 Research
 Open Access
The Choquet integral of logconvex functions
 Hongxia Wang^{1, 2}Email author
https://doi.org/10.1186/s136600181803y
© The Author(s) 2018
 Received: 5 March 2018
 Accepted: 6 August 2018
 Published: 14 August 2018
Abstract
In this paper we investigate the upper bound and the lower bound of the Choquet integral for logconvex functions. Firstly, for a monotone logconvex function, we state the similar Hadamard inequality of the Choquet integral in the framework of distorted measure. Secondly, we estimate the upper bound of the Choquet integral for a general logconvex function, respectively, in the case of distorted Lebesgue measure and in the nonadditive measure. Finally, we present Jensen’s inequality of the Choquet integral for logconvex functions, which can be used to estimate the lower bound of this kind when the nonadditive measure is concave. We provide some examples in the framework of the distorted Lebesgue measure to illustrate all the results.
Keywords
 Choquet integral
 Logconvex function
 Inequality
1 Introduction
It is well known that the concept of nonadditive measure [1] can be used to deal with some uncertainty phenomena which cannot be easily modeled by using additive measure, and the Choquet integral, which also covers the classical Lebesgue integral, is one kind of nonlinear expectations. Many authors developed the Choquet theory with its applications in many areas such as multicriteria decision making, risk measuring, option pricing, and so on. Readers may refer to the references [2–17]. Here we specially mention that Mesiar et al. [9] discussed the integral inequalities known for the Lebesgue integral in the framework of the Choquet integral.
A strong property of convexity is logconvexity. It is known that every logconvex function is also convex. In many applications, assumptions about the logconvexity of a probability distribution allow just enough special structure to yield a workable theory. The logconvexity (logconcavity) of probability densities and their integrals has interesting qualitative implications in many areas of economics, in political science, in biology, and in industrial engineering [18].
Thus the study of the Choquet integral of logconvex functions is an important and interesting topic for further research. It is well known that the Hadamard inequality is a famous and important result, which provides the upper (lower) bound for the mean value of a logconvex (logconcave) function. Abbaszadeh et al. [19] studied the Sugeno fuzzy integral of logconvex functions and showed that the Hadamard inequality is not valid for this kind of Sugeno fuzzy integral. Motivated by [19], we naturally wonder whether the Hadamard inequality still holds for the Choquet integral. If the Hadamard inequality is not valid, then it is necessary to estimate the upper bound and the lower bound of the Choquet integral for logconvex functions.
In this paper, we shall study the upper bound and the lower bound of the Choquet integral for logconvex functions. The rest of this paper is organized as follows. Section 2 is for preliminaries and notations of the nonadditive measure, the Choquet integral, and logconvex function that will be used later. In Sect. 3, the main results are shown. Firstly, we shall point out that the Hadamard inequality is not valid in the framework of distorted Lebesgue measure and shall present a similar Hadamard inequality for monotone logconvex functions. Secondly, the upper bound of the Choquet integral for the general logconvex function is presented. Finally, we shall investigate Jensen’s inequality of the Choquet integral for a logconvex function, which can be used to estimate the lower bound of this kind. At the end of the paper, some conclusions are drawn and some problems for further investigation are suggested.
2 Preliminaries
Throughout the paper, assume that \((X,\mathcal{F})\) is a measurable space and (\(\mathbf{R}^{+}\)) R is the set of all (nonnegative) real numbers.
We first recall some concepts and some elementary results of capacity and the Choquet integral [1, 2].
Definition 2.1
 (1)
\(\mu (\emptyset )=0\);
 (2)
\(\mu (A)\leq \mu (B)\) for any \(A\subseteq B\) and \(A,B\in \mathcal{F}\).
