Inequalities for Generalized Logarithmic Means. (24th January 2010)

• Record Type: Journal Article
• Authors: Chu, Yu-Ming; Xia, Wei-Feng
• Other Names: Cheung Wing-Sum Academic Editor.
• Is Part Of: Journal of inequalities and applications. Volume 2009(2009)
• Publication Date: 2010-01-24
• Subjects: Inequalities (Mathematics) -- Periodicals; 512.97
• DOI: 10.1155/2009/763252
• Languages: English
• ISSNs: 1029-242X

Abstract

Abstract : Forp ∈ ℝ, the generalized logarithmic meanL p of two positive numbersa andb is defined asL p ( a, b ) = a, fora = b, L P ( a, b ) = [ ( b p + 1 − a p + 1 ) / ( p + 1 ) ( b − a ) ] 1 / p, fora ≠ b, p ≠ − 1, p ≠ 0, L P ( a, b ) = ( b − a ) / ( log b − log a ), fora ≠ b, p = − 1, andL P ( a, b ) = ( 1 / e ) ( b b / a a ) 1 / ( b − a ), fora ≠ b, p = 0 . In this paper, we prove thatG ( a, b ) + H ( a, b ) ⩾ 2 L − 7 / 2 ( a, b ), A ( a, b ) + H ( a, b ) ⩾ 2 L − 2 ( a, b ), andL − 5 ( a, b ) ⩾ H ( a, b ) for alla, b > 0, and the constants− 7 / 2, − 2, and− 5 cannot be improved for the corresponding inequalities. HereA ( a, b ) = ( a + b ) / 2 = L 1 ( a, b ), G ( a, b ) = a b = L − 2 ( a, b ), andH ( a, b ) = 2 a b / ( a + b ) denote the arithmetic, geometric, and harmonic means ofa andb, respectively.