Optimal Power Mean Bounds for the Weighted Geometric Mean of Classical Means
نویسندگان
چکیده
منابع مشابه
Optimal Power Mean Bounds for the Weighted Geometric Mean of Classical Means
For p ∈ R, the power mean of order p of two positive numbers a and b is defined by Mp a, b a b /2 , for p / 0, and Mp a, b √ ab, for p 0. In this paper, we answer the question: what are the greatest value p and the least value q such that the double inequality Mp a, b ≤ A a, b G a, b H1−α−β a, b ≤ Mq a, b holds for all a, b > 0 and α, β > 0 with α β < 1? Here A a, b a b /2, G a, b √ ab, and H a...
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For a,b > 0 with a = b , let P = (a− b)/(4arctana/b−π) , A = (a+ b)/2 , G = √ ab denote the Seiffert mean, arithmetic mean, geometric mean of a and b , respectively. In this paper, we present new sharp bounds for Seiffert P in terms of weighted power means of arithmetic mean A and geometric mean G : ( 2 3 A p1 + 3 G p1 )1/p1 < P < ( 2 3 A p2 + 3 G p2 )1/p2 , where p1 = 4/5 and p2 = logπ/2 (3/2)...
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ژورنال
عنوان ژورنال: Journal of Inequalities and Applications
سال: 2010
ISSN: 1025-5834,1029-242X
DOI: 10.1155/2010/905679