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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 righthandside 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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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