Open Access

Fejér-Type Inequalities (I)

Journal of Inequalities and Applications20102010:531976

DOI: 10.1155/2010/531976

Received: 3 May 2010

Accepted: 3 December 2010

Published: 15 December 2010

Abstract

We establish some new Fejér-type inequalities for convex functions.

1. Introduction

Throughout this paper, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq1_HTML.gif be convex, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq2_HTML.gif be integrable and symmetric to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq3_HTML.gif . We define the following functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq4_HTML.gif that are associated with the well-known Hermite-Hadamard inequality [1]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ1_HTML.gif
(1.1)
namely
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ2_HTML.gif
(1.2)

For some results which generalize, improve, and extend the famous integral inequality (1.1), see [26].

In [2], Dragomir established the following theorem which is a refinement of the first inequality of (1.1).

Theorem A.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq5_HTML.gif be defined as above, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq6_HTML.gif be defined on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq7_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ3_HTML.gif
(1.3)
Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq8_HTML.gif is convex, increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq9_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq10_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ4_HTML.gif
(1.4)

In [6], Yang and Hong established the following theorem which is a refinement of the second inequality in (1.1).

Theorem B.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq11_HTML.gif be defined as above, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq12_HTML.gif be defined on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq13_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ5_HTML.gif
(1.5)
Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq14_HTML.gif is convex, increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq15_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq16_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ6_HTML.gif
(1.6)

In [3], Fejér established the following weighted generalization of the Hermite-Hadamard inequality (1.1).

Theorem C.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq17_HTML.gif be defined as above. Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ7_HTML.gif
(1.7)

is known as Fejér inequality.

In this paper, we establish some Fejér-type inequalities related to the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq19_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq20_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq21_HTML.gif introduced above.

2. Main Results

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

Lemma 2.1 (see [4]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq22_HTML.gif be defined as above, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq23_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq24_HTML.gif . Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ8_HTML.gif
(2.1)

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

Theorem 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq25_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq26_HTML.gif be defined as above. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq27_HTML.gif is convex, increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq28_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq29_HTML.gif , one has the following Fejér-type inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ9_HTML.gif
(2.2)

Proof.

It is easily observed from the convexity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq30_HTML.gif that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq31_HTML.gif is convex on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq32_HTML.gif . Using simple integration techniques and under the hypothesis of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq33_HTML.gif , the following identity holds on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq34_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ10_HTML.gif
(2.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq35_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq36_HTML.gif . By Lemma 2.1, the following inequality holds for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq37_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ11_HTML.gif
(2.4)
Indeed, it holds when we make the choice
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ12_HTML.gif
(2.5)

in Lemma 2.1.

Multipling the inequality (2.4) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq38_HTML.gif , integrating both sides over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq39_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq40_HTML.gif and using identity (2.3), we derive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq41_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq42_HTML.gif is increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq43_HTML.gif and then the inequality (2.2) holds. This completes the proof.

Remark 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq44_HTML.gif in Theorem 2.2. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq45_HTML.gif and the inequality (2.2) reduces to the inequality (1.4), where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq46_HTML.gif is defined as in Theorem A.

Theorem 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq47_HTML.gif be defined as above. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq48_HTML.gif is convex, increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq49_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq50_HTML.gif , one has the following Fejér-type inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ13_HTML.gif
(2.6)

Proof.

By using a similar method to that from Theorem 2.2, we can show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq51_HTML.gif is convex on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq52_HTML.gif , the identity
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ14_HTML.gif
(2.7)
holds on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq53_HTML.gif , and the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ15_HTML.gif
(2.8)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ16_HTML.gif
(2.9)

hold for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq54_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq55_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq56_HTML.gif .

By (2.7)–(2.9) and using a similar method to that from Theorem 2.2, we can show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq57_HTML.gif is increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq58_HTML.gif and (2.6) holds. This completes the proof.

The following result provides a comparison between the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq59_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq60_HTML.gif .

Theorem 2.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq61_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq62_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq63_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq64_HTML.gif be defined as above. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq65_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq66_HTML.gif .

Proof.

By the identity
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ17_HTML.gif
(2.10)

on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq67_HTML.gif , (2.3) and using a similar method to that from Theorem 2.2, we can show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq68_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq69_HTML.gif . The details are omited.

Further, the following result incorporates the properties of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq70_HTML.gif .

Theorem 2.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq71_HTML.gif be defined as above. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq72_HTML.gif is convex, increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq73_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq74_HTML.gif , one has the following Fejér-type inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ18_HTML.gif
(2.11)

Proof.

Follows by the identity
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ19_HTML.gif
(2.12)

on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq75_HTML.gif . The details are left to the interested reader.

We now present a result concerning the properties of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq76_HTML.gif .

Theorem 2.7.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq77_HTML.gif be defined as above. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq78_HTML.gif is convex, increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq79_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq80_HTML.gif , one has the following Fejér-type inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ20_HTML.gif
(2.13)

Proof.

By the identity
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ21_HTML.gif
(2.14)

on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq81_HTML.gif and using a similar method to that for Theorem 2.2, we can show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq82_HTML.gif is convex, increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq83_HTML.gif and (2.13) holds.

Remark 2.8.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq84_HTML.gif in Theorem 2.7. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq85_HTML.gif and the inequality (2.13) reduces to (1.6), where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq86_HTML.gif is defined as in Theorem B.

Theorem 2.9.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq87_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq88_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq89_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq90_HTML.gif be defined as above. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq91_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq92_HTML.gif .

Proof.

By the identity
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ22_HTML.gif
(2.15)

on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq93_HTML.gif , (2.12) and using a similar method to that for Theorem 2.2, we can show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq94_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq95_HTML.gif . This completes the proof.

The following Fejér-type inequality is a natural consequence of Theorems 2.2–2.9.

Corollary 2.10.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq96_HTML.gif be defined as above. Then one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ23_HTML.gif
(2.16)

Remark 2.11.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq97_HTML.gif in Corollary 2.10. Then the inequality (2.16) reduces to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ24_HTML.gif
(2.17)

which is a refinement of (1.1).

Remark 2.12.

In Corollary 2.10, the third inequality in (2.16) is the weighted generalization of Bullen's inequality [5]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ25_HTML.gif
(2.18)

Declarations

Acknowledgment

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

Authors’ Affiliations

(1)
Department of Applied Mathematics, Aletheia University
(2)
China University of Science and Technology
(3)
School of Engineering Science, VIC University
(4)
School of Computational and Applied Mathematics, University of the Witwatersrand

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Copyright

© Kuei-Lin Tseng et al. 2010

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