Abstract Convexity and Hermite Hadamard Type Inequalities

Volume 2009, Article ID 943534,13pages doi:10.1155/2009/943534

Research Article

Abstract Convexity and Hermite-Hadamard

Type Inequalities

Gabil R. Adilov

_{and Serap Kemali}

1_{Department of Primary Education, Faculty of Education, Mersin University, 33169 Mersin, Turkey} 2_{Vocational School of Technical Sciences, Akdeniz University, 07058 Antalya, Turkey}

Correspondence should be addressed to Serap Kemali,skemali@akdeniz.edu.tr

Received 24 February 2009; Accepted 8 May 2009

Recommended by Kunquan Lan

The deriving Hermite-Hadamard type inequalities for certain classes of abstract convex functions are considered totally, the inequalities derived for some of these classes before are summarized, new inequalities for others are obtained, and for one class of these functions the results onR2

are

generalized toR_n. By considering a concrete area inRn

, the derived inequalities are illustrated.

Copyrightq2009 G. R. Adilov and S. Kemali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Studying Hermite-Hadamard type inequalities for some function classes has been very important in recent years. These inequalities which are well known for convex functions have been also found in different function classessee1–5.

Abstract convex function is one of this type of function classes. Hermite-Hadamard type inequalities are studied for some important classes of abstract convex functions, and the concrete results are found 6–9. For example, increasing convex-along-rays ICAR functions, which are defined inR2

, are considered in8. First, a correct inequality for these

functions is given. For eachx0, y0∈R2/{0}there exists a numberbx0, y0>0 such that

bx0, y0

min x

x0,

y y0 −

1

≤fx, y−fx0, y0

1.1

for allx, y.

max

x,y∈QDf

x, y≤ 1 AD

D

fx, ydx dy 1.2

inequality is correct, where

QD x, y∈D⊂R2

: _A1_D

D

min

x x,

y

y dx dy1

, 1.3

andADis the area of domainD.

Similar inequalities are found for increasing positively homogenousIPHfunctions in

6, for increasing radiantInRfunctions in9, and for increasing coradiantICRfunctions in7.

In this article, first, the theorem which yields inequality 1.1 is proven for ICAR functions defined on Rn

, then the other inequalities are generalized, which based on this

theorem.

Another generalization is made forQD. A more covering setQkDis considered

and all resultsfor IPH, ICR, InR, ICAR functionsare examined for this set.

2. Abstract Convexity and Hermite-Hadamard

Type Inequalities

LetRbe a real line andR∞R∪ {∞}. Consider a setXand a setHof functionsh:X → R defined onX.A functionf : X → R∞ is called abstract convex with respect toHorH -convexif there exists a setU⊂Hsuch that

fx sup{hx:h∈U} ∀x∈X. 2.1

ClearlyfisH-convex if and only if

fx suphx:h≤f ∀x∈X. 2.2

LetY be a set of functionsf :X → R∞.A setH⊂ Y is called a supremal generator of the setY if each functionf∈Y is abstract convex with respect toH.

2.1. Increasing Positively Homogeneous Functions and

Hermite-Hadamard Type Inequalities

A functionf defined on Rn

{x1, x2, . . . , xn ∈ Rn : x1 ≥ 0, x2 ≥ 0, . . . , xn ≥ 0} is called

The functionfis positively homogeneous of degree one if

fλx λfx 2.3

for allx∈Rn

andλ >0.

LetLbe the set of all min-type functions defined on

Rn{x1, x2, . . . , xn∈Rn:x1 >0, x2>0, . . . , xn>0}, 2.4

that is, the setLconsists of identical zero and all the functions of the form

lx l, xmin

i

xi

li, x∈R n

2.5

with alll∈Rn

.

One has that a functionf : Rn

→ RisL−convex if and only iff is increasing and

positively homogeneous of degree oneIPHfunctionssee10.

The Hermite-Hadamard type inequalities are shown for IPH functions by using the following proposition which is very important for IPH functions.

Proposition 2.1. Letf be an IPH function defined onRn

.Then the following inequality holds for allx, l∈Rn

:

fll, x ≤fx. 2.6

Proposition 2.2can be easily shown by using theProposition 2.1see6.

Proposition 2.2. LetD ⊂ Rn

, f : D → R∞be an IPH function, and letf be integrable on D. Then

fu

D

u, xdx≤

D

fxdx 2.7

for allu∈D,and this inequality is sharp.

Unlike the previous work, inequality 2.7 obtained for IPH functions and inequalities in the type of 2.7 will be obtained for different function classes are going to be inquired for more general theQkDsets not for theQDset.QkDwill be certainly

different for each function class. LetD ⊂Rn

be a closed domain, that is, clintD D,and letkbe positive number.

