Fejér-Type Inequalities (I)

. In this paper, we establish some new Fej´er-type inequalities for convex functions.


Introduction
Throughout this paper, let f : [a, b] → R be convex, and g : [a, b] → [0, ∞) be integrable and symmetric to a+b 2 . We define the following functions on [0, 1] that are associated with the well known Hermite-Hadamard inequality [1] ( namely and 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 f be defined as above, and let H be defined on [0, 1] by Then H is convex, increasing on [0, 1] , and for all t ∈ [0, 1], we have In [6], Yang and Hong established the following theorem which is a refinement of the second inequality in (1.1): Theorem B. Let f be defined as above, and let P be defined on [0, 1] by Then P is convex, increasing on [0, 1] , and for all t ∈ [0, 1], we have In [3], Fejér established the following weighted generalization of the Hermite-Hadamard inequality (1.1).
Theorem C. Let f, g be defined as above. Then is known as Fejér inequality.
In this paper, we establish some Fejér-type inequalities related to the functions I, J, M, N introduced above.

Main Results
In order to prove our main results, we need the following lemma: Lemma 1 (see [4]). Let f be defined as above and let Now, we are ready to state and prove our results.
Theorem 2. Let f, g, I be defined as above. Then I is convex, increasing on [0, 1] , and for all t ∈ [0, 1], we have the following Fejér-type inequality Proof. It is easily observed from the convexity of f that I is convex on [0, 1] . Using simple integration techniques and under the hypothesis of g, the following identity holds on [0, 1] , Let t 1 < t 2 in [0, 1] . By Lemma 1, the following inequality holds for all x ∈ a, a+b 2 : Indeed, it holds when we make the choice: in Lemma 1.
Multipling the inequality (2.3) by g (2x − a), integrating both sides over x on a, a+b 2 and using identity (2.2), we derive I (t 1 ) ≤ I (t 2 ) . Thus I is increasing on [0, 1] and then the inequality (2.1) holds. This completes the proof. Theorem 4. Let f, g, J be defined as above. Then J is convex, increasing on [0, 1] , and for all t ∈ [0, 1], we have the following Fejér-type inequality Proof. By using a similar method to that from Theorem 2, we can show that J is convex on [0, 1] , the identity holds on [0, 1] and the inequalities hold for all t 1 < t 2 in [0, 1] and x ∈ a, 3a+b 4 . By (2.5) − (2.7) and using a similar method to that from Theorem 2, we can show that J is increasing on [0, 1] and (2.4) holds. This completes the proof.
The following result provides a comparison between the functions I and J.
Proof. By the identity Further, the following result incorporates the properties of the function M : Theorem 6. Let f, g, M be defined as above. Then M is convex, increasing on [0, 1] , and for all t ∈ [0, 1], we have the following Fejér-type inequality Proof. Follows by the identity The details are left to the interested reader.
We now present a result concerning the properties of the function N : Theorem 7. Let f, g, N be defined as above. Then N is convex, increasing on [0, 1] , and for all t ∈ [0, 1], we have the following Fejér-type inequality Proof. By the identity Remark 8. Let g (x) = 1 b−a (x ∈ [a, b]) in Theorem 7. Then N (t) = P (t) (t ∈ [0, 1]) and the inequality (2.11) reduces to (1.3) where P is defined as in Theorem B.
Proof. By the identity on [0, 1], (2.10) and using a similar method to that for Theorem 2, we can show that M (t) ≤ N (t) on [0, 1] . This completes the proof.
The following Fejér-type inequality is a natural consequence of Theorems 2 -9.