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Cauchy Means of the Popoviciu Type
Journal of Inequalities and Applications volume 2009, Article number: 628051 (2009)
Abstract
We discuss log-convexity for the differences of the Popoviciu inequalities and introduce some mean value theorems and related results. Also we give the Cauchy means of the Popoviciu type and we show that these means are monotonic.
1. Introduction and Preliminaries
Let and
be two positive real valued functions with
, then from theory of convex means (cf. [1–3]), the well-known Jensen inequality gives that for
or
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ1_HTML.gif)
and vise versa for . In [4], Simic has considered the difference
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ2_HTML.gif)
The following result was given in [4] (see also [5]).
Theorem 1.1.
Let ,
be nonnegative and integrable functions for
, with
, then for
;
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ3_HTML.gif)
Remark 1.2.
For extension of Theorem 1.1 see (cf. [4]).
Popoviciu ([6–8], [9, pages 214-215]) has proved the following results.
Theorem 1.3.
Let be convex and
be continuous, increasing, and convex such that
for
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ4_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ5_HTML.gif)
If is strictly convex, then the equality in (1.4) holds if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ6_HTML.gif)
Theorem 1.4.
Let be continuous and convex, and let
be convex of order
such that
for
.
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ8_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ10_HTML.gif)
If is strictly convex, then equality in (1.7) holds if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ11_HTML.gif)
and equality in (1.8) holds if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ12_HTML.gif)
With the help of the following useful lemmas we prove our results.
Lemma 1.5.
Define the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ13_HTML.gif)
Then , that is,
is convex for
.
The following lemma is equivalent to definition of convex function (see [9, page 2]).
Lemma 1.6.
If is a convex function on
for all
for which
, the following is valid
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ14_HTML.gif)
We quote here another useful lemma from log-convexity theory (cf. [4]).
Lemma 1.7.
A positive function f is log-convex in the Jensen-sense on an open interval , that is, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ15_HTML.gif)
if and only if the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ16_HTML.gif)
holds for each real and
The following lemma given in [10] gives the relation between Beta function and Hypergeometric function
.
Lemma 1.8.
Suppose are such that
and
,
and
are Beta and Hypergeometric functions, respectively. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ17_HTML.gif)
The paper is organized in the following way. After this introduction, in the second section we discuss the log-convexity of differences of the Popoviciu inequalities (1.4), (1.7), and (1.8). In the third section we introduce some mean value theorems and the Cauchy means of the Popoviciu-type and discuss its monotonicity.
2. Main Results
Theorem 2.1.
Let be continuous, increasing, and convex such that
for
, and let
be defined in (1.5) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ18_HTML.gif)
and let be positive.
One has that is log-convex and the following inequality holds for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ19_HTML.gif)
Proof.
Consider the function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ20_HTML.gif)
where ,
is defined by (1.13) and
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ21_HTML.gif)
Therefore, is convex for
. Using Theorem 1.3,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ22_HTML.gif)
since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ23_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ24_HTML.gif)
By Lemma 1.7, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ25_HTML.gif)
that is is log-convex in the Jensen-sense for
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ26_HTML.gif)
This implies is continuous, therefore it is log-convex.
Since is log-convex, that is,
is convex, therefore by Lemma 1.6 for
and taking
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ27_HTML.gif)
which is equivalent to (2.2).
Theorem 2.2.
Let ,
be defined in Theorem 2.1 and let
be real numbers such that
,
,
,
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ28_HTML.gif)
Proof.
In (cf. [9, page 2]), we have the following result for convex function with
,
,
,
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ29_HTML.gif)
Since by Theorem 2.1, is log-convex, we can set in (2.12):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_IEq74_HTML.gif)
and ,
,
,
. We get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ30_HTML.gif)
and after applying exponential function, we get (2.11).
Theorem 2.3.
Let be convex of order
such that
for
, and let
be defined in (1.5) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ31_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ32_HTML.gif)
for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ33_HTML.gif)
and let be positive.
One has that is log-convex and the following inequality holds for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ34_HTML.gif)
Proof.
As in the proof of Theorem 2.1, we use Theorem 1.4 instead of Theorem 1.3.
Theorem 2.4.
Let ,
be defined in Theorem 2.3 and
be real numbers such that
,
,
,
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ35_HTML.gif)
Proof.
Similar to the proof of Theorem 2.2.
Lemma 2.5.
Let be convex of order
such that
for
,
be defined in (1.5) and
be defined in (1.10), and let
and
are Beta and Hypergeometric functions respectively, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ36_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ37_HTML.gif)
for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ38_HTML.gif)
Proof.
