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R E S E A R C H

Open Access

Barnes-Godunova-Levin type inequality of

the Sugeno integral for an

,

m

)

-concave

function

Dong-Qing Li, Yu-Hu Cheng

*

, Xue-Song Wang and Shao-Fei Zang

*Correspondence:

chengyuhucumt204@163.com School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

Abstract

In this paper, a Barnes-Godunova-Levin type inequality for the Sugeno integral based on an (

α

,m)-concave function is proved. Some examples are given to illustrate the validity of these inequalities. Finally, several important results, as special cases of an (

α

,m)-concave function, are also obtained.

MSC: 03E72; 28B15; 28E10; 26D10

Keywords: Barnes-Godunova-Levin type inequality; Sugeno integral; (

α

,m)-concave function

1 Introduction

As a tool for modeling non-deterministic problems, the theory of fuzzy measures and fuzzy integrals was introduced by Sugeno in []. Many authors generalized the Sugeno in-tegral by using some other operators to replace the special operator(s)∨and/or∧and introduced Choquet-like integral [], Shilkret integral [],⊥-integral [], and pseudo-integral []. Suárez and Gil [] presented two families of fuzzy pseudo-integrals, the so-called seminormed fuzzy integral and semiconormed fuzzy integral. Wang and Klir [] provided a general overview on fuzzy measurement and fuzzy integration.

Recently, Flores-Franuličet al.[–] generalized several classical integral inequalities of the Sugeno integral. Agahiet al.[] proved a general Barnes-Godunova-Levin type inequality of the Sugeno integral for a concave function. In [], Mihesan introduced the concept of (α,m)-convex function. For recent results and generalizations concerningm -convex and (α,m)-convex functions, we refer to [–]. The purpose of this paper is to prove a Barnes-Godunova-Levin type inequality for the Sugeno integral based on an (α,m)-concave function. Some examples are given to illustrate the results.

After some preliminaries and summarization of previous known results in Section , Section  deals with a Barnes-Godunova-Levin type inequality for the Sugeno integral, and some examples are given to illustrate the results. Finally, as special cases, some remarks are obtained.

2 Preliminaries

In this section, we recall some basic definitions or properties of a fuzzy integral and an (α,m)-concave function. For details, we refer the reader to Refs. [, , ].

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Suppose that is aσ-algebra of the subsets of X, and let μ:→[,∞) be a non-negative, extended real-valued set function. We say thatμis a fuzzy measure if it satisfies:

() μ(∅) = ;

() E,FandEFimplyμ(E)≤μ(F);

() {En} ⊂,E⊂E⊂ · · ·, implylimn→∞μ(En) =μ(

n=En);

() {En} ⊂,E⊃E⊃ · · ·,μ(E) <∞, implylimn→∞μ(En) =μ(

n=En).

Definition .(Mihesan []) The functionf : [,b]→Ris said to be (α,m)-concave, where (α,m)∈[, ], if for everyx,y∈[,b] andt∈[, ], it satisfies

ftx+m( –t)yf(x) +m –tαf(y). (.)

Note that for (α,m)∈ {(, ), (α, ), (, ), (,m), (, ), (α, )} one obtains the following classes of functions: decreasing, α-starshaped, starshaped,m-concave, concave andα -concave.

Iff is a non-negative real-valued function defined onX, we denote the set{xX:f(x)≥

α}={x∈X:fα}byforα≥. Note that ifαβthen.

Let (X,,μ) be a fuzzy measure space, we denote by M+ the set of all non-negative

measurable functions with respect to.

Definition .(Sugeno []) Let (X,,μ) be a fuzzy measure space,fM+ andA. The Sugeno integral (or the fuzzy integral) off onA, with respect to the fuzzy measureμ, is defined as

(S)

A

f dμ=

α≥

αμ(A), (.)

whenA=X,

(S)

X

f dμ=

α≥

αμ(), (.)

where∨and∧denote the operations sup and inf on [,∞), respectively. The properties of the Sugeno integral are well known and can be found in [].

Proposition . Let(X,,μ)be a fuzzy measure space,A,B℘and f,gM+then:

() (S)Af dμμ(A);

() (S)Ak dμ=kμ(A),kfor a non-negative constant; () (S)Af dμ≤(S)Ag dμforfg;

() (S)ABf dμ≥(S)Af dμ∨(S)Bf dμ; () μ(A∩ {fα})≥α⇒(S)Af dμα; () μ(A∩ {f≥α})≤α⇒(S)Af dμα;

() (S)Af dμ>αthere existsγ >αsuch thatμ(A∩ {f ≥γ}) >α; () (S)Af dμ<αthere existsγ <αsuch thatμ(A∩ {f ≥γ}) <α.

