Optimal one-parameter mean bounds for the convex combination of arithmetic and logarithmic means

Optimal One–parameter Mean Bounds for the Convex Combination of Arithmetic and Logarithmic Means

We find the greatest value p1 = p1(α) and the least value p2 = p2(α) such that the double inequality Jp1 (a,b) <αA(a,b)+(1−α)L(a,b) < Jp2 (a,b) holds for any α ∈ (0,1) and all a,b > 0 with a = b . Here, A(a,b) , L(a,b) and Jp(a,b) denote the arithmetic, logarithmic and p -th one-parameter means of two positive numbers a and b , respectively. Mathematics subject classification (2010): 26E60.

Optimal convex combinations bounds of centrodial and harmonic means for logarithmic and identric means

We find the greatest values $alpha_{1} $ and $alpha_{2} $, and the least values $beta_{1} $ and $beta_{2} $ such that the inequalities $alpha_{1} C(a,b)+(1-alpha_{1} )H(a,b)

The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

and Applied Analysis 3 Lemma 2.1. If α ∈ 0, 1 , then 1 2α log 2 − logα > 3 log 2. Proof. For α ∈ 0, 1 , let f α 1 2α log 2 − logα , then simple computations lead to f ′ α 2 ( log 2 − 1 − 2 logα − 1 α , 2.1 f ′′ α 1 α2 1 − 2α . 2.2 From 2.2 we clearly see that f ′′ α > 0 for α ∈ 0, 1/2 , and f ′′ α < 0 for α ∈ 1/2, 1 . Then from 2.1 we get f ′ α ≤ f ′ ( 1 2 ) 4 ( log 2 − 1 < 0 2.3 for α ∈ 0, 1 ....

The Optimal Convex Combination Bounds of Arithmetic and Harmonic Means for the Seiffert's Mean

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.