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Fejer-type inequalities (I)

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Tseng, Kuei-Lin, Hwang, Shiow-Ru and Dragomir, Sever S (2010) Fejer-type inequalities (I). Journal of Inequalities and Applications, 2010. pp. 1-7. ISSN 1025-5834 (print) 1029-242X (online)

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Journal of Inequalities and Applications Volume 2010, Article ID 531976,7pages doi:10.1155/2010/531976

Research Article

Fej ´er-Type Inequalities (I)

Kuei-Lin Tseng,

¹

Shiow-Ru Hwang,

²

and S. S. Dragomir

^{3, 4}

1Department of Applied Mathematics, Aletheia University, Tamsui 25103, Taiwan

2China University of Science and Technology, Nankang, Taipei 11522, Taiwan

3School of Engineering Science, VIC University, P.O. Box 14428, Melbourne City MC, Victoria 8001, Australia

4School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

Correspondence should be addressed to S. S. Dragomir,sever.dragomir@vu.edu.au Received 3 May 2010; Revised 26 August 2010; Accepted 3 December 2010

Academic Editor: Yeol J. E. Cho

Copyrightq2010 Kuei-Lin Tseng 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 establish some new Fej´er-type inequalities for convex functions.

1. Introduction

Throughout this paper, letf:a, b → Rbe convex, and letg :a, b → 0,∞be integrable and symmetric toab/2. We define the following functions on0,1that are associated with the well-known Hermite-Hadamard inequality1

f ab

2

≤ 1 b−a

_b

a

fxdx≤ fa fb

2 , 1.1

namely

It _b

a

1 2

f

txa

2 1−tab 2

f

txb

2 1−tab 2

gxdx, Jt

_b

a

1 2

f

txa

2 1−t3ab 4

f

txb

2 1−ta3b 4

gxdx,

2 Journal of Inequalities and Applications Mt

_ab/2

a

1 2

f

ta 1−txa 2

f

tab

2 1−txb 2

gxdx

_b

ab/2

1 2

f

tab

2 1−txa 2

f

tb 1−txb 2

gxdx, Nt

_b

a

1 2

f

ta 1−txa 2

f

tb 1−txb 2

gxdx.

1.2

For some results which generalize, improve, and extend the famous integral inequality 1.1, see2–6.

In2, Dragomir established the following theorem which is a refinement of the first inequality of1.1.

Theorem A. Letfbe defined as above, and letHbe defined on0,1by

Ht 1

b−a _b

a

f

tx 1−tab 2

dx. 1.3

Then,His convex, increasing on0,1, and for allt∈0,1, one has

f ab

2

H0≤Ht≤H1 1 b−a

_b

a

fxdx. 1.4

In6, Yang and Hong established the following theorem which is a refinement of the second inequality in1.1.

Theorem B. Letfbe defined as above, and letP be defined on0,1by

Pt 1

2b−a _b

a

f

1t 2

a

1−t 2

x

f

1t 2

b

1−t 2

x

dx. 1.5

Then,Pis convex, increasing on0,1, and for allt∈0,1, one has

1 b−a

_b

a

fxdxP0≤Pt≤P1 fa fb

2 . 1.6

In 3, Fej´er established the following weighted generalization of the Hermite- Hadamard inequality1.1.

Theorem C. Letf, gbe defined as above. Then,

f ab

2

b a

gxdx≤ _b

a

fxgxdx≤ fa fb 2

_b

a

gxdx 1.7

is known as Fej´er inequality.

In this paper, we establish some Fej´er-type inequalities related to the functionsI,J,M, Nintroduced above.

2. Main Results

In order to prove our main results, we need the following lemma.

Lemma 2.1see4. Letfbe defined as above, and leta≤A≤C≤D≤B≤bwithABCD.

Then,

fC fD≤fA fB. 2.1

Now, we are ready to state and prove our results.

