- Research Article
- Open Access
Complementary Inequalities Involving the Stolarsky Mean
© Ovidiu Bagdasar. 2010
- Received: 24 February 2010
- Accepted: 1 May 2010
- Published: 2 June 2010
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.
- Positive Integer
- Real Number
- Partial Derivative
- Simple Computation
- Interior Point
Shisha and Mond  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 , Beck , and Páles .
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.
provided the limit exists.
As an application of Theorem 2.1, the following problem (see [3, pages 70–72]) is solved.
holds true for all positive real numbers
In the following theorem we examine the behavior of when the numbers , in Theorem 2.1, are terms of a sequence with certain properties.
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.
The function attains its maximum at the point if and only if for all
Proof of Lemma 3.1.
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.
attains its maximum.
and find the points where the maximum of is attained in the interval .
which (fortunately) is contained in the interior of
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 .
The value of this is to be called from now on.
Taking the limit in (3.23), we obtain that the limit of as is confined to the interval
Proof of Theorem 2.3.
where , or .
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.
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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