Abstract
We find the greatest value α and least value β in (1/ 2,1) such that the double inequality S ( α a + ( 1 - α ) b , α b + ( 1 - α ) a ) <T ( a , b ) <S ( β a + ( 1 - β ) b , β b + ( 1 - β ) a ) holds for all a,b > 0 with a ≠ b. Here, T(a, b) = (a-b)/[2 arctan((a-b)/(a + b))] and S(a, b) = [(a2 + b2)/2]1/ 2are the Seiffert mean and root mean square of a and b, respectively.2010 Mathematics Subject Classification: 26E60.
Highlights
For a,b > 0 with a ≠ b the Seiffert mean T(a, b) and root mean square S(a, b) are defined by a−b T(a, b) = a−b (1:1)2 arctan a+b and a2 + b2 S(a, b) = (1:2)respectively
We find the greatest value a and least value b in (1/2,1) such that the double inequality
Both mean values have been the subject of intensive research
Summary
T(a, b) = (a-b)/[2 arctan((a-b)/(a + b))] and S(a, b) = [(a2 + b2)/2]1/2 are the Seiffert mean and root mean square of a and b, respectively. M0(a, b) = ab be the arithmetic, geometric, and pth power means of two positive numbers a and b, respectively. In [6], Wang et al answered the question: What are the best possible parameters l and μ such that the double inequality Ll(a,b)
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