi et al. [13–15] proved general Minkowski type inequalities, general extensions of Chebyshev type inequalities and general BarnesGodunovaLevin type inequalities for fuzzy integrals. Caballero and Sadarangani [16, 17] proved Chebyshev type inequalities and Cauchy type inequalities for fuzzy integrals. Kaluszka et al. [18] gave necessary and sufficient conditions guaranteeing the validity of Chebyshev type inequalities for generalized Sugeno fuzzy integrals in the case of functions belonging to a much wider class than the comonotone functions. Wu et al. [19] proved two inequalities for the Sugeno fuzzy integral on abstract spaces generalizing all previous Chebyshev’s inequalities. Mesiar et al. [20] discussed the integral inequalities known for the Lebesgue integral in the framework of the Choquet integral.
A stronger property of convexity is logconvexity. The arithmetic meangeometric mean inequality easily yields that every logconvex function is also convex. The behavior of certain interferencecoupled multiuser systems can be modeled by means of logarithmically convex (logconvex) interference functions [21]. In this paper, the main purpose is to estimate the upper bound of Sugeno fuzzy integral for logconvex functions using the classical Hadamard integral inequality.
The paper is organized as follows. Some necessary preliminaries and summarization of some previous known results are presented in Section 2. In Section 3, the upper bound of the Sugeno fuzzy integral for logconvex functions is investigated. In Section 4, a geometric interpretation is presented to illustrate the results. Convexity associated to means is discussed in Section 5. Finally, a conclusion is given in Section 6.
2 Preliminaries
In this section, we are going to review some wellknown results from the theory of nonadditive measures. Let X be a nonempty set and Σ be a σalgebra of subsets of X.
Definition 2.1
 1.
\(\mu(\emptyset)= 0\).
 2.
\(E, F \in\Sigma\) and \(E \subset F\) imply \(\mu(E) \leq\mu(F)\).
 3.
\(E_{n} \in\Sigma\) (\(n \in\mathbb{N}\)), \(E_{1} \subset E_{2} \subset \cdots\) imply \(\lim_{n \rightarrow\infty} \mu(E_{n})= \mu(\bigcup_{n= 1}^{\infty} E_{n})\) (continuity from below).
 4.
\(E_{n} \in\Sigma\) (\(n \in\mathbb{N}\)), \(E_{1} \supset E_{2} \supset \cdots\) , \(\mu(E_{1}) < \infty\) imply \(\lim_{n \rightarrow\infty} \mu(E_{n})= \mu(\bigcap_{n= 1}^{\infty} E_{n})\) (continuity from above).
The triple \((X, \Sigma, \mu)\) is called a fuzzy measure space.
Definition 2.2
The following properties of the Sugeno fuzzy integral are well known and can be found in [12, 22].
Theorem 2.3
 (1)
\(\fint _{A} f \,\mathrm{d}\mu\leq\mu(A)\).
 (2)
\(\fint _{A} k\, \mathrm{d}\mu= k \wedge\mu(A)\), k nonnegative constant.
 (3)
If \(f \leq g\) on A. then \(\fint _{A} f \,\mathrm{d}\mu \leq \fint _{A} g\, \mathrm{d}\mu\).
 (4)
If \(A \subset B\), then \(\fint _{A} f \,\mathrm{d}\mu\leq \fint _{B} f \,\mathrm{d}\mu\).
 (T_{1}):

\(T(x, 1)= T(1, x)= x\) for any \(x \in[0, 1]\).
 (T_{2}):

For any \(x_{1}, x_{2}, y_{1}, y_{2} \in[0, 1]\) with \(x_{1} \leq x_{2}\) and \(y_{1} \leq y_{2}\), \(T(x_{1}, y_{1}) \leq T(x_{2}, y_{2})\).
 (T_{3}):

\(T(x, y)= T(y, x)\) for any \(x, y \in[0, 1]\).
 (T_{4}):

