SHARP BOUNDS FOR NEUMAN-SÁNDOR MEAN IN TERMS OF THE CONVEX COMBINATION OF QUADRATIC AND FIRST SEIFFERT MEANS

doi:10.1016/S0252-9602(14)60050-3

数学物理学报(英文版) ›› 2014, Vol. 34 ›› Issue (3): 797-806.doi: 10.1016/S0252-9602(14)60050-3

SHARP BOUNDS FOR NEUMAN-SÁNDOR MEAN IN TERMS OF THE CONVEX COMBINATION OF QUADRATIC AND FIRST SEIFFERT MEANS

褚玉明, 赵铁洪, 宋迎清

School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China； Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China； School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China

收稿日期:2013-03-23 修回日期:2013-08-29 出版日期:2014-05-20 发布日期:2014-05-20
基金资助:
This research was supported by the Natural Science Foundation of China under Grants 61374086 and 11371125, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.

SHARP BOUNDS FOR NEUMAN-SÁNDOR MEAN IN TERMS OF THE CONVEX COMBINATION OF QUADRATIC AND FIRST SEIFFERT MEANS

CHU Yu-Ming, ZhAO Tie-Hong, SONG Ying-Qing

School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China； Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China； School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China

Received:2013-03-23 Revised:2013-08-29 Online:2014-05-20 Published:2014-05-20
Supported by:
This research was supported by the Natural Science Foundation of China under Grants 61374086 and 11371125, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.

摘要/Abstract

摘要：

In this article, we prove that the double inequality
αP(a, b) + (1 − α)Q(a, b) < M(a, b) < βP(a, b) + (1 − β)Q(a, b)
holds for any a, b > 0 with a ≠b if and only if α ≥ 1/2 and β ≤ [π(√2 log(1 + √2) −1)]/[(√2π−2) log(1+√2)] = 0.3595 · · · , where M(a, b), Q(a, b), and P(a, b) are the Neuman-S´andor, quadratic, and first Seiffert means of a and b, respectively.

关键词: Neuman-S´andor mean, quadratic mean, first Seiffert mean

Abstract:

Key words: Neuman-S´andor mean, quadratic mean, first Seiffert mean

中图分类号:

26E60

褚玉明, 赵铁洪, 宋迎清. SHARP BOUNDS FOR NEUMAN-SÁNDOR MEAN IN TERMS OF THE CONVEX COMBINATION OF QUADRATIC AND FIRST SEIFFERT MEANS[J]. 数学物理学报(英文版), 2014, 34(3): 797-806.

CHU Yu-Ming, ZhAO Tie-Hong, SONG Ying-Qing. SHARP BOUNDS FOR NEUMAN-SÁNDOR MEAN IN TERMS OF THE CONVEX COMBINATION OF QUADRATIC AND FIRST SEIFFERT MEANS[J]. Acta mathematica scientia,Series B, 2014, 34(3): 797-806.

参考文献

[1] Neuman E, S´andor J. On the Schwab-Borchardt mean. Math Pannon, 2003, 14(2): 253–266

[2] Seiffert H J. Problem 887. Nieuw Arch Wisk, 1993, 11(2): 176–176

[3] H¨ast¨o P A. A monotonicity property of ratios of symmetric homogeneous means. JIPAM J Inequal Pure Appl Math, 2002, 3(5): Article 71, 23 pages

[4] Neuman E, S´andor J. On certain means of two arguments and their extension. Int J Math Math Sci, 2003, 16: 981–993

[5] H¨ast¨o P A. Optimal inequalities between Seiffert’s mean and power mean. Math Inequal Appl, 2004, 7(1): 47–53

[6] Chu Y M, Wang M K, Wang G D. The optimal generalized logarithmic mean bound for Seiffert´s mean. Acta Mathematica Scientia, 2012, 32B(4): 1619–1626

[7] Gao H Y, Guo J L, Li M H. Sharp bounds for the first Seiffert and logarithmic means in terms of generalized Heronian mean. Acta Mathematica Scientia, 2013, 33B(3): 568–572 (in Chinese)

