Abstract
The authors introduce the concept of the s-geometrically convex functions. By the well-known Hölder inequality, they establish some integral inequalities of Hermite-Hadamard type related to the s-geometrically convex functions and apply these inequalities to special means.
1. Introduction
We firstly list several definitions and some known results.
Definition 1.1. A function is said to be convex if for and .
Definition 1.2 (see [1]). A function is said to be -convex if
for some , where , and .
If , the -convex function becomes a convex function on .
Theorem 1.3 ([2], Theorem 2.2). Let be a differentiable mapping on , , . (i) If is a convex function on , then (ii) If is a convex function on , for , then
Theorem 1.4 ([3], Theorems 2.3 and 2.4). Let be differentiable on , , . If is convex on , for , then
Theorem 1.5 ([4], Theorem 1–4). Let be differentiable on , , . If is -convex on , for , then
Theorem 1.6 ([5], Theorem 4). Let be differentiable on , , , and . If is -convex on for some , and , such that then
Theorem 1.7 ([6], Theorems 2.2–2.4). Let be differentiable on , , , and . (i)If is-convex on for some , then (ii)If is a -convex function on for some , then where .(iii)If is s-convex on for some , then Now we introduce the definition of the -geometrically convex function.
Definition 1.8. A function is said to be a geometrically convex function if for and .
Definition 1.9. A function is said to be a -geometrically convex function if
for some , where and .
If , the -geometrically convex function becomes a geometrically convex function on .
In this paper, we will establish some integral inequalities of Hermite-Hadamard type related to the -geometrically convex functions and then apply these inequalities to special means.
2. A Lemma
In order to prove our results, we need the following lemma.
Lemma 2.1. Let be differentiable on , and , with . If , then
Proof. Integrating by part and changing variables of integration yields This completes the proof of Lemma 2.1.
3. Main Results
Theorem 3.1. Let be differentiable on , , with , and. If is -geometrically convex and monotonically decreasing on for and , then where
Proof. (1) Since is -geometrically convex and monotonically decreasing on , from Lemma 2.1 and Hölder inequality, we have
If , then
(i)If , by (3.7), we obtain that
(ii)If , by (3.7), we obtain that
(iii)If , by (3.7), we obtain that
From (3.6) to (3.10), (3.1) holds.
(2) Since is -geometrically convex and monotonically decreasing on , from Lemma 2.1 and Hölder inequality, we have
(i)If , by (3.7), we have
(ii)If , by (3.7), we have
(iii)If , by (3.7), we have
From (3.11) to (3.14), (3.2) holds. This completes the required proof.
Applying Theorem 3.1 to , respectively, results in the following corollary.
Corollary 3.2. Let be differentiable on , with , and . If is -geometrically convex and monotonically decreasing on for , then(i) when , one has (ii) when , one has where , , , are same with (3.3)–(3.5).
Theorem 3.3. Let be differentiable on , , with , and . If is -geometrically convex and monotonically decreasing on , for and , then where and is the same as in (3.4).
Proof. (1) Since is s-geometrically convex and monotonically decreasing on , from Lemma 2.1 and Hölder inequality, we have
(i) If , we have
(ii) If , we have
(iii) If , we have
From (3.20) to (3.23), (3.17) holds.
(2) Since is -geometrically convex and monotonically decreasing on , from Lemma 2.1 and Hölder inequality, we have
From (3.24) and (3.21) to (3.23), (3.18) holds. This completes the proof.
If taking in Theorem 3.3, we can derive the following corollary.
Corollary 3.4. Letbe differentiable on , , with , and . If is geometrically convex and monotonically decreasing on for , then where , , and are the same as in Theorem 3.3.
4. Application to Special Means
Let be the arithmetic, logarithmic, generalized logarithmic means for respectively.
Let ,, and then the function is monotonically decreasing on. For , we have Hence, is -geometrically convex on for .
Theorem 4.1. Let , , and . Then In particular, if , one has
Proof. Let . Then and By Theorem 3.1, Theorem 4.1 is thus proved.
Theorem 4.2. Let ,, and . Then one has
Proof. Let . Then and Using Theorem 3.3, Theorem 4.2 is thus proved.
Acknowledgments
The research was supported by Science Research Funding of Inner Mongolia University for Nationalities (Project no. NMD1103).