- Research Article
- Open access
- Published:
Refinements of Results about Weighted Mixed Symmetric Means and Related Cauchy Means
Journal of Inequalities and Applications volume 2011, Article number: 350973 (2011)
Abstract
A recent refinement of the classical discrete Jensen inequality is given by Horváth and Pečarić. In this paper, the corresponding weighted mixed symmetric means and Cauchy-type means are defined. We investigate the exponential convexity of some functions, study mean value theorems, and prove the monotonicity of the introduced means.
1. Introduction and Preliminary Results
A new refinement of the discrete Jensen inequality is given in [1]. The following notations are also introduced in [1].
Let be a set,
its power set, and
denotes the number of elements in
. Let
and
be fixed integers. Define the functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ1_HTML.gif)
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ2_HTML.gif)
Further, introduce the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ3_HTML.gif)
via
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ4_HTML.gif)
For each , let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ5_HTML.gif)
It is easy to observe from the construction of the functions and
that they do not depend essentially on
, so we can write for short
for
, and so on.
(H1) The following considerations concern a subset of
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ6_HTML.gif)
where and
are fixed integers.
Next, we proceed inductively to define the sets by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ7_HTML.gif)
By (1.6), and this implies that
for
. From (1.6), again, we have
.
For every and for any
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ8_HTML.gif)
Using these sets we define the functions inductively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ9_HTML.gif)
Let be an interval in
, let
, let
such that
and
, and let
be a convex function. For any
, set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ10_HTML.gif)
and associate to each the number
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ11_HTML.gif)
We need the following hypotheses.
(H2)Let and
be positive
-tuples such that
.
(H3) Let be an interval, let
, let
be a positive
-tuples such that
, and let
,
be continuous and strictly monotone functions.
(H4) Let be an interval, let
, and let
be positive
-tuples such that
. Further, let
be a convex function.
Assume (H1) and (H2). The power means of order corresponding to
are given as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ12_HTML.gif)
We also use the means
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ13_HTML.gif)
For ,
, we introduce the mixed symmetric means with positive weights as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ14_HTML.gif)
and, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ15_HTML.gif)
We deduce the monotonicity of these means from the following refinement of the discrete Jensen inequality in [1].
Theorem 1.1.
Assume (H1) and (H4). Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ16_HTML.gif)
where the numbers are defined in (1.10) and (1.11). If
is a concave function, then the inequalities in (1.16) are reversed.
Under the conditions of the previous theorem,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ17_HTML.gif)
Corollary 1.2.
Assume (H1) and (H2). Let ,
such that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ19_HTML.gif)
Proof.
Assume . To obtain (1.18), we can apply Theorem 1.1 to the function
and the
-tuples
to get the analogue of (1.16) and to raise the power
. Equation (1.19) can be proved in a similar way by using
and
and raising the power
.
When or
, we get the required results by taking limit.
Assume (H1) and (H3). Then, we define the quasiarithmetic means with respect to (1.10) and (1.11) as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ20_HTML.gif)
and, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ21_HTML.gif)
The monotonicity of these generalized means is obtained in the next corollary.
Corollary 1.3.
Assume (H1) and (H3). For a continuous and strictly monotone function , one defines
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ22_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ23_HTML.gif)
if either is convex and
is increasing or
is concave and
is decreasing,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ24_HTML.gif)
if either is convex and
is decreasing or
is concave and
is increasing.
Proof.
First, we can apply Theorem 1.1 to the function and the
-tuples
, then we can apply
to the inequality coming from (1.16). This gives (1.23). A similar argument gives (1.24):
and
can be used.
Throughout Examples 1.4-1.5, 1.9–1.12, which are based on examples in [1], the conditions (H2), in the mixed symmetric means, and (H3), in the quasiarithmetic means, will be assumed.
Example 1.4.
Suppose
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ25_HTML.gif)
where means that
divides
. Since
, therefore (1.6) holds. We note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ26_HTML.gif)
where is the largest positive integer not greater than
, and
means the number of positive divisors of
. Then, (1.14) gives for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ27_HTML.gif)
while (1.20) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ28_HTML.gif)
Assume (H4) holds, and consider the set in Example 1.4. Then, Theorem 1.1 implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ29_HTML.gif)
and thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ30_HTML.gif)
Example 1.5.
Let be an integer
, let
, and also let
consist of all sequences
in which the number of occurrences of
is
. Obviously, (1.6) holds, and, by simple calculations, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ31_HTML.gif)
Moreover, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ32_HTML.gif)
Under the above settings, (1.15) can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ33_HTML.gif)
while (1.21) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ34_HTML.gif)
Assume (H4) holds, and consider the set in Example 1.5. Then, Theorem 1.1 yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ35_HTML.gif)
This shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ36_HTML.gif)
The following result is also given in [1].
