Abstract
We use an inequality given by Matić and Pečarić (2000) and obtain improvement and reverse of Slater's and related inequalities.
Journal of Inequalities and Applications volume 2010, Article number: 646034 (2010)
We use an inequality given by Matić and Pečarić (2000) and obtain improvement and reverse of Slater's and related inequalities.
In 1981 Slater has proved an interesting companion inequality to Jensen's inequality [1].
Theorem 1.1.
Suppose that is increasing convex function on interval , for (where is the interior of the interval ) and for with , if , then
When is strictly convex on , inequality (1.1) becomes equality if and only if for some and for all with .
It was noted in [2] that by using the same proof the following generalization of Slater's inequality (1981) can be given.
Theorem 1.2.
Suppose that is convex function on interval , for (where is the interior of the interval ) and for with . Let
then inequality (1.1) holds.
When is strictly convex on , inequality (1.1) becomes equality if and only if for some and for all with .
Remark 1.3.
For multidimensional version of Theorem 1.2 see [3].
Another companion inequality to Jensen's inequality is a converse proved by Dragomir and Goh in [4].
Theorem 1.4.
Let be differentiable convex function defined on interval . If are arbitrary members and with and let
Then the inequalities
hold.
In the case when is strictly convex, one has equalities in (1.4) if and only if there is some such that holds for all with
Matić and Pečarić in [5] proved more general inequality from which (1.1) and (1.4) can be obtained as special cases.
Theorem 1.5.
Let be differentiable convex function defined on interval and let and be stated as in Theorem 1.4. If is arbitrary chosen number, then one has
Also, when is strictly convex, one has equality in (1.5) if and only if holds for all with .
Remark 1.6.
If and are stated as in Theorem 1.4 and we let , also if , then by setting in (1.5), we get Slater's inequality (1.1) and similarly by setting in (1.5), we get (1.4).
The following refinement of (1.4) is also valid [5].
Theorem 1.7.
Let be strictly convex differentiable function defined on interval and let and be stated as in Theorem 1.4 and , then the inequalities
hold.
The equalities hold in (1.6) and in (1.7) if and only if
Remark 1.8.
In [6] Dragomir has also proved Theorem 1.7.
In this paper, we use an inequality given in [5] and derive two mean value theorems, exponential convexity, log-convexity, and Cauchy means. As applications, such results are also deduce for related inequality. We use some log-convexity criterion and prove improvement and reverse of Slater's and related inequalities. We also prove some determinantal inequalities.
Theorem 2.1.
Let , where is closed interval in , and let , , with and . Then there exists such that
Proof.
Since is continuous on , for , where and .
Consider the functions , defined as
Since
for are convex.
Now by applying for in inequality (1.5), we have
From (2.4) we get
and similarly by applying for in (1.5), we get
Since
by combining (2.5) and (2.6), we have
Now using the fact that for there exists such that , we get (2.1).
Corollary 2.2.
Let , where is closed interval in , and let , and be stated as in Theorem 1.4 with and . Then there exists such that
Proof.
By setting in Theorem 2.1, we get (2.9).
Theorem 2.3.
Let , where is closed interval in , and let , and with . Then there exists such that
provided that the denominators are nonzero.
Proof.
Let the function be defined by
where and are defined as
Then, using Theorem 2.1 with , we have
because .
Since as and , therefore, (2.13) gives us
After putting the values of and , we get (2.10).
Corollary 2.4.
Let , where is closed interval in , and , and let , with . Then there exists such that
provided that the denominators are nonzero.
Proof.
By setting in Theorem 2.3, we get (2.15).
Corollary 2.5.
Let with and , . Then for , , there exists , where is positive closed interval, such that
Proof.
By setting and , , in Theorem 2.3, we get (2.16).
Corollary 2.6.
Let , , and with . Then for , , there exists , where is positive closed interval, such that
Proof.
By setting and , , in (2.15), we get (2.17).
Remark 2.7.
Note that we can consider the interval , where ,
Since the function with is invertible, then from (2.16) we have
We will say that the expression in the middle is a mean of .
From (2.17) we have
The expression in the middle of (2.19) is a mean of .
In fact similar results can also be given for (2.10) and (2.15). Namely, suppose that has inverse function, then from (2.10) and (2.15) we have
So, we have that the expression on the right-hand side of (2.20) is also means.
