#### Abstract

Some new integral inequalities of Hermite-Hadamard
type related to the *s*-geometrically convex functions are established and some
applications to special means of positive real numbers are also given.

#### 1. Introduction

In this section, we firstly list several definitions and some known results.

*Definition 1. *Let be an interval in . Then is said to be convex if
for all and .

*Definition 2 (see [1]). *Let . A function is said to be -convex in the second sense if
for all and .

*Definition 3 (see [2]). *A function is said to be a geometrically convex function if
for and .

*Definition 4 (see [2]). *A function is said to be an -geometrically convex function if
for some , where and .

*Example 5. *(i) , is -geometrically convex for all .

(ii) , is -geometrically convex for all , where is a fixed number.

Let . It is easy to show that is -geometrically convex function on with if and only if is -convex function on .

Let be a convex function defined on the interval of real numbers and with . The following double inequality is well known in the literature as Hermite-Hadamard integral inequality: Recently, several integral inequalities connected with inequalities (5) for the -convex functions have been established by many authors; for example, see [1, 3–6]. In [2], the authors have established some integral inequalities connected with inequalities (5) for the -geometrically convex and monotonically decreasing functions. In [7], Tunç has established inequalities for -geometrically and geometrically convex functions which are connected with the famous Hermite-Hadamard inequality holding for convex functions. In [7], Tunç also has given the following result for geometrically convex and monotonically decreasing functions.

Corollary 6. *Let be geometrically convex and monotonically decreasing on , with ; then one has
*

Note that inequalities (6) are also true without the condition monotonically decreasing and inequalities (6) are sharp. Also, inequality (6) obtained for geometrically convex functions is analogous with Hermite-Hadamard inequality (5).

Since is a geometrically convex function if and only if is a geometrically convex function, inequality (6) can be given as follows:

In this paper, the author gives new identities for differentiable functions. A consequence of the identities is that the author establishes some new inequalities connected with inequalities (6) for the -geometrically convex functions.

#### 2. Main Results

In order to prove our results, we need the following lemma.

Lemma 7. *Let be differentiable on and with . If , then
*

*Proof. *Integrating by part and changing variables of integration yield
By the following equality, we obtain inequality (8):
This completes the proof of Lemma 7.

Theorem 8. *Let be differentiable on and with and . If is -geometrically convex on , for and , then
**
where , ,
*

*Proof. *(1) Let . Since is -geometrically convex on , from Lemma 7 and power mean inequality, we have
If , then
(i)If , by (18), we obtain that
(ii)If , by (18), we obtain that
(iii)If , by (18), we obtain that
(iv)If , by (18), we obtain that
From (17) to (22), (12) holds.

(2) Let . Since is -geometrically convex on , from Lemma 7 and Hölder inequality, we have
(i)If , by (18), we obtain that
(ii)If , by (18), we obtain that
(iii)If , by (18), we obtain that
(iv)If , by (18), we obtain that
From (23) to (27), (13) holds. This completes the required proof.

If we take in Theorem 8, we can derive the following corollary.

Corollary 9. *Let be differentiable on and with and . If is geometrically convex on , for , then
**
where , , and are the same as in Theorem 8.*

If we take in Theorem 8, we can derive the following corollary.

Corollary 10. *Let be differentiable on and with and . If is -geometrically convex on , for , then
**
where , , , , and are the same as in Theorem 8.*

Theorem 11. *Let be differentiable on and with and . If is -geometrically convex on , for and , then
**
where , ,
**
and , are the same as in (15).*

*Proof. *(1) Let . Since is -geometrically convex on , from Lemma 7 and Hölder inequality, we have
(i)If , by (18), we have
(ii)If , by (18), we have
(iii)If , by (18), we obtain that
(iv)If , by (18), we obtain that
From (33) to (37), (30) holds.

(2) Let , Since is -geometrically convex on , from Lemma 7 and Hölder inequality, we have
From (38) and (34) to (37), (31) holds.

If taking in Theorem 11, we can derive the following corollary.

Corollary 12. *Let be differentiable on and with and . If is geometrically convex on , for , then
**
where , , and are the same as in Theorem 11.*

#### 3. Application to Special Means

Let us recall the following special means of two nonnegative numbers with .(1)The arithmetic mean (2)The geometric mean (3)The logarithmic mean (4)The -logarithmic mean

Let , and then the function is -geometrically convex on for (see [2]).

Proposition 13. *Let , and . Then
**
where and .*

*Proof. *Let . Then , , . Thus, by Theorem 8, Proposition 13 is proved.

Proposition 14. *Let , and . Then
**
where and .*

*Proof. *Let . Then , , . Thus, by Theorem 11, Proposition 14 is proved.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.