Given a nonadditive measure μ on \((X,\mathcal{F})\), with \(\mu (X)\) finite (i.e., \(\mu (X)<\infty \)), we define its conjugate or dual as the measure μ̄ defined as follows: \(\bar{\mu }(A)= \mu (X)\mu (A^{c})\) for all \(A\in \mathcal{F}\). Here and in the sequel \(A^{c}\) denotes the complement set of A. When a measure μ is additive, it holds that \(\mu (A)=\bar{\mu }(A)\).
First note that the Lebesgue measure λ for an interval \([a, b]\) is defined by \(\lambda ([a, b]) = ba\), and that given a distortion function m, which is increasing (or nondecreasing) and such that \(m(0)=0\), the measure \(\mu (A)=m(\lambda (A))\) is a distorted Lebesgue measure. We denote a Lebesgue measure with distortion m by \(\mu =\mu_{m}\). It is not difficult to know that \(\mu_{m}\) is concave (convex) if m is a concave (convex) function.
The family of all the nonnegative, measurable functions \(f: (X, \mathcal{F})\rightarrow (\mathbf{R}^{+}, \mathcal{B}(\mathbf{R}^{+}))\) is denoted as \(L_{+}^{\infty }\), where \(\mathcal{B}(\mathbf{R}^{+})\) is the Borel σfield of \(\mathbf{R}^{+}\). The concept of the integral with respect to a nonadditive measure was introduced by Choquet [1].
Definition 2.2
The subsequent lemma summarizes the basic properties of Choquet integrals [2].
Lemma 2.1
 (1)
\((C)\int 1_{A}\,d\mu =\mu (A), A\in \mathcal{F}\).
 (2)
(Positive homogeneity) For all \(\lambda \in \mathbf{R}^{+}\), we have \((C) \int \lambda f\,d\mu =\lambda \cdot (C)\int f \,d\mu \).
 (3)
(Translation invariance) For all \(c\in \mathbf{R}\), we have \((C) \int (f+c)\,d\mu =(C)\int f \,d\mu +c\).
 (4)(Monotonicity in the integrand) If \(f\leq g\), then we have(Monotonicity in the set function) If \(\mu \leq \nu \), then we have \((C) \int f \,d\mu \leq (C)\int f \,d\nu \).$$ (C) \int f \,d\mu \leq (C) \int g \,d\mu ; $$
 (5)(Subadditivity) If μ is concave, then(Superadditivity) If μ is convex, then$$ (C) \int (f+g)\,d\mu \leq (C) \int f \,d\mu +(C) \int g \,d\mu ; $$$$ (C) \int (f+g)\,d\mu \geq (C) \int f \,d\mu +(C) \int g \,d\mu . $$
 (6)(Comonotonic additivity) If f and g are comonotonic, thenwhere we say that f and g are comonotonic, if for any \(x,x'\in X\), then$$ (C) \int (f+g)\,d\mu =(C) \int f \,d\mu +(C) \int g \,d\mu , $$$$ \bigl(f(x)f\bigl(x'\bigr)\bigr) \bigl(g(x)g\bigl(x' \bigr)\bigr)\geq 0. $$
We review now the excellent results from the article [20], which permits us to compute the Choquet integral when the nonadditive measure is a distorted Lebesgue measure.
Lemma 2.2
The following lemma, about Jensen’s inequality of the Choquet integral, comes from [9, 15].
Lemma 2.3
In the following subsection, we shall list some preliminaries about the logconvex functions. Concerning more definitions and more results of the logconvexity, readers could refer to Zhang and Jiang’s excellent article [21].
Recall the definition of a logconvex (logconcave) function.
Definition 2.3
3 Main results
In this paper, we shall study the Choquet integral for logconvex (logconcave) functions.
Firstly, we shall investigate whether the Hadamard inequality for the Choquet integral still holds. The following theorem shows that the Hadamard inequality is not valid in the distorted measure theory. However, we can obtain a similar Hadamard inequality under some certain conditions.
Hereafter, let \(m(x)\) and \(f(x)\) be continuously differentiable.