[8] Neuman E. Inequalities for the Schwab-Borchardt mean and their applications. J Math Inequal, 2011, 5(4): 601–609

[9] Chu Y M, Hou S W, Shen Z H. Sharp bounds for Seiffert mean in terms of root mean square. J Inequal Appl, 2012, 2012: 11, 6 pages

[10] Chu Y M, Wang M K, Qiu S L. Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc Indian Acad Sci Math Sci, 2012, 122(1): 41–51

[11] Chu Y M, Wang M K, Wang Z K. Best possible inequalities among harmonic, geometric, logarithmic and Seiffert means. Math Inequal Appl, 2012, 15(2): 415–422

[12] Jiang W D, Qi F. Some sharp inequalities involving Seiffert and other means and their concise proofs. Math Inequal Appl, 2012, 15(4): 1007–1017

[13] Gong W M, Song Y Q, Wang M K, Chu Y M. A sharp double inequality between Seiffert, arithmetic, and geometric means. Abstr Appl Anal, 2012, Article ID 684834, 7 pages

[14] Jiang W D. Some sharp inequalities involving reciprocals of the Seiffert and other means. J Math Inequal, 2012, 6(4): 593–599

[15] Chu Y M, Long B Y, Gong W M, Song Y Q. Sharp bounds for Seiffert and Neuman-S´andor means in terms of generalized logarithmic means. J Inequal Appl, 2013, 2013: 10, 13 pages

[16] Neuman E, S´andor J. On the Schwab-Borchardt mean II. Math Pannon, 2006, 17(1): 49–59

[17] Li YM, Long B Y, Chu YM. Sharp bounds for the Neuman-S´andor mean in terms of generalized logarithmic mean. J Math Inequal, 2012, 6(4): 567–577

[18] Neuman E. A note on a certain bivariate mean. J Math Inequal, 2012, 6(4): 637–643

[19] Zhao T H, Chu YM, Liu B Y. Optimal bounds for Neuman-S´andor mean in terms of the convex combination of harmonic, geometric, quadratic, and contraharmonic means. Abstr Appl Anal, 2012, Article ID 302635, 9 pages

[20] Chu Y M, Long B Y. Bounds of the Neuman-S´andor mean using power and identric means. Abstr Appl Anal, 2013, Article ID 832591, 6 pages

[21] Abramowitz M, Stegun I A. Handbook of Mathematical Functions, New York: Dover Publications, 1970

SHARP BOUNDS FOR NEUMAN-SÁNDOR MEAN IN TERMS OF THE CONVEX COMBINATION OF QUADRATIC AND FIRST SEIFFERT MEANS

SHARP BOUNDS FOR NEUMAN-SÁNDOR MEAN IN TERMS OF THE CONVEX COMBINATION OF QUADRATIC AND FIRST SEIFFERT MEANS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 4

Metrics

本文评价

推荐阅读 0

[1]	赵铁洪, 王淼坤, 褚玉明. ON THE BOUNDS OF THE PERIMETER OF AN ELLIPSE[J]. 数学物理学报(英文版), 2022, 42(2): 491-501.
[2]	杨月英, 钱伟茂, 张宏伟, 褚玉明. SHARP BOUNDS FOR TOADER-TYPE MEANS IN TERMS OF TWO-PARAMETER MEANS[J]. 数学物理学报(英文版), 2021, 41(3): 719-728.
[3]	褚玉明, 王淼坤, 王根娣. THE OPTIMAL GENERALIZED LOGARITHMIC MEAN BOUNDS FOR SEIFFERT'S MEAN[J]. 数学物理学报(英文版), 2012, 32(4): 1619-1626.
[4]	夏卫锋, 褚玉明. THE SCHUR CONVEXITY OF GINI MEAN VALUES IN THE SENSE OF HARMONIC MEAN[J]. 数学物理学报(英文版), 2011, 31(3): 1103-1112.