The Optimal Convex Combination Bounds for Seiffert's Mean
© H. Liu and X.-J. Meng. 2011
Received: 28 November 2010
Accepted: 28 February 2011
Published: 13 March 2011
We derive some optimal convex combination bounds related to Seiffert's mean. We find the greatest values , and the least values , such that the double inequalities and hold for all with . Here, , , , and denote the contraharmonic, geometric, harmonic, and Seiffert's means of two positive numbers and , respectively.
2. Main Results
Firstly, we present the optimal convex combination bounds of contraharmonic and geometric means for Seiffert's mean as follows.
We divide the proof into two cases.
Secondly, we present the optimal convex combination bounds of the contraharmonic and harmonic means for Seiffert's mean as follows.
We divide the proof into two cases.
The authors wish to thank the anonymous referees for their very careful reading of the paper and fruitful comments and suggestions. This research is partly supported by N S Foundation of Hebei Province (Grant A2011201011), and the Youth Foundation of Hebei University (Grant 2010Q24).
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