Definition 3.1 (see [7, page 2]).
A function is convex if
holds for every ,
Lemma 3.2 (see [8]).
Let one define the function
Then , that is, is convex for .
Definition 3.3 (see [9]).
A function is exponentially convex if it is continuous and
for all , and , such that or equivalently
Corollary 3.4 (see [9]).
If is exponentially convex function, then
for every
Corollary 3.5 (see [9]).
If is exponentially convex function, then is a log-convex function that is
Theorem 3.6.
Let , . Consider to be defined by
Then
(i)for every and for every , the matrix is a positive semidefinite matrix; particularly
(ii)the function is exponentially convex;
(iii)if , then the function is log-convex, that is, for , one has
Proof.
Let us consider the function defined by
where for all
Then we have
Therefore, is convex function for . Using in inequality (1.5), we get
so the matrix is positive semi-definite.
Since and , so is continuous for all and we have exponentially convexity of the function .
Let , then by Corollary 3.5 we have that is log-convex, that is, is convex, and by (3.1) for and taking , we get
which is equivalent to (3.9).
Corollary 3.7.
Let , and . Consider to be defined by
Then
(i)for every and for every , the matrix is a positive semi-definite matrix. Particularly
(ii)the function is exponentially convex;
(iii)if , then the function is log-convex, that is, for , one has
Proof.
To get the required results, set in Theorem 3.6.
Let be positive n-tuple and positive real numbers, and let . Let denote the power mean of order , defined by
Let us note that .
By (2.18) we can give the following definition of Cauchy means.
Let with , is positive closed interval, and ,
for are means of . Moreover we can extend these means to the other cases.
So by limit we have
where .
Theorem 3.8.
Let such that , then the following inequality is valid:
Proof.
For convex function it holds that ([7, page 2])
with , , , Since by Theorem 3.6, is log-convex, we can set in (3.21): , , , , and , then we get
From (3.22) we get (3.20) for and .
For and we have limiting case.
Similarly by (2.19) we can give the following definition of Cauchy type means.
Let with , is positive closed interval, and
for are means of . Moreover we can extend these means to the other cases.
So by limit we have
where .
Theorem 3.9.
Let such that , then the following inequality is valid:
Proof.
The proof is similar to the proof of Theorem 3.8.
Let be stated as above, define as
The following improvement and reverse of Slater's inequality are valid.
Theorem 3.10.
Let , . Let be defined by
Then
for and .
for .
where,
Proof.
By setting in (3.7), becomes and for , setting in (3.9), we get
that is,
From (3.32) we get (3.28), and similarly for (3.9) becomes
by the same process we can get (3.28).
For (3.9) becomes
setting in (3.34), we get (3.29).
Theorem 3.11.
Let , .
Then for every and for every , the matrices , are positive semi-definite matrices. Particularly
where is defined by (3.30).
Proof.
By setting and in Theorem 3.6(i), we get the required results.
Remark 3.12.
We note that . So by setting in (3.35), we have special case of (3.28) for , and if and for , and if . Similarly by setting in (3.36), we have special case of (3.29) for if and for if .
Let be stated as above, define as
The following improvement and reverse of inequality (1.6) are also valid.
Theorem 3.13.
Let for all , . Let be defined by
Then
for and .
for ,
where
Proof.
By setting in (3.9), we get (3.39) for , and similarly we can get (3.39) for the case .
For (3.9) becomes
setting in (3.42), we get (3.40).
Theorem 3.14.
Let , .
Then for every and for every , the matrices , are positive semi-definite matrices. Particularly
where is defined by (3.41).
Proof.
By setting and in Theorem 3.6(i), we get the required results.
Remark 3.15.
We note that . So by setting in (3.43), we have special case of (3.39) for if and for , and if . Similarly by setting in (3.44), we have special case of (3.40) for , and if and for , and if .
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The research of the first and second authors was funded by Higher Education Commission, Pakistan. The research of the second author was supported by the Croatian Ministry of Science, Education, and Sports under the Research Grant 117-1170889-0888.
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Adil Khan, M., Pečarić, J. Improvement and Reversion of Slater's Inequality and Related Results. J Inequal Appl 2010, 646034 (2010). https://doi.org/10.1155/2010/646034
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DOI: https://doi.org/10.1155/2010/646034