LetQkDbe the set of all pointsx∗∈Dsuch that

k AD

D

x∗, xdx1, 2.8

In the case ofk1, Q1Dwill be the setQDin8,9.

In6, Proposition 3.2, the proposition has been given forQD, the same proposition is defined forQkDas follows, and its proof is similar.

Proposition 2.3. Letf be an IPH function defined on D.If the setQkDis nonempty andf is

integrable onD, then

sup

x∗_∈QkDfx

∗_≤ k

AD

D

fxdx. 2.9

We had proved a proposition in6by using a function u, x max1≤i≤nxi/ui,

and we get a right-hand side inequality, similar to2.7.

Proposition 2.4. Letfbe an IPH function, and letfbe integrable function onD.Then

D

fxdx≤ inf

u∈D

fu

D

u, x dx

. 2.10

For everyu∈Dthe inequality

D

fxdx≤fu

D

u, x dx 2.11

is sharp.

2.2. Increasing Positively Homogeneous Functions and

Hermite-Hadamard Type Inequalities

A functionf :Rn

→ R∞is called increasing radiantInRfunction if

1fis increasing;

2fis radiant; that is,fλx≤λfxfor allλ∈0,1, andx∈Rn

.

Consider the coupling functionϕdefined onRn

×Rn

ϕu, x

⎧ ⎨ ⎩

0, if u, x<1,

u, x, ifu, x ≥1.

2.12

Denote byϕuthe function defined onRnby the formula

It is known that the set

U

1

cϕu:u∈R

n

, c∈0,∞

2.14

is supremal generator of all increasing radiant functions defined onRn

see9.

Note that forc ∞we getcϕux sup_h>0hϕux.

The very important property for InR functions is given here in after. It can be easily proved.

Proposition 2.5. Letfbe an InR function defined onRn

.Then the following inequality holds for all x, l∈Rn

:

flϕl, x≤fx. 2.15

By using9, Proposition 2.5, the following proposition is proved.

Proposition 2.6. LetD⊂Rn

, f :D → R∞be InR functions and integrable onD. Then

fu

D

ϕu, x≤

D

fxdx 2.16

for allu∈D.This inequality is sharp for anyu∈Dsince one has the inequality in [9] forfx ϕux.

We determine the setQkDfor InR functions. LetQkDbe the set of all pointsx∗∈D

such that

k AD

D

ϕx∗, xdx1, 2.17

which is given in9, Proposition 3.1can be generalized forQkD.

Proposition 2.7. Letf be an InR function defined on Rn.If the setQkDis nonempty andf is

integrable onD,then

sup

x∈QkD

fx≤ k AD

D

fxdx. 2.18

Proof. The proof of the proposition can be made in a similar way to the proof in9, Proposition 3.1.

Now, we will study to achieve right-hand side inequality for InR functions. First, Let us prove the auxiliary proposition.

Proposition 2.8. Letfbe an InR function onD.Then the following inequalities hold for alll, x∈D:

where

ϕxl ⎧ ⎨ ⎩

x, l, if x, l≤1,

∞, if x, l>1.

2.20

Proof. Sincefis InR function onD,then

flϕx≤fx 2.21

for allx, l∈D.From this

fll, x ≤ fx, ifl, x ≥1. 2.22

That is,

fl≤ x, lfx, ifl, x≤1. 2.23

If we consider the definition ofϕxl,then

fl≤ϕxlfx 2.24

for allx, l∈D.

Proposition 2.9. Letfbe an InR function and integrable onD,u∈Dand

Du x∈D| u, x≤1, 2.25

then

Dufxdx≤fu

Duϕ

uxdx 2.26

holds and is sharp since we get equality forfx u, x.

Proof. It follows fromProposition 2.8.

Corollary 2.10. Letfbe an InR function and integrable onD.Ifu∈D andu≥ xfor allx∈ D,

then

D

fxdx≤fu

D

u, xdx 2.27

2.3. Increasing Coradiant Functions and Hermit-Hadamard

Type Inequalities

A functionf :K → R∞defined on a coneK⊂Rnis called coradiant if

fλx≥λfx ∀x∈K, λ∈0,1. 2.28

It is easy to check thatfis coradiant if and only if

fνx≤νfx ∀x∈K, ν≥1. 2.29

We will consider increasing coradiantICRfunction defined on the coneRn

.

Consider the functionΨldefined onRn:

Ψlx ⎧ ⎨ ⎩

l, x, ifl, x ≤1,

1, ifl, x>1, 2.30

wherel∈Rn

.

Recall that the set

H{cΨl:l∈Rn , c∈0,∞} 2.31

is supremal generator of the class ICR functions defined onRn

see10.

The Hermit-Hadamard type inequalities have been obtained for ICR functions by using the following proposition in7.

