Some inequalities associated with the Hermite–Hadamard–Fejér type for convex function

In this paper, we extend some estimates of the right-hand side of a Hermite–Hadamard–Fejér type inequality for functions whose first derivatives’ absolute values are convex. The results presented here would provide extensions of those given in earlier works.


Introduction
Definition 1 The function f : ½a; b & R ! R is said to be convex if the following inequality holds f ðkx þ ð1 À kÞyÞ kf ðxÞ þ ð1 À kÞf ðyÞ for all x; y 2 ½a; b and k 2 0; 1 ½ : We say that f is concave if ðÀf Þ is convex.
The following inequality is well known in the literature as the Hermite-Hadamard integral inequality (see, [2,4]): where f : I & R ! R is a convex function on the interval I of real numbers and a; b 2 I with a\b.
In [1], Dragomir and Agarwal proved the following results connected with the right part of (1.1).
Lemma 1 Let f : I R ! R be a differentiable mapping on I , a; b 2 I with a\b. If f 0 2 L½a; b, then the following equality holds: ðta þ ð1 À tÞbÞdt: ð1:2Þ Theorem 1 Let f : I R ! R be a differentiable mapping on I , a; b 2 I with a\b. If f 0 j j is convex on ½a; b, then the following inequality holds: If the mapping f 0 j j p= pÀ1 ð Þ is convex on a; b ½ , then the following inequality holds: The most well-known inequalities related to the integral mean of a convex function are the Hermite-Hadamard inequalities or its weighted versions, the so-called Hermite-Hadamard-Fejér inequalities (see [5][6][7][8][9][10][11][12][13][14]). In [3], Fejer gave a weighted generalization of the inequalities (1.1) as the following: holds, where w : ½a; b ! R is nonnegative, integrable, and symmetric about x ¼ aþb 2 : In [5], some inequalities of Hermite-Hadamard-Fejer type for differentiable convex mappings were proved using the following lemma.
Lemma 2 Let f : I & R ! R be a differentiable mapping on I , a; b 2 I with a\b, and w : ½a; b ! ½0; 1Þ be a differentiable mapping. If f 0 2 L½a; b, then the following equality holds: for each t 2 ½0; 1; where In this article, using functions whose derivatives' absolute values are convex, we obtained new inequalities of Hermite-Hadamard-Fejer type. The results presented here would provide extensions of those given in earlier works.

Main results
We will establish some new results connected with the right-hand side of (1.5) and (1.1). Now, we prove our main theorems: Theorem 4 Let f : I R ! R be a differentiable mapping on I , a; b 2 I with a\b and let w : a; b ½ ! R be continuous on a; b ½ . If f 0 j j is convex on ½a; b, then for all x 2 a; b ½ ; the following inequalities hold: # ( where a [ 0 and w k k 1 ¼ sup t2 a;b ½ wðtÞ j j: Proof By integration by parts, we have the following equalities: We take absolute value of (2.1) and use convexity of f 0 , we find that for all x 2 a; b ½ : Hence, the proof of theorem is completed.
Corollary 1 Under the same assumptions of Theorem 4 with wðsÞ ¼ 1, then the following inequality holds: for all x 2 a; b ½ : Corollary 2 (Fejer Type Inequality) Under the same assumptions of Theorem 4 with a ¼ 1, then the following inequalities hold: which is proved by Tseng et al. in [8].
Corollary 3 (Weighted Trapezoid Inequality) Let w : a; b ½ ! R be symmetric to aþb 2 and x ¼ aþb 2 in Corollary 2. Then the following inequalities hold: 8 ! which is proved by Tseng et al. in [8].
Theorem 5 Let f : I R ! R be a differentiable mapping on I , a; b 2 I with a\b and let w : a; b ½ ! R be continuous on a; b ½ . If f 0 j j q is convex on ½a; b, q [ 1; then for all x 2 a; b ½ ; the following inequalities hold: where a [ 0; 1 p þ 1 q ¼ 1; and w k k 1 ¼ sup t2 a;b ½ wðtÞ j j: Proof We take absolute value of (2.1). Using Holder's inequality, we find that which this completes the proof.
Corollary 4 Under the same assumptions of Theorem 5 with wðsÞ ¼ 1, then the following inequalities hold: ð2:4Þ Corollary 5 Let the conditions of Corollary 4 hold. If we take a ¼ 1 and x ¼ aþb 2 in (2.4), then the following inequality holds: Corollary 6 (Fejer Type Inequality) Under the same assumptions of Theorem 5 with a ¼ 1, then the following inequalities hold: Corollary 7 (Weighted Trapezoid Inequality) Let w : a; b ½ ! R be symmetric to aþb 2 and x ¼ aþb 2 in Corollary 6. Then the following inequalities hold: Theorem 6 Let f : I R ! R be a differentiable mapping on I , a; b 2 I with a\b and let w : a; b ½ ! R be continuous on a; b ½ . If f 0 j j q is convex on ½a; b, q [ 1; then for all x 2 a; b ½ ; the following inequality holds: Proof We take absolute value of (2.1). Using Holder's inequality and the convexity of f 0 q ; we find that which this completes the proof.
Corollary 8 Under the same assumptions of Theorem 6 with wðsÞ ¼ 1, then the following inequality holds:   Theorem 7 Let f : I R ! R be a differentiable mapping on I , a; b 2 I with a\b and let w : a; b ½ ! R be continuous on a; b ½ . If f 0 j j q is convex on ½a; b, q [ 1; then for all x 2 a; b ½ ; the following inequality holds: Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.