The following double inequality is well known in the literature as the Hermite-Hadamard (H.H) integral inequality

provided thatis a convex function [1, page 137], [2, page 1].

This result for convex functions plays an important role in nonlinear analysis. These classical inequalities have been improved and generalized in a number of ways and applied for special means including Stolarsky type, logarithmic, and-logarithmic means. A generalization of H.H inequalities was obtained in [3–5], [2, page 5], and [1, page 143].

Theorem 1.1.

Let , be positive real numbers and , , , be real numbers such that . Then the inequalities

hold for , , and all continuous convex functions

Remark 1.2.

The inequalities given by (1.2) are strict if is a continuous strictly convex on.

If we keep the assumptions as stated in Theorem 1.1, we also have [1, page 146]

The above inequality is strict, when is strictly convex continuous function.

Let us define for by differences of (1.2) and (1.4)

where , .

Remark 1.3.

It is clear from inequalities (1.2) and (1.4) that if the conditions of Theorem 1.1 are satisfied and ( is continuous convex on ), then

Consider the following means:

where such that and . These means are known as Stolarsky means. Namely, Stolarsky introduced these means in 1975 (see [1, page 120]) and proved that for and one can get

Some simple proofs of inequality (1.8) and related results on means of Stolarsky type are given in [6].

The aim of this paper is to prove the exponential convexity of the functions deduced from (1.5) and apply these functions to generalize the means of Stolarsky type, and at last we prove the monotonicity property of these new means.

We review some necessary definitions and preliminary results.

Definition 1.4 (see [7]).

A functionis exponentially convex if it is continuous and

for each and every , such that , , .

Proposition 1.5 (see [7]).

Let , be a function. Then is exponentially convex if and only ifis continuous and

for  all , and , .

Definition 1.6 (see [1]).

A function, whereis an interval in, is said to belog-convex ifis convex, or equivalently if for all and all , one has

Corollary 1.7 (see [7]).

If is exponentially convex thenislog-convex function.

The following lemma is another way to define convex function [1, page 2].

Lemma 1.8.

If is a convex on an interval , then

holds for each , where .

In Section 2, we prove the exponential and logarithmic convexity of the functions deduced from (1.5). We also prove related mean value theorems of Cauchy type.

The following lemma gives us very important family of convex functions.

Lemma 2.1 (see [7]).

Consider a family of functions , defined as

Then is convex on for each .

Theorem 2.2.

Let , , , , , and be positive real numbers such that

where is defined in Lemma 2.1. Then

matrix is positive semidefinite for each and ; particularly,

the function is exponentially convex on ;

if , then the function is alog-convex onand the following inequality holds forsuch that;

Proof.

(i)Consider the function

for , , where is not identically zero and

This shows that is a convex function for . By setting in (1.5), respectively and from Remark 1.3, we get

or equivalently

Therefore the given matrix is a positive semidefinite. By using well-known Sylvester criterion, we have

(ii)Since for , it follows that is continuous on . Therefore, by Proposition 1.5 for , we get exponential convexity of on .

(iii)Let , then thelog-convexity ofis a simple consequence of Corollary 1.7. By setting , , , in Lemma 1.8, we have

which implies (2.4).

We will use the following lemma in the proof of mean value theorem.

Lemma 2.3 (see [1, page 4]).

Let such that

If one considers the functions , , defined by

then and are convex on .

Proof.

Therefore

that is, for are convex on.

Theorem 2.4.

Let , , , , , andbe real numbers as given in Theorem 1.1. If then there exists such that

Proof.

Since, we can take that. Now in Remark 1.3, replacingby , defined in Lemma 2.3, we have

This gives

Combining (2.16) and (14), we get

By using Remark 1.2

therefore

We get the required result.

Theorem 2.5.

Let , , , , , andbe real numbers as given in Theorem 1.1. If such that do not vanish for any , then there exits such that

Proof.

Define functions , by

where

Then using Theorem 2.4 for , we have

Using Remark 1.2

therefore

which is clearly (2.20).

Corollary 2.6.

If , , , , , and are real numbers as defined in Theorem 1.1 then for , , , and there exists such that

Remark 2.7.

If the inverse of exists, then from (2.20) we get

3. Means of Stolarsky Type

Expression (2.27) gives the means. We can consider

as a means in the broader sense. Moreover we can extend these means in other cases. Consider the following functions to cover all continuous extensions of (3.1):

where .

We have

for . We will use the following lemma to prove the monotonicity of Stolarsky type means.

Lemma 3.1.

Let be log-convex function, and if , , , , then the following inequality is valid:

The proof of this Lemma is given in [1].

Theorem 3.2.

Let , , , , , and be real numbers as defined in Theorem 1.1 and let such that , , then the following inequality is valid:

Proof.

For a convex function , a simple consequence of the definition of convex function is the following inequality [1, page 2]:

As is log-convex we set , , , , in the above inequality and get

which is equivalent to (3.5) for , . By continuity of , (3.5) is valid for , .

Remark 3.3.

If we substitute and replace and in , for , then means of Stolarsky type and related results given in [6] are obtained.

By substiting , , , , , in (2.26), we get

It follows that

To get all continuous extension of (4.2), we consider

For, we define

where is the family of functions defined in Lemma 2.1. Here we have defined as

where , , and .

We have

where .

For , we consider a family of convex functions defined onby

We have , defined as

where , . Now for  

We get means

for.

Theorem 4.1.

Theorem 2.2 is still valid if one sets .

Proof.

The proof is similar to the proof of Theorem 2.2.

Theorem 4.2.

Let , , , , , andare real numbers as defined in Theorem 1.1 also let such that , , then the following inequality is valid:

Proof.

For , in this case we use Lemma 3.1 for , and we have that

for , , , ,  . For , by substituting , , ,   ,   ,   , such that , , , in (4.12), we get

For , by substituting , , , ,   , , , such that ,   , in (4.12) we have

By raising power , to (4.13) and , to (4.14), we get (4.11) for,.

For , since islog-convex function, therefore Lemma 3.1 implies that for , , , , we have

which completes the proof.

Remark 4.3.

If we substitute , , and in the above results, then the results of generalized Stolarsky type means proved in [6] are recaptured.