Fejér-Type Inequalities (I)

Abstract

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

1. Introduction

Throughout this paper, let be convex, and let be integrable and symmetric to . We define the following functions on that are associated with the well-known Hermite-Hadamard inequality [1]

(1.1)

namely

(1.2)

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.

Let be defined as above, and let be defined on by

(1.3)

Then, is convex, increasing on , and for all , one has

(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 be defined as above, and let be defined on by

(1.5)

Then, is convex, increasing on , and for all , one has

(1.6)

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

Theorem C.

Let be defined as above. Then,

(1.7)

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]).

Let be defined as above, and let with . Then,

(2.1)

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

Theorem 2.2.

Let , and be defined as above. Then is convex, increasing on , and for all , one has the following Fejér-type inequality:

(2.2)

Proof.

It is easily observed from the convexity of that is convex on . Using simple integration techniques and under the hypothesis of , the following identity holds on :

(2.3)

Let in . By Lemma 2.1, the following inequality holds for all :

(2.4)

Indeed, it holds when we make the choice

(2.5)

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.

Let be defined as above. Then is convex, increasing on , and for all , one has the following Fejér-type inequality:

(2.6)

Proof.

By using a similar method to that from Theorem 2.2, we can show that is convex on , the identity

(2.7)

holds on , and the inequalities

(2.8)

(2.9)

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.

By the identity

(2.10)

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.

Let be defined as above. Then is convex, increasing on , and for all , one has the following Fejér-type inequality:

(2.11)

Proof.

Follows by the identity

(2.12)

on . The details are left to the interested reader.

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

Theorem 2.7.

Let be defined as above. Then is convex, increasing on , and for all , one has the following Fejér-type inequality:

(2.13)

Proof.

By the identity

(2.14)

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.

By the identity

(2.15)

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.

Let be defined as above. Then one has

(2.16)

Remark 2.11.

Let in Corollary 2.10. Then the inequality (2.16) reduces to

(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]

(2.18)