Theorem 3.1
 (1)Let f be a positive, measurable, decreasing, and logconvex (logconcave) function on \(\mathbf{R}^{+}\) and \(\mu =\mu_{m}\) be a distorted Lebesgue measure. Then there exists \(\xi \in (0,1)\) such that$$ (C) \int_{[0,1]} f(x)\,d\mu_{m}\leq (\geq )m'( \xi )L\bigl(f(0),f(1)\bigr). $$(7)
 (2)Let f be a positive, measurable, increasing, and logconvex (logconcave) function on \(\mathbf{R}^{+}\) and \(\mu =\mu_{m}\) be a distorted Lebesgue measure. Then there exists \(\theta \in (0,1)\) such that$$ (C) \int_{[0,1]} f(x)\,d\mu_{m}\leq (\geq )m'(1\theta )L\bigl(f(0),f(1)\bigr). $$(8)
Proof
(2) In an analogous way as in the proof of (1), we can obtain the desired result. □
Example 3.1
The next theorem is the general case of Theorem 3.1.
Theorem 3.2
 (1)Let f be a positive, measurable, decreasing, and logconvex (logconcave) function on \(\mathbf{R}^{+}\) and \(\mu =\mu_{m}\) be a distorted Lebesgue measure. For \([a,b]\subset \mathbf{R}^{+}\), we know there exists \(\xi \in (0,ba)\) such that$$ \frac{1}{ba}(C) \int_{[a,b]} f(x)\,d\mu_{m}\leq (\geq )m'( \xi )L\bigl(f(a),f(b)\bigr). $$(9)
 (2)Let f be a positive, measurable, increasing and logconvex (logconcave) function on \(\mathbf{R}^{+}\) and \(\mu =\mu_{m}\) be a distorted Lebesgue measure. For \([a,b]\subset \mathbf{R}^{+}\), we know there exists \(\theta \in (0,ba)\) such that$$ \frac{1}{ba}(C) \int_{[a,b]} f(x)\,d\mu_{m}\leq (\geq )m'(ba\theta )L\bigl(f(a),f(b)\bigr). $$(10)
Proof
Example 3.2
Observe that Theorems 3.1 and 3.2 are based on the assumption that the logconvex function is monotone. This suggests an open question: Can we find the upper bound of the Choquet integral when the logconvex function is not monotone? In the following we shall present some results concerning this issue.
Theorem 3.3
Proof
The proof is completed. □
Remark 3.1
In the next theorem, we prove the general case of Theorem 3.3.
Theorem 3.4
Proof
The proof is completed. □
Remark 3.2
Remark 3.3
We need to point out that Theorems 3.3 and 3.4 hold for the general logconvex function; that is to say, the two theorems do not require that f is monotone. Of course, even so, they hold for a monotone logconvex function.
Example 3.3
Example 3.4
In the following, we shall discuss the upper bound of the Choquet integral for the logconvex function in the framework of the general nonadditive measure.
Theorem 3.5
Proof
The proof is completed. □
Remark 3.4
Specially, when \(\mu =\mu_{m}\) is a distorted Lebesgue measure, Theorem 3.5 still holds.
Example 3.5
The next theorem is the general case of Theorem 3.5.
Theorem 3.6
Proof
The proof is completed. □
Example 3.6
The remainder of this paper will be mainly devoted to Jensen’s inequality of the Choquet integral for logconvex functions.
It is well known that there is a result: if \(g: \mathbf{R}^{+}\rightarrow \mathbf{R}^{+}\) is a logconvex function, then g is convex. The proof is sketched for the readers’ convenience.
Proof
Due to the above result and Lemma 2.3, we can easily show the following theorem.
Theorem 3.7
(Jensen’s inequality)
Proof
Similarly we shall present the following Jensen inequality for the Choquet integral of the logconvex function in two dimensions.
