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.
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.
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
(Ralescu and Adams [6]). Suppose that \(\mu: \Sigma\longrightarrow[0, \infty)\) is a set function. We say that μ is a fuzzy measure if it satisfies

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.
Let \((X, \Sigma, \mu)\) be a fuzzy measure space. By \(\mathcal{F}_{+}(X)\) we denote the set
For \(f \in\mathcal{F}_{+}(X)\) and \(\alpha> 0\), \(F_{\alpha}\) and \(F_{\tilde{\alpha}}\) we will denote the following sets:
Note that if \(\alpha\leq\beta\), then \(F_{\beta} \subset F_{\alpha}\) and \(F_{\tilde{\beta}} \subset F_{\tilde{\alpha}}\).
Definition 2.2
(Pap [22], Sugeno [5], Wang and Klir [12]). Let \((X, \Sigma, \mu)\) be a fuzzy measure space, \(f \in\mathcal {F}_{+}(X)\) and \(A \in\Sigma\), then the Sugeno fuzzy integral of f on A with respect to the fuzzy measure μ is defined by
where ∧ is just the prototypical tnorm minimum and ⋁ the prototypical tconorm maximum. When \(A= X\), then
The following properties of the Sugeno fuzzy integral are well known and can be found in [12, 22].
Theorem 2.3
Let \((X, \Sigma, \mu)\) be a fuzzy measure space, \(A, B \in\Sigma\) and \(f, g \in\mathcal{F}_{+}(X)\) then

(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 \(\fint _{A} f \,\mathrm{d}\mu\leq \fint _{B} f \,\mathrm{d}\mu\).
A tnorm [23] is a function \(T: [0, 1] \times[0, 1] \longrightarrow[0, 1]\) satisfying the following conditions:
 (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
The following functions are tnorms:

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
Let T be a tseminorm. Then the seminormed Sugeno fuzzy integral of a function \(f \in\mathcal{F}_{+}(X)\) over \(A \in\Sigma\) with respect to T and the fuzzy measure μ is defined by
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])
Let \((X, \Sigma, \mu)\) be a fuzzy measure space and T be a tseminorm. Then

