Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean

Wei-Dong Jiang

doi:10.1155/2013/842542

Abstract

The authors find the greatest value λ and the least value μ, such that the double inequality holds for all α ∈ (0,1) and a, b > 0 with a ≠ b, where , A(a, b) = (a + b)/2, and denote, respectively, the centroidal, arithmetic, and Toader means of the two positive numbers a and b.

Highlights

In [1], Toader introduced a mean T (a, b) = 2 π π/2 ∫√a2cos2θ + b2sin2θdθ {{{{{{{{{{2aE √1 − (b/a)2 π, a > b, (1)
This conjecture was verified by Qiu and Shen [8] and by Barnard et al [9], respectively
In [10], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows: T (a, b) < Mlog 2/ log(π/2) (a, b), (5)

Summary

Introduction

For r ∈ [0, 1] is the complete elliptic integral of the second kind. There have been plenty of literature, such as [2,3,4,5,6], dedicated to the Toader mean. This conjecture was verified by Qiu and Shen [8] and by Barnard et al [9], respectively. In [10], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows:. Hua and Qi [11] proved that the double inequality αC (a, b) + (1 − α) A (a, b). We present new bounds for the complete elliptic integral of the second kind

Preliminaries and Lemmas

Main Results

Remarks