Remark . Consider the distribution functionF associated to f onA, that is,F(α) =

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(S)Af dμ=α. Thus, from a numerical point of view, the Sugeno integral can be cal-culated solving the equationF(α) =α.

Definition . Functionsf,g:XRare said to be co-monotone if for allx,yX,

f(x) –f(y)g(x) –g(y)≥, (.) andf andgare said to be counter-monotone if for allx,yX,

f(x) –f(y)g(x) –g(y)≤. (.) It is clear that iff andgare co-monotone, then for any real numberss,teitherFsGt

orFtGs.

2.1 Barnes-Godunova-Levin type inequality for the Sugeno integral based on an (α,m)-concave function

The classical Barnes-Godunova-Levin type inequality provides the inequality

b

a

fp(x)dx

p b

a

gq(x)dx

q

B(p,q)

b

a

f(x)g(x)dx, (.)

wherep,q> ,B(p,q) = (b–a)

p+ q–

(+p)

p(+q)q andf,gare non-negative concave functions on [a,b].

Unfortunately, the following example shows that the Barnes-Godunova-Levin type in-equality for the Sugeno integral is not valid.

Example ConsiderX= [, ] andp=q= . Letmbe the Lebesgue measure onX. If we take the functionsf(x) =g(x) =√x, thenf(x),g(x) are two (

, 

)-concave functions. In

fact,

 √

x=f

x

· +  

 – x 

·

x

f() +  

 –

x



f() =

x

. A straightforward calculus shows that

(S)



f(x)dm= (S)



g(x)dm= ,

(S)



f(x)g(x)dm= ., B(, ) = ..

However,

. =

(S)



f(x)dm

(S)



g(x)dm

B(, )(S)



f(x)g(x)dm= ..

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The aim of this work is to show a Barnes-Godunova-Levin type inequality for the Sugeno integral with respect to an (α,m)-concave function.

Theorem . Let X= [, ],α,m∈(, )and f,g be(α,m)-concave functions for all xX.

If m is a Lebesgue measure on X,then Case(i).If f()≤f()and g()≤g(),then

(S)

f(x)g(x)dxtt∧

 –

t–mf()

f() –mf()

α

 –

t–mg()

g() –mg()

α

, (.)

where t= ((S)

fp(x)dx)

p,t

= ((S)

gq(x)dx)

q. Case(ii).If f() >f()and g() >g(),then Case(a).Ifff()()<g()g(),then

Case.If m∈(,ff()()),then

(S)

f(x)g(x)dxtt∧

 –

t–mf()

f() –mf()

α

 –

t–mg()

g() –mg()

α

, (.)

where t= ((S)

 f

p(x)dx)p,t

= ((S)

 g

q(x)dx)q. Case.If m=ff()(),then

(S)

f(x)g(x)dxtt∧∧f()∧

 –

t–mg()

g() –mg()

α

, (.)

where t= ((S)

fp(x)dx)

p,t

= ((S)

gq(x)dx)

q. Case.If m∈(ff()(),g()g()),then

(S)

f(x)g(x)dxtt∧

t–mf()

f() –mf()

α

 –

t–mg()

g() –mg()

α

, (.)

where t= ((S)

 f

p(x)dx)p,t

= ((S)

 g

q(x)dx)q. Case.If m=g()g(),then

(S)

f(x)g(x)dxtt∧

t–mf()

f() –mf()

α

∧∧g(), (.)

where t= ((S)

fp(x)dx)

p,t

= ((S)

gq(x)dx)

q. Case.If m∈(g()g(), ),then

(S)

f(x)g(x)dxtt∧

t–mf()

f() –mf()

α

t–mg()

g() –mg()

α

, (.)

where t= ((S)

fp(x)dx)

p,t

= ((S)

gq(x)dx)

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Case(b).Ifff()()=g()g(),then Case.If m∈(,ff()()),then

(S)

f(x)g(x)dxtt∧

 –

t–mf()

f() –mf()

α

 –

t–mg()

g() –mg()

α

, (.)

