In:
Journal of Applied Mathematics, Hindawi Limited, Vol. 2012 ( 2012), p. 1-14
Abstract:
We prove that α H ( a , b ) + ( 1 − α ) L ( a , b ) 〉 M ( 1 − 4 α ) / 3 ( a , b ) for α ∈ ( 0 , 1 ) and all a , b 〉 0 with a ≠ b if and only if α ∈ [ 1 / 4 , 1 ) and α H ( a , b ) + ( 1 − α ) L ( a , b ) 〈 M ( 1 − 4 α ) / 3 ( a , b ) if and only if α ∈ ( 0 , 3 345 / 80 − 11 / 16 ) , and the parameter ( 1 − 4 α ) / 3 is the best possible in either case. Here, H ( a , b ) = 2 a b / ( a + b ) , L ( a , b ) = ( a − b ) / ( log a − log b ) , and M p ( a , b ) = ( ( a p + b p ) / 2 ) 1 / p ( p ≠ 0 ) and M 0 ( a , b ) = a b are the harmonic, logarithmic, and p th power means of a and b , respectively.
Type of Medium:
Online Resource
ISSN:
1110-757X
,
1687-0042
Language:
English
Publisher:
Hindawi Limited
Publication Date:
2012
detail.hit.zdb_id:
2578385-3
SSG:
17,1
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