Proposition 2.11. Letfbe an ICR function defined onRn

.Then the following inequality holds for allx, l∈Rn

:

flΨlx≤fx. 2.32

Proposition 2.12. LetD ⊂ Rn

, f : D → R∞ be ICR function and integrable onD.Then the following inequality holds for allu∈D:

fu

D

Ψuxdx≤

D

fxdx, 2.33

and it is sharp.

The setQkDis defined for ICR function, namely,QkDdenotes the set of all points

x∗∈Dsuch that

k AD

D

Proposition 2.13. Letfbe an ICR function onD.If the setQkDis nonempty andfis integrable

onD, then

sup

x∈QkD

fx≤ k AD

D

fxdx. 2.35

Let us define a new functionΨ_uxsuch that

Ψ

ux ⎧ ⎨ ⎩

u, x, ifu, x ≥1,

1, ifu, x <1, 2.36

whereu, x is max-type function. By including the new function Ψux, we can achieve

right-hand side inequalities for ICR functions, too.

Proposition 2.14. Let functionfbe an ICR function and integrable onD.Then

D

fxdx≤min

u∈D

fu

D

Ψ

uxdx

, 2.37

and for everyu∈Dthe inequality

D

fxdx≤fu

D

Ψ

uxdx 2.38

is sharp.

2.4. Increasing Convex Along Rays Functions and

Hermit-Hadamard Type Inequalities

The Hermite-Hadamard type inequalities are studied for ICAR functions in8. But only the functions which are defined onR2

are considered.

In this article, the functions which are defined on Rn

are considered, and general

results are found.

LetK ⊂Rn _{be a conic set. A functionf :K → R}

∞is called convex-along-rays if its restriction to each ray starting from zero is a convex function of one variable. In other words, it means that the function

fxt ftx, t≥0 2.39

is convex for eachx∈K.

In this paper we consider increasing convex-along- raysICARsfunctions defined on

KRn

.

It is known that a finite ICAR function is continuous on the Rn

and lower

semicontinuous onRn

in10.

Theorem 2.15. LetHLbe the class of all functionshdefined by

hx l, x −c, 2.40

wherel, xis a min-type function andc∈R.A functionf:Rn

→ R∞isHL-convex if and only if

fis lower semicontinuous and ICAR.

Theorem 2.16. Letfbe ICAR function, and letx∈Rn

\ {0}be a point such that1εx∈domf for someε >0.Then the sup differential

∂Lfx≡

l∈L:l, y− l, x ≤fy−fx 2.41

is not empty and

_u

x :u∈∂fx1

⊂∂Lfx, 2.42

wherefxt ftx.

Now, we can define the following theorem which is important to achieve Hermit-Hadamard type inequalities for ICAR functions.

Theorem 2.17. Letfbe a finite ICAR function defined onRn

.Then for eachy∈Rn\{0}there exists a numberbby>0such that

by, x−1≤fx−fy 2.43

for allx.

Proof. The result follows directly fromTheorem 2.16.

We will applyTheorem 2.17in the study of Hermit-Hadamard type inequalities for ICAR functions.

Proposition 2.18. LetD⊂Rn,f:D → Rbe ICAR function. Then the following inequality holds for allu∈D:

bu

D

u, x −1dxfuAD≤

D

fxdx. 2.44

Proof. It follows fromTheorem 2.17.

Formula2.44can be made simply with the setsQD. LetD⊂Rn

be a bounded set such that clintD Dand

QD≡

x∗∈D| 1

AD

D

x∗, xdx1

Proposition 2.19. Let the setQDbe nonempty, and letfbe a continuous ICAR function defined onD.Then the following inequality holds:

max

u∈QDfu≤

1

AD

D

fxdx. 2.46

Proof. Letu∈QD.It follows from2.43and the definition ofQDthat

0bu

1

AD

D

u, xdx−1

1

AD

D

buu, x −1dx

≤ 1

AD

D

fx−fudx.

2.47

Thus

fu≤ 1 AD

D

fxdx. 2.48

SinceQDis compactsee8andf is continuousfinite ICAR functions is continuous, it follows that the maximum in2.46is attained.

Remark 2.20. Inequalities 2.9, 2.18, 2.35, and 2.46, which are obtained for different convex classes, are actually different, even if they appear to be the same. The reason is that these are determined with the2.8,2.17,2.34, and2.45formulas appropriate for the sets ofQkDand also yielding different sets.

3. Examples

The results of different classes of convex functions are defined for same triangle regionD ⊂ R2

:

Dx1, x2∈R2 : 0< x1≤a,0< x2≤vx1

. 3.1

The inequalities2.7and2.10 have been defined for IPH functions. The inequalities are examined for the regionDin6. If we combine two results, then we get

a32u1v−u2

6u2₁ fu1, u2≤

D

fx1, x2dx≤ a 3

6

u2

u2₁ v2 u2

fu1, u2 3.2

If we study the setQkDfor IPH functions, a pointx₁∗, x₂∗ ∈Dbelongs toQkDif

and only if

x∗₂−3v ak

x∗₁22vx∗₁. 3.3

That is, the setQkDis intersection with the setDand the parabola by formula3.3.