where t= ((S)

 f

p(x)dx)p,t

= ((S)

 g

q(x)dx)q. Case.If m=ff()(),then

(S)

f(x)g(x)dxtt∧f()∧g()∧, (.)

where t= ((S)

fp(x)dx)

p,t

= ((S)

gq(x)dx)

q. Case.If m∈(ff()(), ),then

(S)

f(x)g(x)dxtt∧

t–mf()

f() –mf()

α

t–mg()

g() –mg()

α

, (.)

where t= ((S)

 f

p(x)dx)p,t

= ((S)

 g

q(x)dx)q. Case(c).Ifff()()>g()g(),then

Case.If m∈(,g()g()),then

(S)

f(x)g(x)dxtt∧

 –

t–mf()

f() –mf()

α

 –

t–mg()

g() –mg()

α

, (.)

where t= ((S)

fp(x)dx)

p,t

= ((S)

gq(x)dx)

q. Case.If m=g()g(),then

(S)

f(x)g(x)dxtt∧

 –

t–mf()

f() –mf()

α

g()∧, (.)

where t= ((S)

fp(x)dx)

p,t

= ((S)

gq(x)dx)

q. Case.If m∈(g()g(),ff()()),then

(S)

f(x)g(x)dxtt∧

 –

t–mf()

f() –mf()

α

t–mg()

g() –mg()

α

, (.)

where t= ((S)

fp(x)dx)

p,t

= ((S)

gq(x)dx)

q. Case.If m=ff()(),then

(S)

f(x)g(x)dxtt∧f()∧

t–mg()

g() –mg()

α

, (.)

where t= ((S)

fp(x)dx)

p,t

= ((S)

gq(x)dx)

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Case.If m∈(ff()(), ),then

(S)

f(x)g(x)dxtt∧

t–mf()

f() –mf()

α

t–mg()

g() –mg()

α

, (.)

where t= ((S)

fp(x)dx)

p,t

= ((S)

gq(x)dx)

q. Proof Let p,q ∈ (,∞), ((S)fp(x)dx)p= t

 and ((S)

 g

q(x)dx)q =t

. Since f,g :

[, ]→[,∞) are two (α,m)-concave functions forx∈[, ], we have

f(x) =fm( –x)· +x·≥m –f() +xαf() =h

(x), (.)

g(x) =gm( –x)· +x·≥m –xαg() +xαg() =h(x). (.)

Case (i). If f()≤f() and g()≤g(), then by () of Proposition . and the co-monotonicity ofh(x) andh(x), we have

(S)

f(x)g(x)dx

≥(S)

h(x)h(x)dx

=

β≥

βμ[, ]∩h(x)h(x)≥β

tt∧μ

[, ]∩h(x)h(x)≥tt

tt∧μ

[, ]∩h(x)≥t

h(x)≥t

=tt∧μ

[, ]∩h(x)≥t

μ[, ]∩h(x)≥t

=tt∧μ

[, ]∩h(x)≥t

μ[, ]∩h(x)≥t

=tt∧μ

[, ]∩m –xαf() +xαf()≥t

μ[, ]∩m –xαg() +xαg()≥t

=tt∧μ

[, ]∩

x

t–mf()

f() –mf()

α

μ

[, ]∩

x

t–mg()

g() –mg()

α

=tt∧

 –

t–mf()

f() –mf()

α

 –

t–mg()

g() –mg()

α

. (.)

Case (ii). If f() >f() and g() >g(), then by () of Proposition . and the co-monotonicity ofh(x) andh(x), we have

(S)

f(x)g(x)dx ≥(S)

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=

β≥

βμ[, ]∩h(x)h(x)≥β

tt∧μ

[, ]∩h(x)h(x)≥tt

tt∧μ

[, ]∩h(x)≥t

h(x)≥t

=tt∧μ

[, ]∩h(x)≥t

μ[, ]∩h(x)≥t

=tt∧μ

[, ]∩h(x)≥t

μ[, ]∩h(x)≥t

=tt∧μ

[, ]∩m –xαf() +xαf()≥t

μ[, ]∩m –xαg() +xαg()≥t

=tt∧μ

[, ]∩mf() +f() –mf()t

μ[, ]∩mg() +g() –mg()t

. (.)

