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On Logarithmic Convexity for Power Sums and Related Results
Journal of Inequalities and Applications volume 2008, Article number: 389410 (2008)
Abstract
We give some further consideration about logarithmic convexity for differences of power sums inequality as well as related mean value theorems. Also we define quasiarithmetic sum and give some related results.
1. Introduction and Preliminaries
Let ,
denote two sequences of positive real numbers with
. The well-known Jensen Inequality [1, page 43] gives the following, for
or
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ1_HTML.gif)
and vice versa for .
Simić [2] has considered the difference of both sides of (1.1). He considers the function defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ2_HTML.gif)
and has proved the following theorem.
Theorem 1.1.
For , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ3_HTML.gif)
Anwar and Pečarić [3] have considerd further generalization of Theorem 1.1. Namely, they introduced new means of Cauchy type in [4] and further proved comparison theorem for these means.
In this paper, we will give some results in the case where instead of means we have power sums.
Let be positive -tuples. The well-known inequality for power sums of order
and
, for
(see [1, page 164]), states that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ4_HTML.gif)
Moreover, if is a positive -tuples such that
, then for
(see [1, page 165]), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ5_HTML.gif)
Let us note that (1.5) can also be obtained from the following theorem [1, page 152].
Theorem 1.2.
Let and
be two nonnegative
-tuples such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ6_HTML.gif)
If is an increasing function, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ7_HTML.gif)
Remark 1.3.
Let us note that if is a strictly increasing function, then equality in (1.7) is valid if we have equalities in (1.6) instead of inequalities, that is,
and
The following similar result is also valid [1, page 153].
Theorem 1.4.
Let be an increasing function. If
and if the following hold.
-
(i)
there exists an
such that
(1.8)
where and
, then (1.7) holds.
-
(ii)
If there exists an
such that
(1.9)
then the reverse of inequality in (1.7) holds.
In this paper, we will give some applications of power sums. That is, we will prove results similar to those shown in [2, 3], but for power sums.
2. Main Results
Lemma 2.1.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ10_HTML.gif)
Then is a strictly increasing function for
.
Proof.
Since for
, therefore
is a strictly increasing function for
.
Lemma 2.2 ([2]).
A positive function is log convex in Jensen's sense on an open interval
, that is, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ11_HTML.gif)
if and only if the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ12_HTML.gif)
holds for each real and
.
The following lemma is equivalent to the definition of convex function (see [1, page 2]).
Lemma 2.3.
If is continuous and convex for all
,
,
of an open interval
for which
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ13_HTML.gif)
Theorem 2.4.
Let and
be two positive
-tuples
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ14_HTML.gif)
such that condition (1.6) is satisfied and all 's are not equal. Then
is log-convex, also for
where
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ15_HTML.gif)
Proof.
Since is a strictly increasing function for
and all
's are not equal, therefore by Theorem 1.2 with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ16_HTML.gif)
that is, is a positive-valued function.
Let where
and
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ17_HTML.gif)
This implies that is monotonically increasing.
By Theorem 1.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ18_HTML.gif)
Now by Lemma 2.2, we have that is log-convex in Jensen sense.
Since it follows that
is continuous, therefore it is a log-convex function [1, page 6].
Since is log-convex, that is,
is convex, we have by Lemma 2.3 that, for
with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ19_HTML.gif)
which is equivalent to (2.6).
Similar application of Theorem 1.4 gives the following.
Theorem 2.5.
Let and
be two positive
-tuples
such that
, all
's are not equal and
-
(i)
if
such that condition (1.8) is satisfied, then
is log-convex. Also for
, we have
(2.11)
-
(ii)
moreover if
and (1.9) is satisfied, then we have that
is log-convex and
(2.12)
We will also use the following lemma.
Lemma 2.6.
Let f be a log-convex function and assume that if . Then the following inequality is valid:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ22_HTML.gif)
Proof.
