A Double Inequality for the Combination of Toader Mean and the Arithmetic Mean in Terms of the Contraharmonic Mean
نویسندگان
چکیده
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.
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...
متن کاملOptimal Bounds for Toader Mean in Terms of Arithmetic and Contraharmonic Means
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...
متن کاملRefinements of Bounds for Neuman Means in Terms of Arithmetic and Contraharmonic Means
In this paper, we present the sharp upper and lower bounds for the Neuman means SAC and SCA in terms of the the arithmetic mean A and contraharmonic mean C . The given results are the improvements of some known results. Mathematics subject classification (2010): 26E60.
متن کاملOptimal bounds for arithmetic-geometric and Toader means in terms of generalized logarithmic mean
In this paper, we find the greatest values [Formula: see text] and the smallest values [Formula: see text] such that the double inequalities [Formula: see text] and [Formula: see text] hold for all [Formula: see text] with [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are the arithmetic-geometric, Toader and generalized logarithmic means of two posi...
متن کاملOptimal inequalities for bounding Toader mean by arithmetic and quadratic means
In this paper, we present the best possible parameters [Formula: see text] and [Formula: see text] such that the double inequality [Formula: see text] holds for all [Formula: see text] and [Formula: see text] with [Formula: see text], and we provide new bounds for the complete elliptic integral [Formula: see text] [Formula: see text] of the second kind, where [Formula: see text], [Formula: see ...
متن کامل