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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)
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)
If
, 
, and 
are held fixed while 
, it can be proven that 
(2.2)
, 
, and 
are held fixed while 
, it can be proven that 
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.
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)
, we obtain 
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
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- 2.
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- 3.
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- 4.
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- 7.
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- 8.
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- 9.
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- 10.
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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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Keywords
- Positive Integer
- Real Number
- Partial Derivative
- Simple Computation
- Interior Point