In [1, page 2], we have the following result for convex function , with
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ23_HTML.gif)
Putting , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ24_HTML.gif)
from which (2.13) immediately follows.
Let us introduce the following.
Definition 2.7.
Let and
be two nonnegative -tuples
such that
then for
, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ25_HTML.gif)
Remark 2.8.
Let us note that ,
and
.
Theorem 2.9.
Let such that
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ26_HTML.gif)
Proof.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ27_HTML.gif)
Now taking , where
, and
in Lemma 2.6, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ28_HTML.gif)
Since by substituting
and
, where
, in above inequality, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ29_HTML.gif)
By raising power , we get (2.17) for
.
From Remark 2.8, we get (2.17) is also valid for or
or
or
.
Corollary 2.10.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ30_HTML.gif)
Then for and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ31_HTML.gif)
Proof.
Taking in (2.17), we get (2.22).
3. Mean Value Theorems
Lemma 3.1.
Let , where
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ32_HTML.gif)
Consider the functions and
defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ33_HTML.gif)
Then for
are monotonically increasing functions.
Proof.
We have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ34_HTML.gif)
that is, for
are monotonically increasing functions.
Theorem 3.2.
Let and
be two positive
-tuples
satisfy condition (1.6), all
's are not equal and let
, where
. Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ35_HTML.gif)
Proof.
In Theorem 1.2, setting and
, respectively, as defined in Lemma 3.1, we get the following inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ36_HTML.gif)
Now by combining both inequalities, we get,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ37_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_IEq132_HTML.gif)
is nonzero, it is zero if equalities are given in conditions (1.6), that is, and
.
Now by condition (3.1), there exist , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ38_HTML.gif)
and (3.7) implies (3.4).
Theorem 3.3.
Let and
be two positive
-tuples
satisfy condition (1.6), all
's are not equal and let
, where
. Then there exists
such that the following equality is true:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ39_HTML.gif)
provided that the denominators are nonzero.
Proof.
Let a function be defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ40_HTML.gif)
where and
are defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ41_HTML.gif)
Then, using Theorem 3.2 with , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ42_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ43_HTML.gif)
therefore, (3.11) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ44_HTML.gif)
After putting values, we get (3.8).
Let be a strictly monotone continuous function then quasiarithmetic sum is defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ45_HTML.gif)
Theorem 3.4.
Let and
be two positive
-tuples
, all
's are not equal and let
be strictly monotonic continuous functions,
be positive strictly increasing continuous function, where
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ46_HTML.gif)
Then there exists from
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ47_HTML.gif)
is valid, provided that all denominators are not zero.
Proof.
If we choose the functions and
so that
,
and
. Substituting these in (3.8),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ48_HTML.gif)
Then by setting , we get (3.16).
Corollary 3.5.
Let and
be two nonnegative
-tuples and let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F389410/MediaObjects/13660_2008_Article_1810_Equ49_HTML.gif)
Proof.
If and
are pairwise distinct, then we put
,
and
in (3.16) to get (3.18).
For other cases, we can consider limit as in Remark (2.8).
References
Pečarić JE, Proschan F, Tong YL: Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering. Volume 187. Academic Press, Boston, Mass, USA; 1992:xiv+467.
Simić S: On logarithmic convexity for differences of power means. Journal of Inequalities and Applications 2007, 2007:-8.
Anwar M, Pečarić JE: On logarithmic convexity for differences of power means. to appear in Mathematical Inequalities & Applications
Anwar M, Pečarić JE: New means of Cauchy's type. Journal of Inequalities and Applications 2008, 2008:-10.
Acknowledgment
The authors are really very thankful to Mr. Martin J. Bohner for his useful suggestions.
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Pečarić, J., Rehman, A.U. On Logarithmic Convexity for Power Sums and Related Results. J Inequal Appl 2008, 389410 (2008). https://doi.org/10.1155/2008/389410
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DOI: https://doi.org/10.1155/2008/389410