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Complementary Inequalities Involving the Stolarsky Mean
Journal of Inequalities and Applications volume 2010, Article number: 492570 (2010)
Abstract
Let be a positive integer and
,
,
, and
real numbers satisfying
and
. It is proved that for the real numbers
the maximum of the function
is attained if and only if
of the numbers
are equal to
and the other
are equal to
, while
is one of the values
,
, where
denotes the integer part and
represents the Stolarsky mean of
and
of powers
and
Some asymptotic results concerning
are also discussed.
1. Introduction
Let us begin with some definitions. Given the positive real numbers and
and the real numbers
and
, the difference mean or Stolarsky mean
of
and
is defined by (see, e.g., [1] or [2])

The power mean of power corresponding to the real numbers
is defined by

The relation between the Stolarsky mean and the power mean can be written as

It is well known that for fixed and
, we have the inequality

with equality for (independent of
), or for
(see [3–5] or [6]).
Shisha and Mond [7] obtained a complementary result which examines the upper bounds of (1.4) for weighted versions of the power means. Also, we have a considerable amount of work regarding the complementary means done by many authors, including Diaz and Metcalf [8], Beck [9], and Páles [10].
Returning to our problem, by defining the function

we obtain

Using the inequalities between power means (1.4), if and only if
therefore
if and only if
This condition is more general than
but there are details in the subsequent proofs which would not be satisfied in the other cases.
As the minimum of over
is
(possible only for
), it is natural to question what the maximum of
is, and, eventually, to find the configuration where this is attained. Since
the problem of finding the maximum of
only makes sense when all the variables
of
are restricted to the compact interval
.
The first theorem in the next section, deals with finding the maximum and the corresponding optimal configuration. The result enables one to obtain elegant proofs for some related inequalities. In the end of the present work we obtain some asymptotic limits relative to the configuration where the maximum of is attained.
2. Results
Theorem 2.1.
Given the positive integer , the real numbers
and
. Consider the function
, defined by (1.4). Then the following assertions are true.
-
(1)
The function
attains its maximum at a point
if and only if
of the variables are equal to
while the other
are equal to b, where
can be
(2.1)
-
(2)
If
,
, and
are held fixed while
, it can be proven that
(2.2)
provided the limit exists.
As an application of Theorem 2.1, the following problem (see [3, pages 70–72]) is solved.
Corollary 2.2.
Given the positive integer , determine the smallest value of
such that the inequality

holds true for all positive real numbers
Theorem 2.3.
Given the positive integer , the smallest value of
such that (2.3) holds true for all positive real numbers
is

In the following theorem we examine the behavior of when the numbers
,
in Theorem 2.1, are terms of a sequence with certain properties.
Theorem 2.4.
Consider the sequences and
satisfying
with
and
For each
define
as in (2.1), for the powers
and
Then the
verifies

3. Proofs
Proof of Theorem 2.1.
-
(1)
We first prove that the point
where the maximum of
is attained lies on the boundary of the hypercube
and moreover, it is a vertex. This result is the subject of Lemma 3.1. We then find the configuration where the maximum is realized.
Lemma 3.1.
The function attains its maximum at the point
if and only if
for all
Proof of Lemma 3.1.
Since is continuous on the compact interval
, there is a point
where
attains its maximum. If
is an interior point of
, then
for all
therefore

which implies

for all However, if
, then
which clearly is not the maximum of
Consequently,
lies on the boundary of
. Due to symmetry and since
there exist
and
such that

If then
For this case, consider the function
defined by

If the point where the maximum of
is attained is interior to
, in virtue of Fermat's theorem, we deduce that

for all This is equivalent to

hence

A simple computation shows that

and for this configuration we have

Let us define the function as

and prove it is increasing. Indeed, one finds

where , and
Since
it follows that
so
is increasing and the upper bound is

This finally proves that of the numbers
are equal to
while the other
are equal to
as anticipated. This ends the proof of Lemma 3.1.
The only thing to be done is to find the value of for which the expression

attains its maximum.
To do this, consider the function defined by

and find the points where the maximum of is attained in the interval
.
The critical points of are found from the equation

so they satisfy

As seen in the definition of the Stolarsky mean for this case,

It is finally found that has a single critical point

which (fortunately) is contained in the interior of
Taking into account that the second derivative of is

the extremal point is a point of maximum for
, and also the function
is decreasing on the interval
. Because
, we obtain
for
, and
for
Finally, this means that
is increasing on
and decreasing on
.
We conclude that

The maximum of (3.13) is then attained when takes one of the values
and
, where

The value of this is to be called
from now on.
Remark 3.2.
Because in our case

the Stolarsky mean satisfies the strict inequality , so
-
(2)
Using the properties of the integer part
, we obtain
(3.23)
so

It is then enough to work out the limit

On the other hand we have

Due to symmetry the partial derivatives are equal, so the desired limit is

Taking the limit in (3.23), we obtain that the limit of
as
is confined to the interval
Proof of Theorem 2.3.
Considering and
in Theorem 2.1, we obtain

Out of here, we can immediately obtain the best constant for which

Following the steps mentioned before, the function gets the maximum only when

where , or
.
This proves that the following inequality holds:

so the best constant will be


Remark 3.3.
Although appealing, a result involving arbitrary powers would depend on which the exact value of
is (out of the two possibilities). At the same time, the power
on the righthand-side can only be obtained for
Proof of Theorem 2.4.
To ease the notations we write and
The following relation holds:

Using the notation the limit can be written as

Since the denominator converges to it only remains to examine the limit

which can be written as

It can be proven that the two terms of (3.36) converge to finite limits, and analyze each. From the hypothesis so the limit of the first term is

while second term can be written as

Since

the same argument as above can be used to obtain

where

In the end we obtain

References
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Losonczi L, Páles Zs: Minkowski's inequality for two variable difference means. Proceedings of the American Mathematical Society 1998, 126(3):779–789. 10.1090/S0002-9939-98-04125-2
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Diaz JB, Metcalf FT: Complementary inequalities. I. Inequalities complementary to Cauchy's inequality for sums of real numbers. Journal of Mathematical Analysis and Applications 1964, 9: 59–74. 10.1016/0022-247X(64)90006-X
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Acknowledgments
The author wishs to express his thanks to T. Trif, who provided significant moral and technical support to finish this paper. The author also thanks the reviewers, whose suggestions and "free gifts'' were of great help. Last but not least, his thanks go to the Marie Curie foundation, which gave him the chance to understand Mathematics and its applications from a researcher's perspective.
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Bagdasar, O. Complementary Inequalities Involving the Stolarsky Mean. J Inequal Appl 2010, 492570 (2010). https://doi.org/10.1155/2010/492570
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DOI: https://doi.org/10.1155/2010/492570
Keywords
- Positive Integer
- Real Number
- Partial Derivative
- Simple Computation
- Interior Point