Theorem 2.2. Letf, g, andI be defined as above. ThenIis convex, increasing on0,1, and for all t∈0,1, one has the following Fej´er-type inequality:

f ab

2

b a

gxdxI0≤It≤I1 _b

a

1 2

fxa 2

f xb

2

gxdx. 2.2

Proof. It is easily observed from the convexity off thatI is convex on0,1. Using simple integration techniques and under the hypothesis ofg, the following identity holds on0,1:

It _b

a

1 2

f

txa

2 1−tab 2

gx f

ta2b−x

2 1−tab 2

gab−x

dx

_b

a

1 2

f

txa

2 1−tab 2

f

ta2b−x

2 1−tab 2

gxdx

_ab/2

a

f

tx 1−tab 2

f

tab−x 1−tab 2

g2x−adx.

2.3

4 Journal of Inequalities and Applications Lett1< t2in0,1. ByLemma 2.1, the following inequality holds for allx∈a,ab/2:

f

t1x 1−t1ab 2

f

t1ab−x 1−t1ab 2

≤f

t₂x 1−t₂ab 2

f

t₂ab−x 1−t₂ab 2

.

2.4

Indeed, it holds when we make the choice

At2x 1−t2ab 2 , Ct₁x 1−t₁ab

2 , Dt₁ab−x 1−t₁ab

2 , Bt2ab−x 1−t2ab

2 ,

2.5

inLemma 2.1.

Multipling the inequality2.4byg2x−a, integrating both sides overxona,a b/2and using identity2.3, we deriveIt1≤It2. ThusIis increasing on0,1and then the inequality2.2holds. This completes the proof.

Remark 2.3. Letgx 1/b−a x∈a, binTheorem 2.2. ThenIt Ht t∈0,1and the inequality2.2reduces to the inequality1.4, whereHis defined as in Theorem A.

Theorem 2.4. Let f, g, J be defined as above. Then J is convex, increasing on 0,1, and for all t∈0,1, one has the following Fej´er-type inequality:

f3ab/4 fa3b/4 2

_b

a

gxdxJ0≤Jt≤J1 1

2 _b

a

fxa 2

f xb

2

gxdx.

2.6

Proof. By using a similar method to that fromTheorem 2.2, we can show thatJ is convex on 0,1, the identity

Jt

_3ab/4

a

f

tx 1−t3ab 4

f

t

3ab 2 −x

1−t3ab 4

f

t

xb−a 2

1−ta3b 4

f

tab−x 1−ta3b 4

×g2x−adx

2.7

holds on0,1, and the inequalities

f

t₁x 1−t₁3ab 4

f

t₁

3ab 2 −x

1−t₁3ab 4

≤f

t₂x 1−t₂3ab 4

f

t₂

3ab 2 −x

1−t₂3ab 4

,

2.8

f

t1

xb−a 2

1−t1a3b 4

f

t1ab−x 1−t1a3b 4

≤f

t₂

xb−a 2

1−t₂a3b 4

f

t₂ab−x 1−t₂a3b 4

2.9

hold for allt1< t2in0,1andx∈a,3ab/4.

By2.7–2.9and using a similar method to that fromTheorem 2.2, we can show that Jis increasing on0,1and2.6holds. This completes the proof.

The following result provides a comparison between the functionsIandJ.

Theorem 2.5. Letf,g,I, andJbe defined as above. ThenIt≤Jton0,1.

Proof. By the identity

Jt

_ab/2

a

f

tx 1−t3ab 4

f

tab−x 1−ta3b 4

g2x−adx, 2.10

on0,1,2.3and using a similar method to that fromTheorem 2.2, we can show thatIt≤ Jton0,1. The details are omited.

Further, the following result incorporates the properties of the functionM.

Theorem 2.6. Letf, g, M be defined as above. ThenMis convex, increasing on0,1, and for all t∈0,1, one has the following Fej´er-type inequality:

_b

a

1 2

fxa 2

f xb

2

gxdx M0≤Mt≤M1 1

2

f ab

2

fa fb 2

b a

gxdx.