Case (a). If ff()()<g()g(), then by (.) we obtain Case . Ifm∈(,ff()()), then

(S)

f(x)g(x)dxtt∧

 –

t–mf()

f() –mf()

α

 –

t–mg()

g() –mg()

α

. (.) Case . Ifm=ff()(), then

(S)

f(x)g(x)dxtt∧∧f()∧

 –

t–mg()

g() –mg()

α

. (.)

Case . Ifm∈(ff()(),g()g()), then

(S)

f(x)g(x)dxtt∧

t–mf()

f() –mf()

α

 –

t–mg()

g() –mg()

α

. (.)

Case . Ifm=g()g(), then

(S)

f(x)g(x)dxtt∧

t–mf()

f() –mf()

α

g()∧. (.)

Case . Ifm∈(g()g(), ), then

(S)

f(x)g(x)dxtt∧

t–mf()

f() –mf()

α

t–mg()

g() –mg()

α

. (.)

Case (b). Ifff()()=g()g(), then by (.) we obtain Case . Ifm∈(,ff()()), then

(S)

f(x)g(x)dxtt∧

 –

t–mf()

f() –mf()

α

 –

t–mg()

g() –mg()

α

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Case . Ifm=ff()(), then

(S)

f(x)g(x)dxtt∧f()∧g()∧. (.)

Case . Ifm∈(ff()(), ), then

(S)

f(x)g(x)dxtt∧

t–mf()

f() –mf()

α

t–mg()

g() –mg()

α

. (.)

Case (c). Ifff()()>g()g(), then by (.) we obtain Case . Ifm∈(,g()g()), then

(S)

f(x)g(x)dxtt∧

 –

t–mf()

f() –mf()

α

 –

t–mg()

g() –mg()

α

. (.)

Case . Ifm=g()g(), then

(S)

f(x)g(x)dxtt∧

 –

t–mf()

f() –mf()

α

g()∧. (.)

Case . Ifm∈(g()g(),ff()()), then

(S)

f(x)g(x)dxtt∧

 –

t–mf()

f() –mf()

α

t–mg()

g() –mg()

α

. (.)

Case . Ifm=ff()(), then

(S)

f(x)g(x)dxtt∧∧f()∧

t–mg()

g() –mg()

α

. (.)

Case . Ifm∈(ff()(), ), then

(S)

f(x)g(x)dxtt∧

t–mf()

f() –mf()

α

t–mg()

g() –mg()

α

. (.)

This completes the proof.

Example ConsiderX= [, ] andp= ,q= . If we take the functionsf(x) =√x,g(x) =

x, thenf(x),g(x) are two (,) -concave functions. In fact,√tx=f(x· +

( –x)·)≥

xf() +

( –x

)f() =x fort≥

(9)

straightfor-ward calculus shows that

(S)

f(x)dm= (S)

g(x)dm= ., (S)

f(x)g(x)dm= ..

By Theorem ., we have

. = (S)

f(x)g(x)dx

(S)

f(x)dx

(S)

g(x)dx

 –

(S)

f(x)dx

 ∧

 –

(S)

g(x)dx



= .∧.∧. = .. (.)

Now, we will prove the general cases of Theorem ..

Theorem . Let X= [a,b],α,m∈(, )and f,g be(α,m)-concave functions for all xX.

Ifμis a Lebesgue measure on X,then Case(i).If f(a)≤f(b)and g(a)≤g(b),then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

α

(bma)

 –

t–mg(a)

g(b) –mg(a)

α

, (.)

where t= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

agq(x)dx)

q. Case(ii).If f(a) >f(b)and g(a) >g(b),then Case(a).Ifff(b)(a)<g(b)g(a),then

Case.If m∈(,ff(b)(a)),then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

α

(bma)

 –

t–mg(a)

g(b) –mg(a)

α

, (.)

where t= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

agq(x)dx)

q. Case.If m=ff(b)(a),then

(S)

b

a

f(x)g(x)dxtt∧(ba)∧f(b)

(bma)

 –

t–mg(a)

g(b) –mg(a)

α

, (.)

where t= ((S)

b a f

p(x)dx)p,t

= ((S)

b ag

(10)

(S)

b

a

f(x)g(x)dxtt∧

(bma)

t–mf(a)

f(b) –mf(a)

α

+maa

(bma)

 –

t–mg(a)

g(b) –mg(a)

α

, (.)

where t= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

agq(x)dx)

q. Case.If m=g(b)g(a),then

(S)

b

a

f(x)g(x)dxtt∧g(b)∧(ba)∧

(bma)

t–mf(a)

f(b) –mf(a)

