Sharp Bounds by the Generalized Logarithmic Mean for the Geometric Weighted Mean of the Geometric and Harmonic Means

Sharp Bounds by the Generalized Logarithmic Mean for the Geometric Weighted Mean of the Geometric and Harmonic Means

Sharp Two Parameter Bounds for the Logarithmic Mean and the Arithmetic–geometric Mean of Gauss

For fixed s 1 and t1,t2 ∈ (0,1/2) we prove that the inequalities G(t1a + (1− t1)b,t1b+(1− t1)a)A1−s(a,b) > AG(a,b) and G(t2a+(1− t2)b,t2b+(1− t2)a)A1−s(a,b) > L(a,b) hold for all a,b > 0 with a = b if and only if t1 1/2− √ 2s/(4s) and t2 1/2− √ 6s/(6s) . Here G(a,b) , L(a,b) , A(a,b) and AG(a,b) are the geometric, logarithmic, arithmetic and arithmetic-geometric means of a and b , respectively....

Optimal Convex Combinations Bounds of Centroidal and Harmonic Means for Weighted Geometric Mean of Logarithmic and Identric Means

In this paper, optimal convex combination bounds of centroidal and harmonic means for weighted geometric mean of logarithmic and identric means are proved. We find the greatest value λ(α) and the least value Δ(α) for each α ∈ (0,1) such that the double inequality: λC(a,b)+(1−λ)H(a,b) < Lα (a,b)I1−α (a,b) < ΔC(a,b)+(1−Δ)H(a,b) holds for all a,b > 0 with a = b. Here, C(a,b), H(a,b) , L(a,b) and I...

Sharp Bounds for Seiffert Mean in Terms of Weighted Power Means of Arithmetic Mean and Geometric Mean

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)...

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...