2.11

6 Journal of Inequalities and Applications Proof. Follows by the identity

Mt

_3ab/4

a

fta 1−tx f

tab

2 1−t

3ab 2 −x

f

tab

2 1−t

xb−a 2

ftb 1−tab−x

×g2x−adx,

2.12

on0,1. The details are left to the interested reader.

We now present a result concerning the properties of the functionN.

Theorem 2.7. Letf, g, Nbe defined as above. Then N is convex, increasing on0,1, and for all t∈0,1, one has the following Fej´er-type inequality:

_b

a

1 2

fxa 2

f xb

2

gxdxN0≤Nt≤N1 fa fb 2

_b

a

gxdx.

2.13 Proof. By the identity

Nt

_ab/2

a

fta 1−tx ftb 1−tab−x

g2x−adx 2.14

on0,1and using a similar method to that forTheorem 2.2, we can show thatNis convex, increasing on0,1and2.13holds.

Remark 2.8. Letgx 1/b−a x∈a, binTheorem 2.7. ThenNt Pt t∈0,1and the inequality2.13reduces to1.6, whereP is defined as in Theorem B.

Theorem 2.9. Letf,g,M, andNbe defined as above. ThenMt≤Nton0,1.

Proof. By the identity

Nt

_3ab/4

a

fta 1−tx f

ta 1−t

3ab 2 −x

ftb 1−tab−x f

tb 1−t

xb−a 2

g2x−adx, 2.15 on0,1,2.12and using a similar method to that forTheorem 2.2, we can show thatMt≤ Nton0,1. This completes the proof.

The following Fej´er-type inequality is a natural consequence of Theorems2.2–2.9.

Corollary 2.10. Letf, gbe defined as above. Then one has

f ab

2

b a

gxdx≤ f3ab/4 fa3b/4 2

_b

a

gxdx

≤ _b

a

1 2

fxa 2

f xb

2

gxdx

≤ 1 2

f ab

2

fa fb 2

b a

gxdx

≤ fa fb 2

_b

a

gxdx.

2.16

Remark 2.11. Letgx 1/b−a x ∈ a, binCorollary 2.10. Then the inequality2.16 reduces to

f ab

2

≤ f3ab/4 fa3b/4

2 ≤ 1

b−a _b

a

fxdx

≤ 1 2

f ab

2

fa fb 2

≤ fa fb

2 ,

2.17

which is a refinement of1.1.

Remark 2.12. InCorollary 2.10, the third inequality in2.16is the weighted generalization of Bullen’s inequality5

1 b−a

_b

a

fxdx≤ 1 2

f ab

2

fa fb 2

. 2.18

Acknowledgment

This research was partially supported by Grant NSC 97-2115-M-156-002.

References

1 J. Hadamard, “ ´Etude sur les propri´et´es des fonctions enti`eres en particulier d’une fonction consid´er´ee par Riemann,”Journal de Math´ematiques Pures et Appliqu´ees, vol. 58, pp. 171–215, 1893.

2 S. S. Dragomir, “Two mappings in connection to Hadamard’s inequalities,”Journal of Mathematical Analysis and Applications, vol. 167, no. 1, pp. 49–56, 1992.

3 L. Fej´er, “ ¨Uber die Fourierreihen, II,”Math. Naturwiss. Anz Ungar. Akad. Wiss., vol. 24, pp. 369–390, 1906 Hungarian.

4 D.-Y. Hwang, K.-L. Tseng, and G.-S. Yang, “Some Hadamard’s inequalities for co-ordinated convex functions in a rectangle from the plane,”Taiwanese Journal of Mathematics, vol. 11, no. 1, pp. 63–73, 2007.

5 J. E. Peˇcari´c, F. Proschan, and Y. L. Tong,Convex Functions, Partial Orderings, and Statistical Applications, vol. 187 ofMathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992.

6 G.-S. Yang and M.-C. Hong, “A note on Hadamard’s inequality,”Tamkang Journal of Mathematics, vol.

28, no. 1, pp. 33–37, 1997.