Theorem 3.8
(Jensen’s inequality in two dimensions)
Proof
Remark 3.5
 (1)
Here the two discussed Jensen inequalities of the Choquet integral for logconvex functions are valid whenever the considered nonadditive measure μ is concave, which compares also the articles [9] and [15] where all considered inequalities for Choquet integrals hold whenever μ is concave.
In particular, when the nonadditive measure μ is a plausibility function or the concave distortion measure \(\mu =\mu_{m}\), where m is a concave function, both inequalities above hold.
 (2)
In fact, for the Choquet integral of logconvex functions, we might investigate the lower bound, when the considered nonadditive measure μ is concave, using Jensen’s inequality. We shall provide an easy example below.
Example 3.7
Consider the function \(L(x)=2^{(x+1)^{2}}\) on \(\mathbf{R}^{+}\). \(L(x)=g\circ f\) is the compound function of \(g(u)\) and \(f(x)\), where \(g(u)=2^{u^{2}}\) and \(u=f(x)=x+1\). Evidently \(g(u)\) is a logconvex function.
4 Conclusions and problems for further investigation

We firstly pointed out that the Hadamard inequality is not valid in the nonadditive measure theory, and stated the similar Hadamard inequality of the Choquet integral for monotone logconvex function in the framework of distorted measure.

We secondly estimated the upper bound of the Choquet integral for the general logconvex functions both in the case of distorted measure and in the case of nonadditive measure.

Finally, we obtained Jensen’s inequality and Jensen’s inequality in two dimensions of the Choquet integral for logconvex functions.
These results are extensions of the Choquet theory.
For further investigation, since we have already had a clear understanding of the upper bound and Jensen’s inequality of the Choquet integral for logconvex functions, it is natural to consider how to use them to estimate unsolvable integrals of this kind. Thus the study of their applications is an interesting topic for further research. On the other hand, we shall continue to explore some other inequalities for the Choquet integral of the logconvex functions and also investigate their applications in some areas.
5 Methods
The aim of this paper is to study the upper bound and the lower bound of the Choquet integral for logconvex functions. It is well known that the Hadamard inequality provides the upper (lower) bound for the mean value of a logconvex (logconcave) function, so we want to know whether the Hadamard inequality still holds for the Choquet integral. If the Hadamard inequality is not valid, then how to estimate the upper bound and the lower bound of the Choquet integral for logconvex functions? This is the author’s studying route.
In the paper, we attained the similar Hadamard inequality of the Choquet integral for a monotone logconvex function in the framework of distorted measure by Lemma 2.2 (which permits us to compute the Choquet integral when the nonadditive measure is a distorted Lebesgue measure). Then we estimated the upper bound of the Choquet integral for a general logconvex function, respectively, in the case of distorted Lebesgue measure and in the nonadditive measure using the basic properties of Choquet integrals. Finally, we studied Jensen’s inequality of the Choquet integral for logconvex functions by Lemma 2.3 (Jensen’s inequality of the Choquet integral).
Declarations
Acknowledgements
Not applicable.
Availability of data and materials
Not applicable.
Funding
This work is supported by the Scientific Research Foundation for Doctors of Henan University of Economics and Law, and by the University Key Research Project of Henan Province, China (18A110011).
Authors’ contributions
HW is the sole author of this paper. She wrote the manuscript, read and approved the final manuscript.
Competing interests
The author confirms that this work is original and has not been published elsewhere nor is it currently under consideration for publication elsewhere and there are no known conflicts of interest associated with this publication.