\(T (T(x, y), z )= T (x, T(y ,z) )\) for any \(x, y, z \in[0, 1]\).
A function \(S: [0, 1] \times[0, 1] \longrightarrow[0, 1]\) is called a tconorm [24] if there is a tnorm T such that \(S(x, y)= 1 T(1 x, 1 y)\).
Example 2.4
 1.
\(T_{M} (x, y)= x \wedge y\),
 2.
\(T_{P} (x, y)= x \cdot y\),
 3.
\(T_{L} (x, y)= (x+ y 1) \vee0\).
Remark 2.5
A binary operator T on \([0, 1]\) is called a tseminorm [23] if it satisfies the above conditions (T_{1}) and (T_{2}). Notice that if T is a tseminorm, for any \(x, y \in[0, 1]\), we have \(T(x, y) \leq T(x, 1)= x\) and \(T(x, y) \leq T(1, y)= y\), and consequently, \(T(x, y) \leq T_{M} (x, y)\).
By using the concept of tseminorm, García and Álvarez [23] proposed the following family of fuzzy integrals.
Definition 2.6
Notice that the Sugeno fuzzy integral of \(f \in\mathcal{F}_{+}(X)\) over \(A \in\Sigma\) is the seminormed Sugeno fuzzy integral of f over \(A \in\Sigma\) with respect to the tseminorm \(T_{M}\).
Proposition 2.7
(García and Álvarez [23])
 1.For any \(A \in\Sigma\) and \(f, g \in\mathcal{F}_{+}(X)\) with \(f \leq g\), we have$$\int_{T, A} f \,\mathrm{d}\mu\leq\int_{T, A} g \,\mathrm{d}\mu. $$
 2.For \(A, B \in\Sigma\) with \(A \subset B\) and any \(f \in\mathcal {F}_{+}(X)\),$$\int_{T, A} f \,\mathrm{d}\mu\leq\int_{T, B} f \,\mathrm{d}\mu. $$
3 The main results
The following example shows that the Hadamard inequality (2) is not valid in the fuzzy context.
Example 3.1
In the sequel, we will establish an upper bound on the Sugeno fuzzy integral of logconvex functions. Some specific examples will be given to illustrate the results.
Theorem 3.2
Proof
Remark 3.3
Corollary 3.4
Example 3.5
Proposition 3.6
Proof
Example 3.7
In the next theorem, we prove the general case of Theorem 3.2.
Theorem 3.8
Proof
Remark 3.9
Corollary 3.10
Example 3.11
4 Geometric interpretation
Assume that \(X= \mathbb{R}\), Σ is the Borel field, μ is the Lebesgue measure and \(f: A \subseteq\mathbb{R} \longrightarrow(0, \infty)\) is a continuous function. Then the geometric significance of \(\fint _{A} f \,\mathrm{d}\mu\) is the edge’s length of the largest square between the curve of \(f(x)\) and the xaxis.
5 Convexity associated to means

The arithmetic mean$$A= A(a, b):= \frac{a+ b}{2},\quad a, b > 0. $$

The geometric mean$$G= G(a, b):= \sqrt{a b},\quad a, b > 0. $$
The following result provides an upper bound on the righthand side of the inequality (3) (Theorem 3.8) in the case that f is continuous.
Theorem 5.1
Proof
Corollary 5.2
Example 5.3
6 Conclusion
In this paper, we have established an upper bound on the Sugeno fuzzy integral of logconvex functions which is a useful tool to estimate unsolvable integrals of this kind. In many applications, assumptions about the logconvexity of a probability distribution allow just enough special structure to yield a workable theory. The logconcavity or logconvexity of probability densities and their integrals has interesting qualitative implications in many areas of economics, in political science, in biology, and in industrial engineering. As we know, fuzzy measures have been introduced by Sugeno in the early seventies in order to extend probability measures by relaxing the additivity property. Thus the study of the Sugeno fuzzy integral for logconvex functions is an important and interesting topic for further research.
Declarations
Acknowledgements
The authors are very grateful to the Spanish Government for its support of this research through Grant DPI201230651. The authors are also grateful to the Basque Government through Grant IT 37810.
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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