New Inequalities Of Hermite-Hadamard Type For Convex Functions With Applications

In this paper, some new inequalities of the Hermite-Hadamard type for functions whose modulus of the derivatives are convex and applications for special means are given. Finally, some error estimates for the trapezoidal formula are obtained.

One of the most famous inequality for convex functions is so called Hermite-Hadamard's inequality as follows: Let f : I ⊆ R → R be a convex function defined on the interval I of real numbers and a, b ∈ I, with a < b. Then : In [3], the following theorem which was obtained by Dragomir and Agarwal contains the Hermite-Hadamard type integral inequality.
Theorem 1. Let f : I • ⊆ R → R be a differentiable mapping on I • , a, b ∈ I • with a < b. If |f ′ | is convex on [a, b], then the following inequality holds: In [4] Kırmacı, Bakula,Özdemir and Pečarić proved the following theorem.
Theorem 2. Let f : I → R, I ⊂ R, be a differentiable function on I • such that For recent results and generalizations concerning Hermite-Hadamard's inequality see [1]- [4] and the references therein.

THE NEW HERMITE-HADAMARD TYPE INEQUALITIES
In order to prove our main theorems, we first prove the following lemma: , then the following inequality holds: Proof. We note that Integrating by parts, we get Using the Lemma 1 the following result can be obtained.
, then the following inequality holds: for each x ∈ [a, b] .
Proof. Using Lemma 1 and taking the modulus, we have Since |f ′ | is convex, then we get which completes the proof.
Remark 1. In Corollary 1, using the convexity of |f ′ |, we have which is the inequality in (1.2).
and for some fixed p > 1, then the following inequality holds: for each x ∈ [a, b] and q = p p−1 .
Proof. From Lemma 1 and using the well-known Hölder integral inequality, we have which completes the proof.
Theorem 5. Let f : I ⊆ R → R be a differentiable mapping on I • such that , for some fixed q > 1, then the following inequality holds: Proof. As in Theorem 4, using Lemma 1 and the well-known Hölder integral inequality for q > 1 and p = q q−1 , we have Since |f ′ | q is concave on [a, b], we can use the Jensen's integral inequality to obtain: Analogously, Combining all the obtained inequalities, we get which completes the proof.

Remark 2. In Theorem 5, if we choose
which is the inequality in (1.3).
Theorem 6. Let f : , for some fixed q ≥ 1, then the following inequality holds: Proof. Suppose that q ≥ 1. From Lemma 1 and using the well-known power-mean inequality, we have Since |f ′ | q is convex, therefore we have Combining all the above inequalities gives the desired result.
Corollary 3. In Theorem 6, choosing x = a+b 2 and then using the convexity of |f ′ | q , we have Theorem 7. Let f : , for some fixed q ≥ 1, then the following inequality holds: Proof. First, we note that by the concavity of |f ′ | q and the power-mean inequality, we have Hence, so |f ′ | is also concave. Accordingly, using Lemma 1 and the Jensen integral inequality, we have Corollary 4. In Theorem 7, if we choose x = a+b 2 , we have

APPLICATIONS TO SPECIAL MEANS
Recall the following means which could be considered extensions of arithmetic, logarithmic and generalized logarithmic from positive to real numbers.
(1) The arithmetic mean: (2) The logarithmic mean: The generalized logarithmic mean: Now using the results of Section 2, we give some applications to special means of real numbers. Proof. The assertion follows from Corollary 2 and 3 for f (x) = x n , x ∈ R, n ∈ Z, |n| ≥ 2. Proposition 5. Let f : I ⊆ R → R be a differentiable mapping on I • such that f ′ ∈ L [a, b], where a, b ∈ I with a < b. If |f ′ | q is concave on [a, b], for some fixed q ≥ 1.Then in (4.1), for every division d of [a, b] , the trapezoidal error estimate satisfies Proof. The proof is similar to that of Proposition 3 and using Corollary 4.