Acta mathematica scientia, Series B >
SHARP BOUNDS FOR NEUMAN-SÁNDOR MEAN IN TERMS OF THE CONVEX COMBINATION OF QUADRATIC AND FIRST SEIFFERT MEANS
Received date: 2013-03-23
Revised date: 2013-08-29
Online 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.
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
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 . DOI: 10.1016/S0252-9602(14)60050-3
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