α

+maa

, (.)

where t= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

agq(x)dx)

q. Case.If m∈(g(b)g(a), ),then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

t–mf(a)

f(b) –mf(a)

α

+maa

(bma)

t–mg(a)

g(b) –mg(a)

α

+maa

, (.)

where t= ((S)

b a f

p(x)dx)p,t

= ((S)

b ag

q(x)dx)q. Case(b).Ifff(b)(a)=g(b)g(a),then

Case.If m∈(,ff(b)(a)),then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

α

(bma)

 –

t–mg(a)

g(b) –mg(a)

α

, (.)

where t= ((S)

b a f

p(x)dx)p,t

= ((S)

b ag

q(x)dx)q. Case.If m=ff(b)(a),then

(S)

b

a

f(x)g(x)dxtt∧f(b)∧g(b)∧(ba), (.)

where t= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

agq(x)dx)

q. Case.If m∈(ff(a)(b), ),then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

t–mf(a)

f(b) –mf(a)

α

+maa

(bma)

t–mg(a)

g(b) –mg(a)

α

+maa

, (.)

where t= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

agq(x)dx)

(11)

Case(c).Ifff(b)(a)>g(b)g(a),then Case.If m∈(,g(b)g(a)),then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

α

(bma)

 –

t–mg(a)

g(b) –mg(a)

α

, (.)

where t= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

agq(x)dx)

q. Case.If m=g(a)g(b),then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

α

g(b)∧(ba),

(.)

where t= ((S)

b a f

p(x)dx)p,t

= ((S)

b ag

q(x)dx)q. Case.If m∈(g(b)g(a),ff(a)(b)),then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

α

(bma)

t–mg(a)

g(b) –mg(a)

α

+maa

, (.)

where t= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

agq(x)dx)

q. Case.If m=ff(b)(a),then

(S)

b

a

f(x)g(x)dxtt∧(ba)∧f(b)∧

(bma)

t–mg(a)

g(b) –mg(a)

α

+maa

, (.)

where t= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

agq(x)dx)

q. Case.If m∈(f(b)f(a), ),then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

t–mf(a)

f(b) –mf(a)

α

+maa

(bma)

t–mg(a)

g(b) –mg(a)

α

+maa

, (.)

where t= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

agq(x)dx)

q. Proof Let p,q ∈ (,∞), ((S)abfp(x)dx)p =t

 and ((S)

b

a gq(x)dx)

q = t

. Since f,g :

(12)

f(x) =f

m

 –xma

bma

·a+xma

bma·b

m

 –

xma bma

α f(a) +

xma bma

α

f(b) =h(x), (.)

g(x) =g

m

 –xma

bma

·a+xma

bma·b

m

 –

xma bma

α g(a) +

xma bma

α

g(b) =h(x). (.)

Case (i). If f(a)≤f(b) and g(a)≤g(b), then by () of Proposition . and the co-monotonicity ofh(x) andh(x), we have

(S)

b

a

f(x)g(x)dx

≥(S)

b

a

h(x)h(x)dx

=

β≥

βμ[a,b]∩h(x)h(x)≥β

tt∧μ

[a,b]∩h(x)h(x)≥tt

tt∧μ

[a,b]∩h(x)≥t

h(x)≥t

=tt∧μ

[a,b]∩h(x)≥t

μ[a,b]∩h(x)≥t

=tt∧μ

[a,b]∩h(x)≥t

μ[a,b]∩h(x)≥t

=tt∧μ

[a,b]∩

m

 –

xma bma

α f(a) +

xma bma

α

f(b)≥t

μ

[a,b]∩

m

 –

xma bma

α g(a) +

xma bma

α

g(b)≥t

=tt∧μ

[a,b]∩

x

t–mf(a)

f(b) –mf(a)

α

(bma) +ma

μ

[a,b]∩

x

t–mg(a)

g(b) –mg(a)

α

(bma) +ma

=tt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

α

(bma)

 –

t–mg(a)

g(b) –mg(a)

α

. (.)

