Elsevier

Fuzzy Sets and Systems

Volume 430, 28 February 2022, Pages 126-143
Fuzzy Sets and Systems

Generalized pseudo-integral Jensen's inequality for ((⊕1,⊗1),(⊕2,⊗2))-pseudo-convex functions

https://doi.org/10.1016/j.fss.2021.06.007Get rights and content

Abstract

It is remarked that the generalization of Jensen's inequality for pseudo-integrals (Pap and Štrboja [14]) is not a complete generalization of the classical Jensen's inequality, and a generalized Jensen's inequality for pseudo-integral with respect to (,)-pseudo-convex function is given in [28]. The present article is a continuation of the previous work. A new notion of (1,1),(2,2)-pseudo-convex function is introduced, which generalize the notion of (,)-pseudo-convex function and many other previously generalizations. Motivated by the work of Kaluszka et al. [6], related to Jensen's inequality with respect to different generalized fuzzy integrals, a new generalized Jensen's inequality between a pseudo-integral and general fuzzy integral, as well as between two different pseudo-integrals, with respect to ((1,1),(2,2))-pseudo-convex functions are proved. These results cover all previously obtained Jensen's inequalities for pseudo-integrals (Zhang and Pap [28]) as well as the classical Jensen's inequality.

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 φ:[a,b]R is convex and functions f:[0,1][a,b] and φf:[0,1]R are integrable, thenφ([0,1]f(x)dx)[0,1]φ(f(x))dx.

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 φ:[a,b][a,b] be a convex and nonincreasing function. If a generator g:[a,b][a,b] of the pseudo-additionand the pseudo-multiplicationis a convex and strictly increasing function, then for any measurable function f:[0,1][a,b], we haveφ([0,1]f(x)dx)[0,1]φ(f(x))dx.

Unfortunately there are three shortcomings:

  • (i)

    the range of g should be [0,] (not [a,b]);

  • (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 ([a,b],,) be a semiring having pseudo-inverses and m(X)=1. If φ:[a,b][a,b] is a continuous (,)-pseudo-convex function, then for any measurable function f:X[a,b], we haveφ(Xfdm)X(φf)dm.

Although the obtained Jensen's inequality (1.3) cover the classical one and the above shortcomings are overcome, it is only for single pseudo-integral. It is well known that Jensen's inequality with two different generalized fuzzy integrals is given by Kaluszka et al. [6]. Motivated by this work, we investigate Jensen's inequality with two different pseudo-integrals in this paper. A new notion of (1,1),(2,2)-pseudo-convex functions is introduced, which covers the (,)-pseudo-convex function and many other generalizations. It is proved that there exists a Jensen's inequality between pseudo-integral and general fuzzy integral with respect to ((1,1),(2,2))-pseudo-convex functions. In order to further elaborate the work of Kaluszka et al. [6], a generalized Jensen's inequality with two different pseudo-integrals with respect to ((1,1),(2,2))-pseudo-convex functions is obtained, which renders all previously obtained Jensen's inequalities for pseudo-integrals [28], as well as the classical Jensen's inequality as very special case. Hence the paper can be viewed as a continuation of our previous work [28].

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 ((1,1),(2,2))-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 ((1,1),(2,2))-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 ((1,1),(2,2))-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 [a,b] be a closed (in some cases can be considered semiclosed) subinterval of [,]. The full order on [a,b] will be denoted by ⪯. A binary operation ⊕ on [a,b] is pseudo-addition, if it is commutative, nondecreasing (with respect to ⪯), associative and with a zero (neutral) element denoted by 0. Let [a,b]+[a,b] with 0x. A binary operation ⊗ on [a,b] is

((1,1),(2,2))-pseudo-convex function

In this section, we introduce the notion of ((1,1),(2,2))-pseudo-convex functions from a semiring to another semiring.

Let ([a,b],,) be a semiring. A subset B of [a,b] is said to be (,)-pseudo-convex if and only if for all x,yB and λ,μ[a,b]+ with λμ=1 imply λxμyB. It was shown that any sub-interval of [a,b] is (,)-pseudo-convex for Case II, see [5].

Definition 3.1

Let ([ai,bi],i,i)(i=1,2) be semirings with [a1,b1][a2,b2] and B[a1,b1] a (1,1)-pseudo-convex set. A function φ:B[a2,b2] 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 Si=([ai,bi],i,i)(i=1,2) be semirings with [a1,b1][a2,b2] and a1=a2=01. Suppose that full orders 1 and 2 coincide on [a1,b1] and pseudo-inverses in S1 exist and let m1:A[a1,b1] be a σ-1-measure with m1(X)=11 and φ:[a1,b1][a2,b2] be a continuous function.

  • (i)

    (Generalized Jensen's inequality I) If φ is nondecreasing and ((1,1),(2,2))-pseudo-convex, then for any measurable function f:X[a1,b1

Modification of ((1,1),(2,2))-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 [a1,b1][a2,b2].

Definition 5.1

Let ([ai,bi],i,i)(i=1,2) be semirings endowed with generators g1 and g2, respectively. Let B[a1,b1] be a (1,1)-pseudo-convex set. A function φ:B[a2,b2] is ((1,1),(2,2))-pseudo-convex (resp., pseudo-concave), if there holds:φ(λ1x1μ1y)2(g21g1)(λ)2φ(x)2(g21g1)(μ)2φ(y),(resp., φ(λ1x1μ1y)2(g21g1)(λ)2φ(x)2(g21g1)(μ)2φ(y)) for all x,

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 ([ai,bi],i,i)(i=1,2) be semirings and mi:A[ai,bi]

Concluding remarks

We have introduced a new concept of (1,1),(2,2)-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).

References (30)

Cited by (5)

  • Classical inequalities for pseudo-integral

    2022, Georgian Mathematical Journal
View full text