Optimal Convex Combination Bounds for Toader Mean

Optimal convex combination bounds of geometric and Neuman means for Toader-type mean

In this paper, we prove that the double inequalities [Formula: see text] hold for all [Formula: see text] with [Formula: see text] if and only if [Formula: see text], [Formula: see text] , [Formula: see text] and [Formula: see text] , where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text] are the Toader, geometric, arithmetic and two Neu...

Sharp Generalized Seiffert Mean Bounds for Toader Mean

and Applied Analysis 3 2. Lemmas In order to establish ourmain result, we need several formulas and lemmas, whichwe present in this section. The following formulas were presented in 10, Appendix E, pages 474-475 : Let r ∈ 0, 1 , then

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.

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.