Abstract and Applied Analysis

Sharp Geometric Mean Bounds for Neuman Means

Yan Zhang, Yu-Ming Chu, and Yun-Liang Jiang

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Abstract

We find the best possible constants α 1 , α 2 , β 1 , β 2 [ 0,1 / 2 ] and α 3 , α 4 , β 3 , β 4 [ 1 / 2,1 ] such that the double inequalities G ( α 1 a + ( 1 - α 1 ) b , α 1 b + ( 1 - α 1 ) a ) < N A G ( a , b ) < G ( β 1 a + ( 1 - β 1 ) b , β 1 b + ( 1 - β 1 ) a ) , G ( α 2 a + ( 1 - α 2 ) b , α 2 b + ( 1 - α 2 ) a ) < N G A ( a , b ) < G ( β 2 a + ( 1 - β 2 ) b , β 2 b + ( 1 - β 2 ) a ) , Q ( α 3 a + ( 1 - α 3 ) b , α 3 b + ( 1 - α 3 ) a ) < N Q A ( a , b ) < Q ( β 3 a + ( 1 - β 3 ) b , β 3 b + ( 1 - β 3 ) a ) , Q ( α 4 a + ( 1 - α 4 ) b , α 4 b + ( 1 - α 4 ) a ) < N A Q ( a , b ) < Q ( β 4 a + ( 1 - β 4 ) b , β 4 b + ( 1 - β 4 ) a ) hold for all a , b > 0 with a b , where G , A , and Q are, respectively, the geometric, arithmetic, and quadratic means and N A G , N G A , N Q A , and N A Q are the Neuman means.

Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 949815, 6 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412276970

Digital Object Identifier
doi:10.1155/2014/949815

Mathematical Reviews number (MathSciNet)
MR3208577

Zentralblatt MATH identifier
07023379

Citation

Zhang, Yan; Chu, Yu-Ming; Jiang, Yun-Liang. Sharp Geometric Mean Bounds for Neuman Means. Abstr. Appl. Anal. 2014 (2014), Article ID 949815, 6 pages. doi:10.1155/2014/949815. https://projecteuclid.org/euclid.aaa/1412276970


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References

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