Let us consider the InR functions for same regionD.The inequality2.16has been examined forD,and the following inequality has been obtained in9:

va3

3u1 −

vu2 1 3 u2

2u1

3 −

a

2 −

a3

6u2 1

fu1, u2≤

D

fx1, x2dx1dx2 3.4

for allu1, u2∈D.

Let us study on the right-hand side inequality2.26, which is obtained in this article, for same regionD, which has been defined as follows:

Du x∈D:u, x≤1 3.5

for allu∈D.

We will separate two sets:

D1u

x∈D: x2

u2 ≤

x1

u1 ≤

1

x∈D: 0≤x1≤u1,0≤x2≤ u2

u1x1

,

D2u

x∈D:x1

u1 ≤

x2

u2 ≤

1

x∈D: 0≤x2≤u2, x2

v ≤x1≤ u1

u2x2

,

3.6

such thatDu D1u∪D2u.

In this case, we get

Duu, x _dx

1dx2

1

u1

D1u

x1dx1dx2

1

u2

D2u

x2dx1dx2

1 u1

_u1

0

_u2/u1x1

0

x1dx1dx2

1

u2 _u1

0

_u1/u2x2

0

x2dx1dx2

2vu1u2−u22

3v .

3.7

Thus, the inequality2.26becomes

Du

fx1, x2dx1dx2 ≤

2vu1u2−u22

3v fu1, u2 3.8

The setQkDcan be defined for InR functions such that, a pointx∗belongs toQkD

if and only if

x∗₂

1 3

a2

x∗₁2− 4 a3

x∗₁3 vx∗₁

2− 3

akx

∗ 1−

2

a3

x₁∗3 . 3.9

The inequalities 2.33 and 2.35 had been obtained for ICR functions. If these inequalities are examined for the same triangle region D, then the following inequality is obtained in7:

1 6

2u1u23a2v−vu21−3au2

fu1, u2≤

D

fx1, x2dx1dx2fu1, u2

≤ 6vu21u2

a3_vu2

22vu31u22a3v3u21−u21u32

3.10

for allu∈D.

The setQkDhas been obtained for ICR functions as formula2.34. Formula2.34

becomes formula3.11for the triangle regionD.That is a pointx∗belongs toQkDif and

only if

2x∗₁x₂∗−3ax₂∗−vx₁∗2 3va

2

k . 3.11

Lastly, formula2.44has been defined for ICAR functions. Now, we will define the same formula for the triangle regionD:

bu1, u2

a3_2u

1v−u2

6u2 1

− a2v

2

a2v

2 fu1, u2≤

D

fx1, x2dx1dx2, 3.12

and the inequality is held for allu∈D, wherebu1, u2is parameter which depends onfsee

10.

ICAR functions had been studied for the setQDwhich is determined by formula

2.45. Ifk 1 inQkD, then the setQDis special case of the given formula2.8. Then a

pointx∗∈Dbelongs toQDif and only if

x2−3v

a x

2

12vx1. 3.13

4. Conclusion

Hermite-Hadamard type inequalities are investigated for specific functions classes. One of these functions classes is abstract convex functions. The deriving Hermite-Hadamard type inequalities for IPH, InR, ICR, and ICAR functions, which are important classes of abstract convex functions, are investigated by different authors6–10.

In this article, this problem is considered entirely; findings from 6–10 are summarized; new results are found for some classes; results of some classes are generalized. For example, all results are found for more generalQkDcase, not all forQD. Even though

the results, 2.9, 2.18, 2.35, 2.46, are similar in appearance, they represent different inequalities, since the sets, which are defined with formulas2.8,2.17,2.34, and2.45, for different classes, are different.

Right-hand side inequalities, which are found for InR functions classes in 9, are considered here as well; more general results are found with the support ofϕuxfunctions

and explained asProposition 2.9.

ICAR functions, which are studied in8, are investigated onR2

here, and results are

explained inProposition 2.18. The inequality, which is explained in formula2.44, is a new inequality for these functions classes.

Finally, all the results are explained for the same region given onR2

. Formulas3.2, 3.8,3.10, and3.12are concrete results of Hermit-Hadamard type inequalities of different abstract convex function classes on given triangle region. Formulas3.3,3.9,3.11, and

3.13are concrete explanations ofQkDsets in this region.

Acknowledgments

The first author was supported by the Scientific Research Project Administration Unit of Mersin University Turkey. The second author was supported by the Scientific Research Project Administration Unit of Akdeniz UniversityTurkey.

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