Theorem 1.6.
Assume (H1) and (H4), and suppose for any
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ37_HTML.gif)
and thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ38_HTML.gif)
If is a concave function then the inequalities (1.38) are reversed.
Under the conditions of the previous theorem, we have, from (1.38), that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ39_HTML.gif)
Assume (H1) and (H2), and suppose for any
. In this case, the power means of order
corresponding to
has the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ40_HTML.gif)
Now, for ,
and
, we introduce the mixed symmetric means with positive weights related to (1.37) as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ41_HTML.gif)
Corollary 1.7.
Assume (H1) and (H2), and suppose for any
. Let
,
such that
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ42_HTML.gif)
Proof.
The proof comes from Corollary 1.2.
Assume (H1) and (H3), and suppose for any
. We define for
the quasiarithmetic means with respect to (1.37) as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ43_HTML.gif)
Corollary 1.8.
Assume (H1) and (H3), and suppose for any
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ44_HTML.gif)
where either is convex and
is increasing or
is concave and
is decreasing,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ45_HTML.gif)
where either is convex and
is decreasing or
is concave and
is increasing.
Proof.
The proof is a consequence of Corollary 1.3.
Example 1.9.
If we set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ46_HTML.gif)
then , that is, (1.6) is satisfied for
. It comes easily that
,
, and for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ47_HTML.gif)
In this case, (1.41) becomes for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ48_HTML.gif)
and (1.43) has the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ49_HTML.gif)
Equation (1.48) is a weighted mixed symmetric mean and (1.49) is a generalized mean, as given in [2]. Therefore, Corollaries 1.7 and 1.8 are more general than the Corollaries 1.2 and 1.3 given in [2].
Assume (H4) holds, and consider the set in Example 1.9. Then, Theorem 1.6 shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ50_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ51_HTML.gif)
Example 1.10.
If we set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ52_HTML.gif)
then and thus (1.6) is satisfied. It is easy to see that
,
, and for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ53_HTML.gif)
Under these settings (1.41) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ54_HTML.gif)
and (1.43) has the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ55_HTML.gif)
Equation (1.54) represents weighted mixed symmetric means, and (1.55) defines generalized means, as given in [2]. Therefore, Corollaries 1.7 and 1.8 are more general than the Corollaries 1.9 and 1.10 given in [2].
Assume (H4) holds, and consider the set in Example 1.10. Then, it follows from Theorem 1.6 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ56_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ57_HTML.gif)
This yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ58_HTML.gif)
Example 1.11.
We set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ59_HTML.gif)
Then, , hence (1.6) holds. It is not hard to see that
, and for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ60_HTML.gif)
Therefore, under these settings, for , (1.41) leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ61_HTML.gif)
and (1.43) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ62_HTML.gif)
respectively.
Assume (H4) holds, and consider the set in Example 1.11. Then, Theorem 1.6 implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ63_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ64_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ65_HTML.gif)
Example 1.12.
Let and let
consist of all sequences
of
distinct numbers from
. Then,
, hence (1.6) holds. It is immediate that
, and for every
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ66_HTML.gif)
Therefore under these settings, for , (1.41) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ67_HTML.gif)
and (1.43) has the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ68_HTML.gif)
respectively.
Assume (H4) holds, and consider the set in Example 1.12. Then, Theorem 1.6 yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ69_HTML.gif)
where for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ70_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ71_HTML.gif)
2. Main Results
We have seen that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ72_HTML.gif)
From now on, (H1) and (H4) are assumed if we consider , further, the hypothesis
for any
is also assumed if we consider
. The numbers
are generated by concrete examples, and (H4) is assumed.
We need the following subclass of convex functions (see [3]).
Definition 2.1.
A function is exponentially convex if it is continuous and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ73_HTML.gif)
for all and all choices
and
.
We quote here useful propositions from [3].
Proposition 2.2.
Let be a function. Then, the following statements are equivalent
(i) is exponentially convex.
(ii) is continuous and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ74_HTML.gif)
for every and every
.
Proposition 2.3.
If is an exponentially convex function, then
is log-convex which means that for every
,
and all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ75_HTML.gif)
First, we introduce a special class of functions.
(H5) Let be a set of twice differentiable convex functions such that the function
is exponentially convex for every fixed
.
As examples, consider two classes of functions defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ76_HTML.gif)
and defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ77_HTML.gif)
It is easy to see that the sets of functions and
satisfy (H5).
Assume (H5). If is replaced by
in (2.1), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ78_HTML.gif)
Especially,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ79_HTML.gif)
In this paper we prove the exponential convexity of the functions , and we give mean value theorems for
. We also define the respective means of Cauchy type and study their monotonicity. The results for
are special cases of the results for
, and the results for
are special cases of results for
. Especially, the results for
are also given in [2].
