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

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 ....

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)

Optimal Lower Power Mean Bound for the Convex Combination of Harmonic and Logarithmic Means

and Applied Analysis 3 Alzer and Qiu 27 found the sharp bound of 1/2 L a, b I a, b in terms of the power mean as follows: Mc a, b < 1 2 L a, b I a, b 1.8 for all a, b > 0 with a/ b, with the best possible parameter c log 2/ 1 log 2 . The main purpose of this paper is to find the least value λ ∈ 0, 1 and the greatest value p p α such that αH a, b 1 − α L a, b > Mp a, b for α ∈ λ, 1 and all a, b ...

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