The Choquet integral of log-convex functions

Hongxia Wang

doi:10.1186/s13660-018-1803-y

Hongxia Wang

Open Access

https://doi.org/10.1186/s13660-018-1803-y

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Abstract

In this paper we investigate the upper bound and the lower bound of the Choquet integral for log-convex functions. Firstly, for a monotone log-convex 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 log-convex function, respectively, in the case of distorted Lebesgue measure and in the non-additive measure. Finally, we present Jensen’s inequality of the Choquet integral for log-convex functions, which can be used to estimate the lower bound of this kind when the non-additive measure is concave. We provide some examples in the framework of the distorted Lebesgue measure to illustrate all the results.

Highlights

It is well known that the concept of non-additive measure [1] can be used to deal with some uncertainty phenomena which cannot be modeled by using additive measure, and the Choquet integral, which covers the classical Lebesgue integral, is one kind of nonlinear expectations
The upper bound of the Choquet integral for the general logconvex function is presented
Observe that Theorems 3.1 and 3.2 are based on the assumption that the log-convex function is monotone. This suggests an open question: Can we find the upper bound of the Choquet integral when the log-convex function is not monotone? In the following we shall present some results concerning this issue

Summary

Introduction

It is well known that the concept of non-additive measure [1] can be used to deal with some uncertainty phenomena which cannot be modeled by using additive measure, and the Choquet integral, which 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. 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 log-convexity. It is known that every log-convex function is convex. Assumptions about the log-convexity of a probability distribution allow just enough special structure to yield a workable theory. The log-convexity (log-concavity) 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]

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