Generalized pseudo-integral Jensen's inequality for ((⊕1,⊗1),(⊕2,⊗2))-pseudo-convex functions
Introduction
It is well-known that Jensen's inequality is a part of the classical mathematical analysis [17], and plays an important role in other fields of mathematics, such as probability theory, optimization, control theory, etc. The classical Jensen's inequality is as follows: Theorem 1.1 [17] If the given function is convex and functions and are integrable, then
Recently, Jensen's inequality is generalized with respect to various kinds of non-additive integrals, such as Sugeno integral [1], [16], generalized Sugeno integral [3], [6], extremal universal integrals [11], Choquet integral [8], [22], integrals for fuzzy-interval-valued functions [4] and fuzzy set-valued functions [25]. It is important to stress that pseudo-analysis [12], [13], [14] as the generalization of the classical analysis unify the treatment of three important problems (usually nonlinear, under uncertainty, requiring optimization) in many different fields such as systems, optimization, decision making, control theory, differential equations, etc. The pseudo-integral [5], [9], [12], [13], [14], [19] is one of the main tools in pseudo-analysis. Recently, a generalization of Jensen's inequality for pseudo-integrals was managed by Pap and Štrboja [14]: Theorem 1.2 Theorem 4, [14] Let be a convex and nonincreasing function. If a generator of the pseudo-addition ⊕ and the pseudo-multiplication ⊗ is a convex and strictly increasing function, then for any measurable function , we have
Unfortunately there are three shortcomings:
- (i)
the range of g should be (not );
- (ii)
the inequality (1.2) is not a generalization of the classical Jensen's inequality (1.1), although the pseudo-integral is an extension of Lebesgue integral, which is the sufficient condition, the convex function φ is supposed to be nonincreasing, which is not necessary in the classical case;
- (iii)
the relation of φ and pseudo-operations does not need to be established.
To overcome the above shortcomings, in our previous paper [28], a notion of -pseudo-convex function was introduced, which covers the classical convex function, then Jensen's inequality for pseudo-integral was shown as follows: Theorem 1.3 Theorem 5.1, [28] Let be a semiring having pseudo-inverses and . If is a continuous -pseudo-convex function, then for any measurable function , we have
The rest of this paper consists of six sections. After the introduction, some basic concepts on pseudo-integrals are shown in preliminaries, Section 2. Then a -pseudo-convex function is introduced and investigated in Section 3. As one of the main results, in Section 4, a generalized Jensen's inequality between pseudo-integrals and general fuzzy integrals for -pseudo-convex functions is obtained. Then, in Section 5, a modification of the Jensen's inequality with respect to generators is introduced. Generalized Jensen's inequality with two different pseudo-integrals with respect to -pseudo-convex functions is obtained in Section 6.
Section snippets
Preliminaries
In this section, we recall some notions related to semirings and pseudo-integrals, which are adopted from [12], [13], [14]. Let be a closed (in some cases can be considered semiclosed) subinterval of . The full order on will be denoted by ⪯. A binary operation ⊕ on is pseudo-addition, if it is commutative, nondecreasing (with respect to ⪯), associative and with a zero (neutral) element denoted by 0. Let with . A binary operation ⊗ on is
-pseudo-convex function
In this section, we introduce the notion of -pseudo-convex functions from a semiring to another semiring.
Let be a semiring. A subset B of is said to be -pseudo-convex if and only if for all and with imply . It was shown that any sub-interval of is -pseudo-convex for Case II, see [5].
Definition 3.1 Let be semirings with and a -pseudo-convex set. A function is
A generalized Jensen's inequality for pseudo-integral and general fuzzy integral
We shall prove a generalized Jensen's inequality for pseudo-integral and general fuzzy integral. Theorem 4.1 Let be semirings with and . Suppose that full orders and coincide on and pseudo-inverses in exist and let be a σ--measure with and be a continuous function. (Generalized Jensen's inequality I) If φ is nondecreasing and -pseudo-convex, then for any measurable function
Modification of -pseudo-convex functions with generators
Suppose that the semirings are Cases I and II. We modify Definition 3.1 so that in this case, we do not need the condition . Definition 5.1 Let be semirings endowed with generators and , respectively. Let be a -pseudo-convex set. A function is -pseudo-convex (resp., pseudo-concave), if there holds: for all
Generalized Jensen's inequality with generators
Since Jensen's inequality for Sugeno integral has been proved earlier [1], [16], and Case III will be described in the subsequent sections, we suppose that all semirings are Cases I and II (it is possible to extend the obtained results for many interesting cases introduced in [2]). In this section, generalized Jensen's inequality is shown, and as its corollary, Jensen's inequality is obtained, these results cover Theorem 1.1, Theorem 1.2. Theorem 6.1 Let be semirings and
Concluding remarks
We have introduced a new concept of -pseudo-convex function, which generalize the notion of -pseudo-convex function [28] and some previously obtained generalizations. Based on this, generalized Jensen's inequalities between pseudo-integral and generalized fuzzy integral, as well as between pseudo-integral and pseudo-integral are obtained. They do not only generalize the classical Jensen's inequalities, but also generalizations of Pap and Štrboja's [14] and Zhang and Pap's
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to express their sincere thank to the editors-in-chief, the area editor and the unknown referees for their kind help. This study was supported by the National Natural Science Fund of China (No. 11271062) and the Natural Science Fund of Jilin Province (No. 20190201014JC) (for the first author) and supported by the Science Fund of the Republic of Serbia, #Grant No. 6524105, AI-ATLAS. (for the second author).
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