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On Hadamard-Type Inequalities Involving Several Kinds of Convexity

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Set, Erhan, Ozdemir, Emin and Dragomir, Sever S (2010) On Hadamard-Type Inequalities Involving Several Kinds of Convexity. Journal of Inequalities and Applications, 2010. pp. 1-12. ISSN 1025-5834 (print) 1029-242X (online)

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Volume 2010, Article ID 286845,12pages doi:10.1155/2010/286845

Research Article

On Hadamard-Type Inequalities Involving Several Kinds of Convexity

Erhan Set,

¹

M. Emin ¨ Ozdemir,

¹

and Sever S. Dragomir

^{2, 3}

1Department of Mathematics, K.K. Education Faculty, Atat ¨urk University, Campus, 25240 Erzurum, Turkey

2Research Group in Mathematical Inequalities & Applications, School of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne City, VIC 8001, Australia

3School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

Correspondence should be addressed to Erhan Set,erhanset@yahoo.com Received 14 May 2010; Accepted 23 August 2010

Academic Editor: Sin E. I. Takahasi

Copyrightq2010 Erhan Set et al. 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.

We do not only give the extensions of the results given by Gill et al.1997for log-convex functions but also obtain some new Hadamard-type inequalities for log-convexm-convex, andα, m-convex functions.

1. Introduction

The following inequality is well known in the literature as Hadamard’s inequality:

f ab

2

≤ 1 b−a

_b

a

fxdx≤ fa fb

2 , 1.1

wheref : I → Ris a convex function on the intervalI of real numbers anda, b ∈ I with a < b.This inequality is one of the most useful inequalities in mathematical analysis. For new proofs, note worthy extension, generalizations, and numerous applications on this inequality;

see1–6where further references are given.

LetIbe on interval inR. Thenf :I → R is said to be convex if, for allx, y ∈ Iand λ∈0,1,

f

λx 1−λy

≤λfx 1−λf y

1.2 see5, Page 1. Geometrically, this means that ifK, L, andMare three distinct points on the graph offwithLbetweenKandM, thenLis on or below chordKM.

Recall that a functionf:I → 0,∞is said to be log-convex function if, for allx, y∈I andt∈0,1, one has the inequalitysee5, Page 3

f

tx 1−ty

≤ fxt

f y_1−t

. 1.3

It is said to be log-concave if the inequality in1.3is reversed.

In7, Toader definedm-convexity as follows.

Definition 1.1. The functionf :0, b → R,b >0 is said to bem-convex, wherem∈0,1, if one has

f

txm1−ty

≤tfx m1−tf y

1.4

for allx, y∈0, bandt∈0,1.We say thatfism-concave if−fism-convex.

Denote byK_mbthe class of allm-convex functions on 0, bsuch thatf0 ≤ 0if m < 1. Obviously, if we choose m 1,Definition 1.1 recaptures the concept of standard convex functions on0, b.

In8, Mihes¸an definedα, m-convexity as in the following:

Definition 1.2. The functionf :0, b → R,b >0, is said to beα, m-convex, whereα, m∈ 0,1², if one has

f

txm1−ty

≤t^αfx m1−t^αf y

1.5

for allx, y∈0, bandt∈0,1.

Denote byK^α_mbthe class of allα, m-convex functions on0, bfor whichf0≤0.

It can be easily seen that forα, m 1, m,α, m-convexity reduces tom-convexity and for α, m 1,1,α, m-convexity reduces to the concept of usual convexity defined on0, b, b >0.

For recent results and generalizations concerning m-convex and α, m-convex functions, see9–12.

In the literature, the logarithmic mean of the positive real numbersp, qis defined as the following:

L p, q

p−q lnp−lnq

p /q

1.6 forpq, we putLp, p p.

In13, Gill et al. established the following results.

Theorem 1.3. Letfbe a positive,log-convex function ona, b. Then

1 b−a

_b

a

ftdt≤L

fa, fb

, 1.7

whereL·,·is a logarithmic mean of the positive real numbers as in1.6.

Forfa positivelog-concave function, the inequality is reversed.

