Optimal Bounds for Toader Mean in Terms of Arithmetic and Contraharmonic Means
نویسندگان
چکیده
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)/2 , and
منابع مشابه
A Double Inequality for the Combination of Toader Mean and the Arithmetic Mean in Terms of the Contraharmonic Mean
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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