Optimal bounds of the arithmetic mean in terms of new Seiffert-like means

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 Convex Combination Bounds of Seiffert and Geometric Means for the Arithmetic Mean

We find the greatest value α and the least value β such that the double inequality αT (a,b) + (1−α)G(a,b) < A(a,b) < βT (a,b) + (1− β)G(a,b) holds for all a,b > 0 with a = b . Here T (a,b) , G(a,b) , and A(a,b) denote the Seiffert, geometric, and arithmetic means of two positive numbers a and b , respectively. Mathematics subject classification (2010): 26E60.

Optimal Bounds for Seiffert Mean in terms of One-Parameter Means

Sharp Bounds for Seiffert Mean in Terms of Contraharmonic Mean

and Applied Analysis 3 2. Proof of Theorem 1.1 Proof of Theorem 1.1. Let λ 1 √ 4/π − 1 /2 and μ 3 √3 /6. We first proof that the inequalities T a, b > C λa 1 − λ b, λb 1 − λ a , 2.1 T a, b < C ( μa ( 1 − μb, μb 1 − μa 2.2 hold for all a, b > 0 with a/ b. From 1.1 and 1.2 we clearly see that both T a, b and C a, b are symmetric and homogenous of degree 1. Without loss of generality, we assume th...

Bounds for the Combinations of Neuman-Sándor, Arithmetic, and Second Seiffert Means in terms of Contraharmonic Mean

and Applied Analysis 3 If f(x)/g(x) is strictly monotone, then the monotonicity in the conclusion is also strict. Lemma 2. Let u, α ∈ (0, 1) and f u,α (x) = ux 2 − (1 − α) ( x arctanx − 1) . (12) Then f u,α (x) > 0 for all x ∈ (0, 1) if and only if u ≥ (1 − α)/3 andf u,α (x) < 0 for allx ∈ (0, 1) if and only if u ≤ (1−α)(4/π− 1). Proof. From (12), one has f u,α (0 + ) = 0, (13) f u,α (1 − ) = u...