Theorem 2.4.
Assume (H5), and suppose that the functions are continuous. The following statements hold for
.
(a)For every and
, the matrix
is positive semidefinite. Particularly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ80_HTML.gif)
(b)The function is exponentially convex.
Proof.
Fix .
(a)Let , and define the functions
by
for
, where
. Then
is a convex function since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ81_HTML.gif)
By taking in (2.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ82_HTML.gif)
This means that the matrix is positive semidefinite, that is, (2.9) is valid.
(b)It is assumed that the function is continuous. By using Poposition 2.2 and (a), we get the exponential convexity of the function
.
Since the functions are continuous
, we have the following.
Corollary 2.5.
The function are exponentially convex. This remains valid if we replace
by
in (2.8).
3. Cauchy Means
In this section, first, we are interested in mean value theorems.
Theorem 3.1.
Assume and
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ83_HTML.gif)
Theorem 3.2.
Assume ,
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ84_HTML.gif)
provided that the denominators are nonzero.
The idea of the proofs of Theorems 3.1 and 3.2 is the same as the proofs of Theorems 2.3 and 2.4 in [2].
Corollary 3.3.
Let ,
,
and
. Then, for distinct real numbers
and
, different from 0 and 1, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ85_HTML.gif)
Proof.
Theorem 3.2 can be applied.
Remark 3.4.
Suppose the conditions of Corollary 3.3 are satisfied.
(a)Since the function ,
is invertible, then we get, from (3.3), that for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ86_HTML.gif)
-
(b)
By choosing
and
, we can see that the expression between
and
in (3.4) defines a mean.
Corollary 3.5.
Assume (H1) and (H2), and suppose . In (3.6), it is also supposed that
for any
. Then, for distinct real numbers
,
, and
, all are different from 0 and 1, there exists
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ87_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ88_HTML.gif)
Proof.
We can apply Theorem 3.2 to the functions ,
,
, and
, and the
-tuples
.
Remark 3.6.
Suppose the conditions of Corollary 3.5 are satisfied.
(a)Since the function is invertible, then we get, from (3.5) and (3.6) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ89_HTML.gif)
(b)As in Remark 3.4 (b), the expressions in (3.7) define means.
By Remark 3.4 (b), we can define Cauchy means for as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ90_HTML.gif)
Moreover, we have continuous extensions of these means in other cases. By taking the limit, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ91_HTML.gif)
Now, we deduce the monotonicity of these means in the form of Dresher's inequality as follows.
Theorem 3.7.
Let such that
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ92_HTML.gif)
Proof.
Fix . Corollary 2.5 shows that the function
is exponentially convex, and hence, by Proposition 2.3, it is log-convex. Therefore, the function
is convex, which implies (see [4]) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ93_HTML.gif)
This gives (3.10) if and
. The other cases come from this by taking limit.
By Remark 3.6 (b), we can define Cauchy means in the following form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ94_HTML.gif)
where . By taking the limit, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ95_HTML.gif)
where .
Now, we give the monotonicity of these new means.
Theorem 3.8.
Let such that
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ96_HTML.gif)
Proof.
Suppose , 2 is fixed. The function
is exponentially convex, and hence, by Proposition 2.3, it is log-convex. Therefore, the function
is convex, which implies (as in the proof of Theorem 3.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F350973/MediaObjects/13660_2010_Article_2334_Equ97_HTML.gif)
If , set
in (3.15) to obtain (3.14).
For , we get the required result by limit.
Funding
This research was partially funded by the Higher Education Commission, Pakistan.
References
Horváth L, Pečarić J: A refinement of the discrete Jensen's inequality. to appear in Mathematical Inequalities and Applications
Khan KA, Pečarić J, Perić I: Differences of weighted mixed symmetric means and related results. journal of Inequalities and Applications 2010, 2010:-16.
Anwar M, Jakšetić J, Pečarić J, Ur Rehman A: Exponential convexity, positive semi-definite matrices and fundamental inequalities. Journal of Mathematical Inequalities 2010,4(2):171–189.
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.
Acknowledgments
The research of L. Horváth was supported by the Hungarian National Foundations for Scientific Research, Grant no. K73274, and that of J. Pecărić was supported by the Croatian Ministry of Science, Education, and Sports under the research Grant 117-1170889-0888.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
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.
About this article
Cite this article
Horváth, L., Khan, K. & Pečarić, J. Refinements of Results about Weighted Mixed Symmetric Means and Related Cauchy Means. J Inequal Appl 2011, 350973 (2011). https://doi.org/10.1155/2011/350973
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2011/350973