Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean
نویسندگان
چکیده
منابع مشابه
Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean
The authors find the greatest value λ and the least value μ, such that the double inequality C(λa + (1-λb), λb+(1-λ)a) < αA(a, b) + (1-α)T(a,b) < C(μa + (1 - μ)b, μb + (1 - μ)a) holds for all α ∈ (0, 1) and a, b > 0 with a ≠ b, where C(a, b) = 2(a² + ab + b²)/3(a + b), A(a, b) = (a + b)/2, and T(a, b) = (a + b)/2, and T(a, b) = (2/π) ∫₀(π/2) √a²cos²θ + b²sin²θdθ denote, respectively, the centro...
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We find the greatest value λ and the least value μ such that the double inequality C(λa + (1 − λ)b, λb+ (1 − λ)a) < αA(a, b) + (1 − α)T (a, b) < C(μa + (1 − μ)b, μb+ (1− μ)a) holds for all α ∈ (0, 1) and a, b > 0 with a 6= b, where C(a, b), A(a, b), and T (a, b) denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers a and b.
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We find the greatest value α1 and α2 , and the least values β1 and β2 , such that the double inequalities α1C(a,b)+(1−α1)A(a,b) < T (a,b) < β1C(a,b)+(1−β1)A(a,b) and α2/A(a,b)+(1−α2)/C(a,b) < 1/T (a,b) < β2/A(a,b)+(1−β2)/C(a,b) hold for all a,b > 0 with a = b . As applications, we get new bounds for the complete elliptic integral of the second kind. Here, C(a,b) = (a2 +b2)/(a+b) , A(a,b) = (a+b...
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and Applied Analysis 3 2. Lemmas In order to establish ourmain result, we need several formulas and lemmas, whichwe present in this section. The following formulas were presented in 10, Appendix E, pages 474-475 : Let r ∈ 0, 1 , then
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ژورنال
عنوان ژورنال: The Scientific World Journal
سال: 2013
ISSN: 1537-744X
DOI: 10.1155/2013/842542