First, we solve these three integrals
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ39_HTML.gif)
Take
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ40_HTML.gif)
Substitute
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ41_HTML.gif)
and limits, when then
, when
then
. So,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ42_HTML.gif)
By using Lemma 1.8 with ,
,
,
,
such that
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ43_HTML.gif)
Take second integral
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ44_HTML.gif)
using integration by parts, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ45_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ46_HTML.gif)
By using same substitution (A) as above, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ47_HTML.gif)
By using Lemma 1.8 with ,
,
,
,
such that
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ48_HTML.gif)
Now, take third integral
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ49_HTML.gif)
Using integration by parts, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ50_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ51_HTML.gif)
By using same substitution (A) as above, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ52_HTML.gif)
By using Lemma 1.8 with ,
,
,
,
such that
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ53_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ54_HTML.gif)
By using same substitution (A) as above, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ55_HTML.gif)
By using Lemma 1.8 with ,
,
,
,
such that
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ56_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ57_HTML.gif)
For ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ58_HTML.gif)
Using , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ59_HTML.gif)
For ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ60_HTML.gif)
Using , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ61_HTML.gif)
For ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ62_HTML.gif)
Using , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ63_HTML.gif)
Theorem 2.6.
Let be convex of order
such that
for
,
be defined in (1.5) and let
defined in (2.20) be positive.
One has that is log-convex and the following inequality holds for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ64_HTML.gif)
Proof.
As in the proof of Theorem 2.1, we use Theorem 1.4 instead of Theorem 1.3.
Theorem 2.7.
Let ,
be defined in Theorem 2.6 and
be real numbers such that
,
,
,
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ65_HTML.gif)
Proof.
Similar to the proof of Theorem 2.2.
3. Cauchy Means
Let us note that (2.11) has the form of some known inequalities between means (e.g., Stolarsky's means, etc.). Here we prove that expressions on both sides of (2.11) are also means.
Lemma 3.1.
Let be such that
is bounded, that is,
Then the functions
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ66_HTML.gif)
are convex functions.
Theorem 3.2.
Let
is a compact interval in
and
be a continuous, increasing and convex such that
for
,
be defined in (1.5) then
,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ67_HTML.gif)
Proof.
Suppose for
. Then by applying
and
defined in Lemma 3.1 for
in (1.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ68_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ69_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ70_HTML.gif)
By combining (3.4) and (3.5) and using the fact that for there exists
such that
we get (3.2).
Theorem 3.3.
Let and satisfy (3.2), f be a continuous, increasing and convex such that
for
,
be defined in (1.5), and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ71_HTML.gif)
then there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ72_HTML.gif)
Provided that denominators are non-zero.
Proof.
Consider the linear functionals and
such that
for some function
and
. Consider the following linear combination
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ73_HTML.gif)
where and
are defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ74_HTML.gif)
Since and satisfy (3.2), therefore
as linear combination of
and
should also satisfy (3.2).
Let be defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ75_HTML.gif)
Obviously, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ76_HTML.gif)
On the other hand, there is an such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ77_HTML.gif)
By using the linearity property of the operator
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ78_HTML.gif)
Now and
, we have from the last equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ79_HTML.gif)
After putting values, we get (3.7).
Corollary 3.4.
Let f be a continuous, increasing and convex such that for
, then for
there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ80_HTML.gif)
where and
.
Proof.
Set and
in (3.7) we get (3.15).
Remark 3.5.
Since the function is invertible, therefore from (3.15) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ81_HTML.gif)
In fact, similar result can also be given for (3.7). Namely, suppose that has inverse function. Then from (3.7) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ82_HTML.gif)
The expression on the right-hand side of (3.17) is also a mean.
From the inequality (3.16), we can define means as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ83_HTML.gif)
for . Moreover we can extend these means in other cases. By limit we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ84_HTML.gif)
where denotes
,
denotes
,
denotes
,
denotes
,
denotes
, and
denotes
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ85_HTML.gif)
In our next result we prove that this new mean is monotonic.
Theorem 3.6.
Let , then the following inequality is valid
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F628051/MediaObjects/13660_2008_Article_1984_Equ86_HTML.gif)
Proof.
Since is log-convex, therefore by (2.11) we get (3.21).
Remark 3.7.
Similar results of the Cauchy means and related results can also proved for Theorems 2.3 and 2.6.
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Acknowledgments
This research work is funded by Higher Education Commission Pakistan. The research of the third author is supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 117-1170889-0888.
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Anwar, M., Latif, N. & Pečarić, J. Cauchy Means of the Popoviciu Type. J Inequal Appl 2009, 628051 (2009). https://doi.org/10.1155/2009/628051
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DOI: https://doi.org/10.1155/2009/628051