Generalized Lazarevic's Inequality and Its Applications—Part II
© Ling Zhu. 2009
Received: 21 July 2009
Accepted: 30 November 2009
Published: 24 December 2009
A generalized Lazarevic's inequality is established. The applications of this generalized Lazarevic's inequality give some new lower bounds for logarithmic mean.
Moreover, the inequality (1.1) can be extended as follows.
2. Three Lemmas
3. A Concise Proof of Theorem 1.3
Let , then and . Let for we have that and is decreasing for so is decreasing for and is decreasing on by Lemma 2.2. Hence is decreasing on and is decreasing on by Lemma 2.1. Thus is decreasing on by Lemma 2.1.
Let , then . Let for we have that and is decreasing for so is decreasing for and is decreasing on by Lemma 2.2. Hence is increasing on and is decreasing on by Lemma 2.1. Thus is decreasing on by Lemma 2.1.
the proof of Theorem 1.3 is complete.
4. Some New Lower Bounds for Logarithmic Mean
- Lazarević I: Neke nejednakosti sa hiperbolickim funkcijama. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika 1966, 170: 41–48.Google Scholar
- Mitrinović DS: Analytic Inequalities. Springer, New York, NY, USA; 1970:xii+400.View ArticleMATHGoogle Scholar
- Zhu L: On Wilker-type inequalities. Mathematical Inequalities & Applications 2007,10(4):727–731.MathSciNetView ArticleMATHGoogle Scholar
- Zhu L: Generalized Lazarevic's inequality and its applications—part I. submitted submittedGoogle Scholar
- Vamanamurthy MK, Vuorinen M: Inequalities for means. Journal of Mathematical Analysis and Applications 1994,183(1):155–166. 10.1006/jmaa.1994.1137MathSciNetView ArticleMATHGoogle Scholar
- Anderson GD, Qiu S-L, Vamanamurthy MK, Vuorinen M: Generalized elliptic integrals and modular equations. Pacific Journal of Mathematics 2000,192(1):1–37. 10.2140/pjm.2000.192.1MathSciNetView ArticleGoogle Scholar
- Pinelis I: L'Hospital type results for monotonicity, with applications. Journal of Inequalities in Pure and Applied Mathematics 2002,3(1, article 5):1–5.MathSciNetMATHGoogle Scholar
- Pinelis I: On L'Hospital-type rules for monotonicity. Journal of Inequalities in Pure and Applied Mathematics 2006,7(2, article 40):1–19.MathSciNetMATHGoogle Scholar
- Biernacki M, Krzyż J: On the monotonity of certain functionals in the theory of analytic functions. Annales Universitatis Mariae Curie-Skłodowska 1955, 9: 135–147.MathSciNetMATHGoogle Scholar
- Ponnusamy S, Vuorinen M: Asymptotic expansions and inequalities for hypergeometric functions. Mathematika 1997,44(2):278–301. 10.1112/S0025579300012602MathSciNetView ArticleMATHGoogle Scholar
- Alzer H, Qiu S-L: Monotonicity theorems and inequalities for the complete elliptic integrals. Journal of Computational and Applied Mathematics 2004,172(2):289–312. 10.1016/j.cam.2004.02.009MathSciNetView ArticleMATHGoogle Scholar
- Kuang JC: Applied Inequalities. 3rd edition. Shangdong Science and Technology Press, Jinan City, China; 2004.Google Scholar
- Sándor J: On the identric and logarithmic means. Aequationes Mathematicae 1990,40(2–3):261–270.MathSciNetView ArticleMATHGoogle Scholar
- Alzer H: Ungleichungen für Mittelwerte. Archiv der Mathematik 1986,47(5):422–426. 10.1007/BF01189983MathSciNetView ArticleMATHGoogle Scholar
- Stolarsky KB: The power and generalized logarithmic means. The American Mathematical Monthly 1980,87(7):545–548. 10.2307/2321420MathSciNetView ArticleMATHGoogle Scholar
- Zhu L: From chains for mean value inequalities to Mitrinovic's problem II. International Journal of Mathematical Education in Science and Technology 2005,36(1):118–125. 10.1080/00207390412331314971View ArticleGoogle Scholar
- Ostle B, Terwilliger HL: A comparison of two means. Proceedings of the Montana Academy of Sciences 1957, 17: 69–70.Google Scholar
- Leach EB, Sholander MC: Extended mean values. II. Journal of Mathematical Analysis and Applications 1983,92(1):207–223. 10.1016/0022-247X(83)90280-9MathSciNetView ArticleMATHGoogle Scholar
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