Corollary 1.4. Letfbe positivelog-convex functions ona, b. Then

1 b−a

_b

a

ftdt≤ min

x∈a,b

x−aL

fa, fx

b−xL

fx, fb

b−a . 1.8

Iffis a positivelog-concave function, then

1 b−a

_b

a

fxdx≥ max

x∈a,b

x−aL

fa, fx

b−xL

fx, fb

b−a . 1.9

For some recent results related to the Hadamard’s inequalities involving two log- convex functions, see14and the references cited therein. The main purpose of this paper is to establish the general version of inequalities1.7and new Hadamard-type inequalities involving two log-convex functions, twom-convex functions, or twoα, m-convex functions using elementary analysis.

2. Main Results

We start with the following theorem.

Theorem 2.1. Letfi :I ⊂R → 0,∞ i1,2, . . . , nbelog-convex functions onI anda, b ∈I witha < b. Then the following inequality holds:

1 b−a

_b

a n i1

f_ixdx≤L _n

i1

f_ia, ⁿ

i1

f_ib

, 2.1

whereLis a logarithmic mean of positive real numbers.

Forfa positivelog-concave function, the inequality is reversed.

Proof. Sincefii1,2, . . . , nare log-convex functions onI, we have

f_ita 1−tb≤ f_ia_t

f_ib_1−t 2.2

for alla, b ∈ I and t ∈ 0,1.Writing 2.2fori 1,2, . . . , nand multiplying the resulting inequalities, it is easy to observe that

n i1

f_ita 1−tb≤ _n

i1

f_ia _{t n}

i1

f_ib _1−t

ⁿ

i1

f_ib _n

i1

f_ia fib

_t 2.3

for alla, b∈Iandt∈0,1.

Integrating inequality2.3on0,1overt, we get ₁

0 n i1

f_ita 1−tbdt≤ ⁿ

i1

f_ib ₁

0

_n

i1

f_ia f_ib

t

dt. 2.4

As

₁

0 n i1

f_ita 1−tbdt 1 b−a

_b

a n i1

f_ixdx, 2.5

₁

0

_n

i1

fia fib

_t

dt 1

n i1fib

L _n

i1

f_ia, ⁿ

i1

f_ib

, 2.6

the theorem is proved.

Remark 2.2. By takingi1 andf₁finTheorem 2.1,we obtain1.7.

Corollary 2.3. Letfi :I ⊂R → 0,∞ i1,2, . . . , nbelog-convex functions onIanda, b∈I witha < b. Then

1 b−a

_b

a n i1

fixdx

≤ min

x∈a,b

x−aL_n

i1f_ia,_n

i1f_ix

b−xL_n

i1f_ix,_n

i1f_ib

b−a .

2.7

Iffii1,2, . . . , nare positivelog-concave functions, then

1 b−a

_b

a n i1

f_ixdx

≥ max

x∈a,b

x−aL_n

i1fia,_n

i1f_ix

b−xL_n

i1fix,_n

i1fib

b−a .

2.8

Proof. Letfii 1,2, . . . , nbe positive log-convex functions. Then byTheorem 2.1we have that

_b

a n i1

f_itdt _x

a n i1

f_itdt _b

x n i1

f_itdt

≤x−aL _n

i1

f_ia, ⁿ

i1

f_ix

b−xL _n

i1

f_ix, ⁿ

i1

f_ib

,

2.9

for allx∈a, b, whence2.7. Similarly we can prove2.8.

Remark 2.4. By taking i 1 and f₁ f in 2.7 and 2.8, we obtain the inequalities of Corollary 1.4.

We will now point out some new results of the Hadamard type for log-convex,m- convex, andα, m-convex functions, respectively.

Theorem 2.5. Letf, g:I → 0,∞belog-convex functions onIanda, b∈Iwitha < b.Then the following inequalities hold:

f ab

2

g ab

2

≤ 1 2

1 b−a

_b

a

fxfab−x gxgab−x dx

≤ fafb gagb

2 .

2.10

Proof. We can write

ab

2 ta 1−tb

2 1−tatb

2 . 2.11

Using the elementary inequalitycd ≤ 1/2c²d² c, d ≥ 0 reals and equality2.11, we have

f ab

2

g ab

2

≤ 1 2

f² ab

2

g² ab

2

1 2

f²

ta 1−tb

2 1−tatb 2

g²

ta 1−tb

2 1−tatb 2

≤ 1 2

fta 1−tb_1/22

f1−tatb_1/22 gta 1−tb_1/22g1−tatb_1/22 1

2

fta 1−tbf1−tatb gta 1−tbg1−tatb .

