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,
1Shiow-Ru Hwang,
2and S. S. Dragomir
3, 41Department 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,[email protected]
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−t
ab 2
f
txb
2 1−t
ab 2
gxdx,
Jt
b
a 1 2
f
txa
2 1−t
3ab 4
f
txb
2 1−t
a3b 4
Mt
ab/2
a 1 2
fta 1−txa 2
f
tab
2 1−t
xb 2
gxdx
b
ab/2 1 2
f
tab
2 1−t
xa 2
f
tb 1−txb 2
gxdx,
Nt
b
a 1 2
fta 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
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−t
ab 2
gx f
ta2b−x
2 1−t
ab 2
gab−x
dx
b
a 1 2
f
txa
2 1−t
ab 2
f
ta2b−x
2 1−t
ab 2
gxdx
ab/2
a
f
tx 1−tab 2
f
tab−x 1−tab 2
g2x−adx.
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−t2 ab
2
f
t2ab−x 1−t2 ab
2
.
2.4
Indeed, it holds when we make the choice
At2x 1−t2ab 2 ,
Ct1x 1−t1 ab
2 ,
Dt1ab−x 1−t1 ab
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
holds on0,1, and the inequalities
f
t1x 1−t1 3ab
4
f
t1
3ab
2 −x
1−t1 3ab
4
≤f
t2x 1−t2 3ab
4
f
t2
3ab
2 −x
1−t2 3ab
4
,
2.8
f
t1
xb−a 2
1−t1a3b 4
f
t1ab−x 1−t1a3b 4
≤f
t2
xb−a 2
1−t2
a3b
4
f
t2ab−x 1−t2 a3b
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.
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−xg2x−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.
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,
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