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 \(f \in\mathcal {F}_{+}(X)\),
$$\int_{T, A} f \,\mathrm{d}\mu\leq\int_{T, B} f \,\mathrm{d}\mu. $$
A stronger property of convexity is logconvexity. A positive function f defined on a real interval \([a, b]\) (or, more generally, on a convex subset of some vector space) is called logconvex if logf is a convex function of x; equivalently, if for all \(x, y \in[a, b]\) and \(\lambda\in[0, 1]\) we have
It will be convenient to invoke the logarithmic mean \(L(x, y)\) of two positive numbers \(x, y\), which is given by
The following Hadamard inequality provides an upper bound for the mean value of a logconvex function \(f: [a, b] \longrightarrow\mathbb{R}\) (see [25]):
The main results
Hereafter, we assume that \((X, \Sigma, \mu)\) is a general fuzzy measure space. To simplify the calculation of the fuzzy integral, for a given \(f \in\mathcal{F}_{+}(X)\) and \(A \in\Sigma\), we write
It is easy to see that
The following example shows that the Hadamard inequality (2) is not valid in the fuzzy context.
Example 3.1
Let \(X= [0, 1]\) and μ be the Lebesgue measure on \(\mathbb{R}\). We take the positive and logconvex function \(f(x)= 2^{x^{2}  3}\). We have
In this expression, \(1 \sqrt{\frac{\ln(\alpha)}{\ln(2)}+ 3}\) may be negative, but it is a decreasing continuous function of α when \(\alpha\geq0\). Hence, the supremum will be attained at the point which is one of the solutions of the equation
that is, at \(\alpha\approx0.196\). So, we have
On the other hand, \(L (f(0), f(1) ) \approx0.180\). This proves that the Hadamard inequality (2) is not satisfied in the fuzzy context.
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
Let \(f: [0, 1] \longrightarrow(0, \infty)\) be a logconvex function with \(f(0)\neq f(1)\). Then
where \(\Gamma= [f(0), f(1) )\) for \(f(1) > f(0)\) and \(\Gamma= [f(1), f(0) )\) for \(f(0) > f(1)\).
Proof
Using the logconvexity of f, for \(x \in[0, 1]\), we have
and by (3) of Theorem 2.3, we get
For calculating the integral in the righthand part of the last inequality, we consider the distribution function G given by
If \(f(1)> f(0)\), then
Thus, \(\Gamma= [f(0), f(1) )\), and we only need to consider \(\alpha\in [f(0), f(1) )\).
If \(f(1)< f(0)\), then
Thus, \(\Gamma= [f(1), f(0) )\) and we only need to consider \(\alpha\in [f(1), f(0) )\).
Taking into account (1) of Theorem 2.3, we get
and the proof is completed. □
Remark 3.3
In the case \(f(0)= f(1)\) in Theorem 3.2, we have \(g(x)= f(0)\), and using (2) and (3) of Theorem 2.3, we get
Corollary 3.4
Let \(f: [0, 1] \longrightarrow(0, \infty)\) be a logconvex function with \(f(0)\neq f(1)\), Σ be the Borel field and μ be the Lebesgue measure on \(X= \mathbb{R}\). Then
Example 3.5
Let μ be the Lebesgue measure on \(\mathbb{R}\). Consider the function \(f(x)= 2^{x^{2}  \frac{1}{2}}\) on \(X= [0, 1]\). This function is, obviously, logconvex and positive on \([0, 1]\). As \(f(0)= \sqrt {2}/2\), \(f(1)= \sqrt{2}\), using Corollary 3.4 we can get the following estimate:
In this expression, \(\log_{2} \frac{\sqrt{2}}{\alpha}\) is a decreasing continuous function of α when \(\alpha\in [\sqrt{2}/2, \sqrt {2} )\). Hence, the supremum will be attained at the point which is one of the solutions of the equation
that is, at \(\alpha\approx0.808\). Consequently, we have
Proposition 3.6
Let \(f: [0, 1] \longrightarrow(0, 1]\) be a logconvex function with \(f(0)\neq f(1)\), Σ be the Borel field and μ be the Lebesgue measure on \(X= \mathbb{R}\). Then
Proof
For a logconvex function \(f: [0, 1] \longrightarrow(0, 1]\) with \(f(0)\neq f(1)\), according to (1) of Proposition 2.7 and Corollary 3.4 with tnorm \(T_{P}\), we have
□
Example 3.7
Let μ be the Lebesgue measure on \(\mathbb{R}\). Consider the function \(f(x)= 2^{x^{2}  1}\) on \(X= [0, 1]\). This function is, obviously, logconvex and positive on \([0, 1]\). As \(f(0)= 1/2\) and \(f(1)= 1\), using Proposition 3.6, we can get the following estimate:
In the next theorem, we prove the general case of Theorem 3.2.
Theorem 3.8
Suppose that \(f: [a, b] \longrightarrow(0, \infty)\) is a logconvex function with \(f(a)\neq f(b)\). Then
where \(t= \frac{x a}{b a}\), \(\Gamma= [f(a), f(b) )\) for \(f(b) > f(a)\) and \(\Gamma= [f(b), f(a) )\) for \(f(a) > f(b)\).
Proof
As f is logconvex, for \(x \in[a, b]\), we have
where \(t= \frac{x a}{b a}\). By (3) of Theorem 2.3, we get
A similar argument as in the proof of Theorem 3.2 yields
where \(t= \frac{x a}{b a}\), \(\Gamma= [f(a), f(b) )\) for \(f(b) > f(a)\) and \(\Gamma= [f(b), f(a) )\) for \(f(a) > f(b)\). This completes the proof. □
Remark 3.9
In the case \(f(a)= f(b)\) in Theorem 3.8, we have \(g(x)= f(a)\) and using (2) and (3) of Theorem 2.3, we get
Corollary 3.10
Let \(f: [a, b] \longrightarrow(0, \infty)\) be a logconvex function with \(f(a)\neq f(b)\), Σ be the Borel field and μ be the Lebesgue measure on \(X= \mathbb{R}\). Then
Example 3.11
Let μ be the Lebesgue measure on \(\mathbb{R}\). Consider the function \(f(x)= x 2^{x^{2}  2}\) on \(X= [1, 2]\). This function is, obviously, positive and logconvex on \([1, 2]\). As \(f(1)= 1/2\), \(f(2)= 8\), using Corollary 3.10, we may approximate the upper bound of the Sugeno fuzzy integral of f on \([1, 2]\) by
In this expression, \(\log_{16} \frac{8}{\alpha}\) is a decreasing continuous function of α when \(\alpha\in [1/2, 8 )\). Hence, the supremum will be attained at the point which is one of the solutions of the equation
that is, at \(\alpha\approx0.822\). Therefore,
It should be noted that the exact solution of \(\fint _{1}^{2} x 2^{x^{2}  2} \,\mathrm{d}\mu\) cannot be easily calculated. But surely the exact solution is less than or equal to 0.822.
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.
In Example 3.5, for the real logconvex function \(f(x)= 2^{x^{2}  \frac{1}{2}}\) on \([0, 1]\), there exists the real function
such that
Geometric interpretation of (4) is shown in Figure 1. The lengths of the lines 1 and 2 are the solutions of the integrals in left and righthand sides of (4), respectively. We have a similar geometric interpretation for Example 3.11.
Convexity associated to means
Let us recall the following means for two positive numbers:

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 inequality \(G \leq A\) is well known in the literature.
The definition of convexity and logconvexity can be embedded into a more general framework by taking two regular means M and N (on the intervals I and J respectively) and calling a function \(f: I \longrightarrow J\) to be \((M, N)\)midpoint convex if it satisfies
for every \(x, y \in I\) (see [26–28]). If f is continuous, this yields the \((M, N)\)convexity of f; that is,
for every \(x, y \in I\) and every \(\lambda\in[0, 1]\). For example, if f is continuous, the inequality (1) can be introduced as \((A, G)\)convexity.
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
Let \(f: [a, b] \longrightarrow(0, \infty)\) be a continuous and logconvex function with \(f(a)\neq f(b)\). Then
where \(t= \frac{x a}{b a}\), \(\Gamma= [f(a), f(b) )\) for \(f(b) > f(a)\) and \(\Gamma= [f(b), f(a) )\) for \(f(a) > f(b)\).
Proof
By the geometric meanarithmetic mean inequality (\(G \leq A\)), we have
where \(t= \frac{x a}{b a}\), which yields
Hence, the assertion of this theorem is true in view of Definition 2.2 and Theorem 3.8. □
Corollary 5.2
Let \(f: [a, b] \longrightarrow(0, \infty)\) be a continuous and logconvex function with \(f(a)\neq f(b)\), Σ be the Borel field and μ be the Lebesgue measure on \(X= \mathbb{R}\). Then, for \(f(b)> f(a)\),
If \(f(b)< f(a)\), then
Example 5.3
Consider again the function \(f(x)= x 2^{x^{2}  2}\) on \(X= [1, 2]\), see Example 3.11. In virtue of Corollary 5.2 and Example 3.11, we have
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.
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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.
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Abbaszadeh, S., Eshaghi, M. & de la Sen, M. The Sugeno fuzzy integral of logconvex functions. J Inequal Appl 2015, 362 (2015). https://doi.org/10.1186/s1366001508626
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DOI: https://doi.org/10.1186/s1366001508626
MSC
 26A51
 28E10
 39B62
Keywords
 Sugeno fuzzy integral
 the Hadamard inequality
 logconvex function
 seminormed Sugeno fuzzy integral