2.12

Sincef, gare log-convex functions, we obtain 1

2

fta 1−tbf1−tatb gta 1−tbg1−tatb

≤ 1 2

fa_t

fb_1−t

fa_1−t fb_t

ga_t

gb_1−t

ga_1−t

gb_t

fafb gagb

2

2.13

for alla, b∈Iandt∈0,1.

Rewriting2.12and2.13, we have

f ab

2

g ab

2

≤ 1 2

fta 1−tbf1−tatb gta 1−tbg1−tatb , 2.14 1

2

fta 1−tbf1−tatb gta 1−tbg1−tatb

≤ fafb gagb

2 .

2.15 Integrating both sides of2.14and2.15on0,1overt, respectively, we obtain

f ab

2

g ab

2

≤ 1 2

1 b−a

_b

a

fxfab−x gxgab−x dx

,

1 2

1 b−a

_b

a

fxfab−x gxgab−x dx

≤ fafb gagb

2 .

2.16

Combining2.16, we get the desired inequalities2.10. The proof is complete.

Theorem 2.6. Letf, g:I → 0,∞belog-convex functions onIanda, b∈Iwitha < b.Then the following inequalities hold:

2f ab

2

g ab

2

≤ 1 b−a

_b

a

f²x g²x dx

≤ fa fb

2 L

fa, fb

ga gb

2 L

ga, gb ,

2.17

whereL·,·is a logarithmic mean of positive real numbers.

Proof. From inequality2.14, we have

f ab

2

g ab

2

≤ 1 2

fta 1−tbf1−tatb gta 1−tbg1−tatb .

2.18

for alla, b∈Iandt∈0,1.

Using the elementary inequalitycd≤1/2c²d² c, d ≥0 realson the right side of the above inequality, we have

f ab

2

g ab

2

≤ 1 4

f²ta 1−tb f²1−tatb g²ta 1−tb g²1−tatb .

2.19

Sincef, gare log-convex functions, then we get

f²ta 1−tb f²1−tatb g²ta 1−tb g²1−tatb

≤

fa2t

fb_2−2t

fa_2−2t

fb2t

ga2t

gb_2−2t

ga_2−2t

gb2t

f²b

fa fb

2t

f²a fb

fa 2t

g²b ga

gb 2t

g²a gb

ga 2t

.

2.20

Integrating both sides of2.19and2.20on0,1overt, respectively, we obtain

2f ab

2

g ab

2

≤ 1 b−a

_b

a

f²x g²x dx,

1 b−a

_b

a

f²x g²x dx

≤ 1 2

f²b

₁

0

fa fb

2t

dtf²a ₁

0

fb fa

2t

dt

g²b ₁

0

ga gb

2t

dtg²a ₁

0

gb ga

2t

dt

1 2

⎛

⎝f²b fa/fb_2t 2 logfa/fb

1 0

f²a fb/fa_2t 2 logfb/fa

1 0

g²b ga/gb2t

2 logga/gb 1

0

g²a gb/ga2t

2 loggb/ga 1

0

⎞

⎠

1 2

f²a−f²b 2

logfa−logfb f²b−f²a 2

logfb−logfa

g²a−g²b 2

logga−loggb g²b−g²a 2

loggb−logga

1 2

fa fb

2 L

fa, fb

fa fb

2 L

fb, fa

ga gb

2 L

ga, gb

ga gb

2 L

gb, ga

fa fb

2 L

fa, fb

ga gb

2 L

ga, gb .

2.21

Combining2.21, we get the required inequalities2.17. The proof is complete.