Case (ii). If f(a) >f(b) and g(a) >g(b), then by () of Proposition . and the co-monotonicity ofh(x) andh(x), we have

(S)

b

a

f(x)g(x)dx

≥(S)

b

a

(13)

=

β≥

βμ[a,b]∩h(x)h(x)≥β

tt∧μ

[a,b]∩h(x)h(x)≥tt

tt∧μ

[a,b]∩h(x)≥t

h(x)≥t

=tt∧μ

[a,b]∩h(x)≥t

μ[a,b]∩h(x)≥t

=tt∧μ

[a,b]∩h(x)≥t

μ[a,b]∩h(x)≥t

=tt∧μ

[a,b]∩

m

 –

xma bma

α f(a) +

xma bma

α

f(b)≥t

μ

[a,b]∩

m

 –

xma bma

α g(a) +

xma bma

α

g(b)≥t

=tt∧μ

[a,b]∩

mf(a) +f(b) –mf(a)xma

bma α

t

μ

[a,b]∩

mg(a) +g(b) –mg(a)x–ma

bma α

t

. (.)

Case (a). If ff(b)(a) <g(a)g(b), then by (.) we obtain Case . Ifm∈(,ff(b)(a)), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

α

(bma)

 –

t–mg(a)

g(b) –mg(a)

α

. (.)

Case . Ifm=ff(b)(a), then

(S)

b

a

f(x)g(x)dxtt∧(ba)∧f(b)

(bma)

 –

t–mg(a)

g(b) –mg(a)

α

. (.)

Case . Ifm∈(ff(b)(a),g(b)g(a)), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

t–mf(a)

f(b) –mf(a)

α

+maa

(bma)

 –

t–mg(a)

g(b) –mg(a)

α

. (.)

Case . Ifm=g(b)g(a), then

(S)

b

a

f(x)g(x)dxtt∧g(b)∧(ba)

(bma)

t–mf(a)

f(b) –mf(a)

α

+maa

(14)

Case . Ifm∈(g(b)g(a), ), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

t–mf(a)

f(b) –mf(a)

α

+maa

(bma)

t–mg(a)

g(b) –mg(a)

α

+maa

. (.)

Case (b). Ifff(b)(a)=g(b)g(a), then by (.) we obtain Case . Ifm∈(,ff(b)(a)), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

α

(bma)

 –

t–mg(a)

g(b) –mg(a)

α

. (.)

Case . Ifm=ff(b)(a), then

(S)

b

a

f(x)g(x)dxtt∧f(b)∧g(b)∧(ba). (.)

Case . Ifm∈(ff(b)(a), ), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

t–mf(a)

f(b) –mf(a)

α

+maa

(bma)

t–mg(a)

g(b) –mg(a)

α

+maa

. (.)

Case (c). Ifff(b)(a)>g(b)g(a), then by (.) we obtain Case . Ifm∈(,g(a)g(b)), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

α

(bma)

 –

t–mg(a)

g(b) –mg(a)

α

. (.)

Case . Ifm=g(b)g(a), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

α

(15)

Case . Ifm∈(g(b)g(a),ff(b)(a)), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

α

(bma)

t–mg(a)

g(b) –mg(a)

α

+maa

. (.)

Case . Ifm=f(b)f(a), then

(S)

b

a

f(x)g(x)dxtt∧f(b)∧(ba)∧

(bma)

t–mg(a)

g(b) –mg(a)

α

+maa

. (.)

Case . Ifm∈(ff(b)(a), ), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

t–mf(a)

f(b) –mf(a)

α

+maa

(bma)

t–mg(a)

g(b) –mg(a)

α

+maa

. (.)

This completes the proof.

Example ConsiderX= [, ] andp= ,q= . Letmbe the Lebesgue measure onX. If we take the functionf(x) =√ –x,g(x) =√ –x, thenf(x),g(x) are two (,

 ) -concave

functions. In fact,

 –x=f

  ·

 –x

  ×

 –

  ×

 +

x

  ×

 –

  ×

·

x

  ×

 –

  ×

 

f() +

 

 –

x

  ×

 –

  ×

 

f()

=( –

)x+ √

 –√ (.)

and

 √

 –x=g

  ·

 –x

  ×

 –

  ×

× +

x

  ×

 –

  ×

×

x

  ×

 –√  ×

 

g() +

 

 –

x

  ×

 –√  ×

 

g()

=( –

)x+ √ – 

(16)

A straightforward calculus shows that

(S)

f(x)dm= (S)

g(x)dm= , (S)

f(x)g(x)dm= ..

