Acta mathematica scientia,Series B ›› 2014, Vol. 34 ›› Issue (3): 797-806.doi: 10.1016/S0252-9602(14)60050-3

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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   

  1. 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.

Abstract:

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.

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

CLC Number: 

  • 26E60
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