Some Integral Inequalities of Hermite-Hadamard Type for Convex Functions with Applications to Means
Bo-Yan Xi1and Feng Qi2,3
Academic Editor: Lars Diening
Received21 Feb 2012
Accepted21 May 2012
Published30 Jul 2012
Abstract
The authors establish some new inequalities for differentiable convex functions, which are similar to the celebrated Hermite-Hadamard's integral inequality for convex functions, and apply these inequalities to construct inequalities for special means of two positive numbers.
1. Introduction
In [1], the following Hermite-Hadamard type inequalities for differentiable convex functions were proved.
Theorem 1.1 (see [1, Theorem 2.2]). Let be a differentiable mapping on , with . If is convex on , then
Theorem 1.2 (see [1, Theorem 2.3]). Let be a differentiable mapping on , with , and let . If the new mapping is convex on , then
In [2], the above inequalities were generalized as follows.
Theorem 1.3 (see [2, Theorems 1 and 2]). Let be differentiable on , with , and let . If is convex on , then
In [3], the above inequalities were further generalized as follows.
Theorem 1.4 (see [3, Theorems 2.3 and 2.4]). Let be differentiable on , with , and let . If is convex on , then
In [4], an inequality similar to the above ones was given as follows.
Theorem 1.5 (see [4, Theorem 3]). Let be an absolutely continuous mapping on whose derivative belongs to . Then
where and .
Recently, the following inequalities were obtained in [5].
Theorem 1.6. Let be differentiable on , with , and . If for is convex on , then
In this paper, we will establish some new Hermite-Hadamard type integral inequalities for differentiable functions and apply them to derive some inequalities of special means.
2. Lemmas
For establishing new integral inequalities of Hermite-Hadamard type, we need the lemmas below.
Lemma 2.1. Let be differentiable on , with . If and , then
Proof. Integrating by part and changing variable of definite integral yield
Similarly, we have
Adding these two equations leads to Lemma 2.1.
Lemma 2.2. For and , one has
Proof. This follows from a straightforward computation of definite integrals.
3. Some Integral Inequalities of Hermite-Hadamard Type
Now we are in a position to establish some new integral inequalities of Hermite-Hadamard type for differentiable convex functions.
Theorem 3.1. Let be a differentiable function on , with , , and . If for is convex on , then
Proof. For , by Lemma 2.1, the convexity of on , and the noted Hölder's integral inequality, we have
In virtue of Lemma 2.2, a direct calculation yields
Substituting the above two equalities into the inequality (3.2) and utilizing Lemma 2.2 result in the inequality (3.1) for . For , from Lemmas 2.1 and 2.2 it follows that
which is just equivalent to (3.1) for . Theorem 3.1 is proved.
If taking in Theorem 3.1, we derive the following corollary.
Corollary 3.2. Let be differentiable on , with , , and . If is convex on for , then
If letting , respectively, in Theorem 3.1, we can deduce the inequalities below.
Corollary 3.3. Let be differentiable on , with , and . If is convex on for , then
If setting in Corollary 3.3, then one has the following.
Corollary 3.4. Let be differentiable on , with , and . If is convex on , then
Theorem 3.5. Let be differentiable on , with , 0 ≤ λ, μ ≤ 1, and . If is convex on for , then
Proof. For , by the convexity of on , Lemma 2.1, and Hölder's integral inequality, it follows that
By Lemma 2.2, we have
Substituting these two equalities into the inequality (3.9) yields (3.8) for . For , the proof is the same as the deduction of (3.4). Thus, Theorem 3.5 is proved.
As the derivation of corollaries of Theorem 3.1, we can obtain the following corollaries of Theorem 3.5.
Corollary 3.6. Let be differentiable on , with , , and . If is convex for on , then
Corollary 3.7. Let be differentiable on , with , and . If is convex for on , then
Corollary 3.8. Let be differentiable on , with , and . If is convex on , then
Corollary 3.9. Let be differentiable on , with , and . If is convex for on and
then
In particular, when , one has
4. Applications to Means
For two positive numbers and , define
for . These means are, respectively, called the arithmetic, geometric, harmonic, generalized logarithmic, identric, and Heronian means of two positive number and .
Applying Theorem 3.1 to for and leads to the following inequalities for means.
Theorem 4.1. Let , , either and or , then
In particular, if or , then
Taking for in Theorem 3.1 results in the following inequalities for means.
Theorem 4.2. For and , one has
In particular,
Finally, we can establish an inequality for Heronian mean as follows.
Theorem 4.3. For , , and or , one has
Proof. Let for and . Then
In virtue of Corollary 3.3, it follows that
On the other hand, we have
The proof is complete.
Remark 4.4. Some inequalities of Hermite-Hadamard type were also obtained in [6–9] by the authors.
Acknowledgment
The first author was supported in part by the National Natural Science Foundation of China under Grant no. 10962004.
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