Theorem 2.7. Letf, g:0,∞ → 0,∞be such thatfgis inL¹a, b, where0≤a < b <∞. If fis nonincreasingm1-convex function andgis nonincreasingm2-convex function ona, bfor some fixedm₁, m₂∈0,1,then the following inequality holds:

1 b−a

_b

a

fxgxdx≤min{S1, S2}, 2.22

where

S1 1 6

f²a g²a

m1faf b

m₁

m2gag b

m₂

m²₁f² b

m₁

m²₂g² b

m₂

, 2.23 S2 1

6

f²b g²b

m1fbf a

m1

m2gbg a

m2

m²₁f²

a m1

m²₂g²

a m2

2.24

Proof. Sincefism₁-convex function andgism₂-convex function, we have

fta 1−tb≤tfa m₁1−tf b

m₁

, gta 1−tb≤tga m₂1−tg

b m₂

2.25

for allt∈0,1. It is easy to observe that _b

a

fxgxdx b−a ₁

0

fta 1−tbgta 1−tbdt. 2.26

Using the elementary inequalitycd≤1/2c²d² c, d≥0 reals,2.25on the right side of 2.26and making the charge of variable and sincef, gis nonincreasing, we have

_b

a

fxgxdx

≤ 1 2b−a

₁

0

fta 1−tb ²

gta 1−tb ² dt

≤ 1 2b−a

₁

0

tfa m11−tf b

m₁ 2

tga m21−tg b

m₂ 2

dt

1 2b−a

1

3f²a 1 3m²₁f²

b m₁

1

3m1faf b

m₁

1

3g²a 1 3m²₂g²

b m₂

1

3m2gag b

m2

b−a 6

f²a g²a

m₁faf b

m1

m₂gag b

m2

m²₁f²

b m1

m²₂g² b

m₂

.

2.27

Analogously we obtain _b

a

fxgxdx

≤b−a 6

f²bg²b

m1fbf a

m1

m2gbg a

m2

m²₁f²

a m1

m²₂g²

a m2

. 2.28

Rewriting2.27and2.28, we get the required inequality in2.22. The proof is complete.

Theorem 2.8. Letf, g :0,∞ → 0,∞be such thatfg is inL¹a, b, where0≤ a < b <∞.

Iff is nonincreasingα1, m1-convex function andg is nonincreasingα2, m2-convex function on a, bfor some fixedα₁, m₁, α₂, m₂∈0,1.Then the following inequality holds:

1 b−a

_b

a

fxgxdx≤min{E1, E₂}, 2.29

where

E₁ 1 2

1

2α₁1f²a 2α²₁

α112α11m²₁f² b

m₁

2α₁

α112α11m₁faf b

m₁

1

2α₂1g²a 2α²₂

α212α21m²₂g² b

m₂

2α₂

α212α21m₂gag b

m₂

,

2.30

E₂ 1 2

1

2α₁1f²b 2α²₁

α112α11m²₁f² a

m₁

2α₁

α112α11m₁fbf a

m₁

1

2α₂1g²b 2α²₂

α212α21m²₂g² a

m₂

2α₂

α212α21m₂gbg a

m₂

.

2.31

Proof. Sincefisα1, m1-convex function andgisα2, m2-convex function, then we have

fta 1−tb≤t^α1fa m₁1−t^α1f b

m1

, gta 1−tb≤t^α2ga m₂1−t^α2g

b m₂

2.32

for allt∈0,1. It is easy to observe that _b

a

fxgxdx b−a ₁

0

fta 1−tbgta 1−tbdt. 2.33

Using the elementary inequalitycd≤1/2c²d² c, d≥0 reals,2.32on the right side of 2.33and making the charge of variable and sincef, gis nonincreasing, we have

_b

a

fxgxdx≤ 1 2b−a

₁

0

fta 1−tb ²

gta 1−tb ² dt

≤ 1 2b−a

₁

0

t^α1fa m₁1−t^α1f b

m1

₂

t^α2ga m21−t^α2g b

m2

₂ dt

1 2b−a

1

2α11f²a 2α²₁

α112α11m²₁f² b

m1

2α₁

α112α11m1faf b

m1

1

2α21g²a 2α²₂

α212α21m²₂g² b

m₂

2α2

α212α21m2gag b

m₂ 2.34

Analogously we obtain _b

a

fxgxdx

≤ 1 2b−a

1

2α11f²b 2α²₁

α112α11m²₁f² a

m1

2α₁

α112α11m₁fbf a

m₁

1

2α₂1g²b 2α²₂

α212α21m²₂g² a

m2

2α₂

α212α21m₂gbg a

m2

.

2.35

Rewriting2.34and2.35, we get the required inequality in2.29. The proof is complete.

Remark 2.9. InTheorem 2.8, if we chooseα₁α₂1, we obtain the inequality ofTheorem 2.7.

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