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Generalization of Stolarsky Type Means
Journal of Inequalities and Applications volume 2010, Article number: 720615 (2010)
Abstract
We generalize means of Stolarsky type and show the monotonicity of these generalized means.
1. Introduction and Preliminaries
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.
2. Main Results
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.
4. Generalized Means of Stolarsky Type
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.
References
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Acknowledgment
This research was partially funded by Higher Education Commission, Pakistan. The research of the _rst author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant 117-1170889-0888.
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Pečarić, J., Roqia, G. Generalization of Stolarsky Type Means. J Inequal Appl 2010, 720615 (2010). https://doi.org/10.1155/2010/720615
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DOI: https://doi.org/10.1155/2010/720615