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Complementary Inequalities Involving the Stolarsky Mean


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 [35] 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. (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

  1. (2)

    If , , 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. (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




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

  1. (2)

    Using the properties of the integer part , we obtain




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




the same argument as above can be used to obtain




In the end we obtain



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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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Correspondence to Ovidiu Bagdasar.

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Bagdasar, O. Complementary Inequalities Involving the Stolarsky Mean. J Inequal Appl 2010, 492570 (2010).

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