- Research Article
- Open Access

# Fejér-Type Inequalities (I)

- Kuei-Lin Tseng
^{1}, - Shiow-Ru Hwang
^{2}and - SS Dragomir
^{3, 4}Email author

**2010**:531976

https://doi.org/10.1155/2010/531976

© Kuei-Lin Tseng et al. 2010

**Received:**3 May 2010**Accepted:**3 December 2010**Published:**15 December 2010

## Abstract

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

## Keywords

- Convex Function
- Natural Consequence
- Integration Technique
- Integral Inequality
- Simple Integration

## 1. Introduction

For some results which generalize, improve, and extend the famous integral inequality (1.1), see [2–6].

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

Theorem A.

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

Theorem B.

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

Theorem C.

is known as Fejér inequality.

In this paper, we establish some Fejér-type inequalities related to the functions , , , introduced above.

## 2. Main Results

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

Lemma 2.1 (see [4]).

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

Theorem 2.2.

Proof.

in Lemma 2.1.

Multipling the inequality (2.4) by , integrating both sides over on and using identity (2.3), we derive . Thus is increasing on and then the inequality (2.2) holds. This completes the proof.

Remark 2.3.

Let in Theorem 2.2. Then and the inequality (2.2) reduces to the inequality (1.4), where is defined as in Theorem A.

Theorem 2.4.

Proof.

hold for all in and .

By (2.7)–(2.9) and using a similar method to that from Theorem 2.2, we can show that is increasing on and (2.6) holds. This completes the proof.

The following result provides a comparison between the functions and .

Theorem 2.5.

Let , , , and be defined as above. Then on .

Proof.

on , (2.3) and using a similar method to that from Theorem 2.2, we can show that on . The details are omited.

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

Theorem 2.6.

Proof.

on . The details are left to the interested reader.

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

Theorem 2.7.

Proof.

on and using a similar method to that for Theorem 2.2, we can show that is convex, increasing on and (2.13) holds.

Remark 2.8.

Let in Theorem 2.7. Then and the inequality (2.13) reduces to (1.6), where is defined as in Theorem B.

Theorem 2.9.

Let , , , and be defined as above. Then on .

Proof.

on , (2.12) and using a similar method to that for Theorem 2.2, we can show that on . This completes the proof.

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

Corollary 2.10.

Remark 2.11.

which is a refinement of (1.1).

Remark 2.12.

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

## References

- Hadamard J: Étude sur les propriétés des fonctions entières en particulier d'une fonction considérée par Riemann.
*Journal de Mathématiques Pures et Appliquées*1893, 58: 171–-215.Google Scholar - Dragomir SS: Two mappings in connection to Hadamard's inequalities.
*Journal of Mathematical Analysis and Applications*1992, 167(1):49–56. 10.1016/0022-247X(92)90233-4MathSciNetView ArticleMATHGoogle Scholar - Fejér L: Über die Fourierreihen, II.
*Math. Naturwiss. Anz Ungar. Akad. Wiss.*1906, 24: 369–390.Google Scholar - Hwang D-Y, Tseng K-L, Yang G-S: Some Hadamard's inequalities for co-ordinated convex functions in a rectangle from the plane.
*Taiwanese Journal of Mathematics*2007, 11(1):63–73.MathSciNetMATHGoogle Scholar - Pečarić JE, Proschan F, Tong YL:
*Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering*.*Volume 187*. Academic Press, Boston, Mass, USA; 1992:xiv+467.Google Scholar - Yang G-S, Hong M-C: A note on Hadamard's inequality.
*Tamkang Journal of Mathematics*1997, 28(1):33–37.MathSciNetMATHGoogle Scholar

## Copyright

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