By Theorem ., we have

. = (S)

f(x)g(x)dx

(S)

f(x)dx

(S)

g(x)dx

 –

  ×

((S)f(x)dx)

 

  f()

f() –

  f()

+

  × – 

 –

  ×

((S)g(x)dx)

 

  g()

g() –

 g()

= .∧.∧. = .. (.) As some special cases of (α,m)-concave functions in Theorem ., we have the following results.

Remark . LetX= [a,b],α=m=  andf,gbe two decreasing functions for allxX. If

μis a Lebesgue measure onX, then

(S)

b

a

f(x)g(x)dx

(S)

b

a

fp(x)dx

p

(S)

b

a

gq(x)dx

q

f(b)∧g(b)∧(ba).

(.)

Remark . LetX= [a,b],α= ,m=  andf,gbe two starshaped functions for allxX. Ifμis a Lebesgue measure onX, then

(S)

b

a

f(x)g(x)dxtt∧b

 – t

f(b)

b

 – t

g(b)

, (.)

wheret= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

a gq(x)dx)

q.

Remark . LetX= [a,b],α= ,m∈(, ) andf,gbe twom-concave functions for all

xX. Ifμis a Lebesgue measure onX, then Case (i). Iff(a)≤f(b) andg(a)≤g(b), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

(bma)

 –

t–mg(a)

g(b) –mg(a)

, (.)

wheret= ((S)

b a f

p(x)dx)p,t

= ((S)

b a g

q(x)dx)q.

(17)

Case . Ifm∈(,ff(b)(a)), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

(bma)

 –

t–mg(a)

g(b) –mg(a)

, (.)

wheret= ((S)

b a f

p(x)dx)p,t

= ((S)

b a g

q(x)dx)q.

Case . Ifm=ff(b)(a), then

(S)

b

a

f(x)g(x)dxtt∧(ba)∧f(b)∧

(bma)

 –

t–mg(a)

g(b) –mg(a)

, (.)

wheret= ((S)

b a f

p(x)dx)p,t

= ((S)

b a g

q(x)dx)q.

Case . Ifm∈(ff(b)(a),g(b)g(a)), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

t–mf(a)

f(b) –mf(a)

+maa

(bma)

 –

t–mg(a)

g(b) –mg(a)

, (.)

wheret= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

a gq(x)dx)

q.

Case . Ifm=g(b)g(a), then

(S)

b

a

f(x)g(x)dxtt∧g(b)∧(ba)∧

(bma)

t–mf(a)

f(b) –mf(a)

+maa

, (.) wheret= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

a gq(x)dx)

q.

Case . Ifm∈(g(b)g(a), ), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

t–mf(a)

f(b) –mf(a)

+maa

(bma)

t–mg(a)

g(b) –mg(a)

+maa

, (.)

wheret= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

a gq(x)dx)

q.

Case (b). Ifff(b)(a)=g(b)g(a), then Case . Ifm∈(,ff(b)(a)), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

(bma)

 –

t–mg(a)

g(b) –mg(a)

, (.)

wheret= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

a gq(x)dx)

(18)

Case . Ifm=ff(b)(a), then

(S)

b

a

f(x)g(x)dxtt∧f(b)∧g(b)∧(ba), (.)

wheret= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

a gq(x)dx)

q.

Case . Ifm∈(ff(b)(a), ), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

t–mf(a)

f(b) –mf(a)

+maa

(bma)

t–mg(a)

g(b) –mg(a)

+maa

, (.)

wheret= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

a gq(x)dx)

q.

Case (c). Ifff(b)(a)>g(b)g(a), then Case . Ifm∈(,g(a)g(b)), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

(bma)

 –

t–mg(a)

g(b) –mg(a)

, (.)

wheret= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

a gq(x)dx)

q.

Case . Ifm=g(b)g(a), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

g(b)∧(ba), (.)

wheret= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

a gq(x)dx)

q.

Case . Ifm∈(g(b)g(a),ff(b)(a)), then

(S)

b

a

f(x)g(x)dxtt∧

(bma)

 –

t–mf(a)

f(b) –mf(a)

(bma)

t–mg(a)

g(b) –mg(a)

+maa

, (.)

wheret= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

a gq(x)dx)

q.

Case . Ifm=f(b)f(a), then

(S)

b

a

f(x)g(x)dxtt∧(ba)∧f(b)

(bma)

t–mg(a)

g(b) –mg(a)

+maa

, (.)

wheret= ((S)

b

a fp(x)dx)

p,t

= ((S)

b

a gq(x)dx)

q.

References

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