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.
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.
Proof of Lemma 3.1.
attains its maximum.
Proof of Theorem 2.3.
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.
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.
- Stolarsky KB: Generalizations of the logarithmic mean. Mathematics Magazine 1975, 48: 87–92. 10.2307/2689825MathSciNetView ArticleMATHGoogle Scholar
- 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/S0002-9939-98-04125-2MathSciNetView ArticleMATHGoogle Scholar
- Drâmbe MO: Inequalities—Ideas and Methods. Zalău, Romania, Gil; 2003.Google Scholar
- Mitrinović DS, Pečarić JE, Fink AM: Classical and New Inequalities in Analysis, Mathematics and Its Applications (East European Series). Volume 61. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xviii+740.View ArticleMATHGoogle Scholar
- Mitrinović DS: Analytic Inequalities. Volume 2. Springer, London, UK; 1970:xii+395.View ArticleGoogle Scholar
- Hardy GH, Litllewod JE, Polya G: Inequalities. Cambridge University Press, Cambridge, UK; 1967.Google Scholar
- Shisha O, Mond B: Differences of means. Bulletin of the American Mathematical Society 1967, 73: 328–333. 10.1090/S0002-9904-1967-11737-3MathSciNetView ArticleMATHGoogle Scholar
- 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-XMathSciNetView ArticleMATHGoogle Scholar
- Beck E: Über komplementäre Ungleichungen mit drei Mittelwerten. Monatshefte für Mathematik 1975, 80: 13–29. 10.1007/BF01487800View ArticleMATHGoogle Scholar
- Páles Zs: On complementary inequalities. Publicationes Mathematicae Debrecen 1983, 30(1–2):75–88.MathSciNetMATHGoogle Scholar
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