Optimal convex combination bounds of Seiffert and geometric means for the arithmetic 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.

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

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.

Sharp Power Mean Bounds for the Combination of Seiffert and Geometric Means

and Applied Analysis 3 The following sharp lower power mean bounds for 1/3 G a, b 2/3 H a, b , 2/3 G a, b 1/3 H a, b , and P a, b can be found in 4, 6 : 1 3 G a, b 2 3 H a, b > M−2/3 a, b , 2 3 G a, b 1 3 H a, b > M−1/3 a, b , P a, b > Mlog 2/ logπ a, b 1.8 for all a, b > 0 with a/ b. The purpose of this paper is to answer the question: for α ∈ 0, 1 , what are the greatest value p and the least...

Optimal bounds for Neuman-Sándor mean in terms of the convex combination of the logarithmic and the second Seiffert means

In the article, we prove that the double inequality [Formula: see text] holds for [Formula: see text] with [Formula: see text] if and only if [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] denote the Neuman-Sándor, logarithmic and second Seiffert means of two positive numbers a and b, respectively.