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
References
 Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953) MathSciNetView ArticleMATHGoogle Scholar
 Denneberg, D.: NonAdditive Measure and Integral. Kluwer Academic Publishers, Boston (1994) View ArticleMATHGoogle Scholar
 Denneberg, D.: Conditioning (updating) nonadditive measures. Ann. Oper. Res. 52, 21–42 (1994) MathSciNetView ArticleMATHGoogle Scholar
 Chen, Z., Kulperger, R.: Minimax pricing and Choquet pricing. Insur. Math. Econ. 38, 518–528 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Grabisch, M., Labreuche, C.: A decade of application of the Choquet and Sugeno integrals in multicriteria decision aid. Q. J. Oper. Res. 6, 1–44 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Huber, P., Strassen, V.: Minimax tests and the Neyman–Pearson lemma for capacities. Ann. Stat. 1, 251–263 (1973) MathSciNetView ArticleMATHGoogle Scholar
 Kadane, J., Wasserman, L.: Symmetric, coherent, Choquet capacities. Ann. Stat. 24(1), 1250–1264 (1996) MathSciNetMATHGoogle Scholar
 Klement, E.P., Mesiar, R., Pap, E.: A universal integral as common frame for Choquet and Sugeno integral. IEEE Trans. Fuzzy Syst. 18, 178–187 (2010) View ArticleGoogle Scholar
 Mesiar, R., Li, J., Pap, E.: The Choquet integral as Lebesgue integral and related inequalities. Kybernetika 46, 931–934 (2010) MathSciNetMATHGoogle Scholar
 Mesiar, R., Li, J., Pap, E.: Pseudoconcave integrals. In: Nonlinear Mathematics for Uncertainty and Its Applications, pp. 43–49. Springer, Berlin (2011) View ArticleGoogle Scholar
 Murofushi, T., Sugeno, M.: An interpretation of fuzzy measure and the Choquet integral as an integral with respect to a fuzzy measure. Fuzzy Sets Syst. 29, 201–227 (1989) MathSciNetView ArticleMATHGoogle Scholar
 Pap, E.: NullAdditive SetFunctions. Kluwer, Dordrecht (1995) MATHGoogle Scholar
 Wang, Z., Yan, J.: A selective overview of applications of Choquet integrals. In: Advanced Lectures in Mathematics, pp. 484–515. Springer, Berlin (2007). (2007) Google Scholar
 Wang, S., Young, V., Panjer, H.: Axiomatic characterization of insurance prices. Insur. Math. Econ. 21, 17–183 (2007) MathSciNetMATHGoogle Scholar
 Wang, R.S.: Some inequalities and convergence theorems for Choquet integral. J. Appl. Math. Comput. 35, 305–321 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Wasserman, L., Kadane, J.: Bayes’ theorem for Choquet capacities. Ann. Stat. 18, 1328–1339 (1990) MathSciNetView ArticleMATHGoogle Scholar
 Yan, J.: A short presentation of Choquet integral. In: Duan, J., Luo, S., Wang, C. (eds.) Recent Developments in Stochastic Dynamics and Stochastic Analysis. Interdisciplinary Mathematical Sciences, vol. 8, pp. 269–291. World Scientific, Singapore (2009). https://doi.org/10.1142/9789814277266_0017 View ArticleGoogle Scholar
 Bagnoli, M., Bergstrom, T.: Logconcave probability and its applications. Econ. Theory 26(2), 445–469 (2005) MathSciNetView ArticleMATHGoogle Scholar
 Abbaszadeh, S., Eshaghi, M., Sen, M.: The Sugeno fuzzy integral of logconvex functions. J. Inequal. Appl. 2015, Article ID 362 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Sugeno, M.: A note on derivatives of functions with respect to fuzzy measures. Fuzzy Sets Syst. 222, 1–17 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, X., Jiang, W.: Some properties of logconvex function and applications for the exponential function. Comput. Math. Appl. 63, 1111–1116 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Gill, P.M., Pearce, C.E.M., Pec̆arić, J.: Hadamard’s inequality for rconvex functions. J. Math. Anal. Appl. 215, 461–470 (1997) MathSciNetView ArticleMATHGoogle Scholar
 Dieudonne, J.: Foundations of Modern Analysis, Enlarged and Corrected Printing. Academic Press, San Diego (1969) MATHGoogle Scholar
 Kallenberg, O.: Foundation of Modern Probability, 2nd edn. Springer, Berlin (2002) View ArticleMATHGoogle Scholar