Optimal Bounds for Gaussian Arithmetic-Geometric Mean with Applications to Complete Elliptic Integral

Hua Wang,Yu-Ming Chu,Wei-Mao Qian

doi:10.1155/2016/3698463

Abstract

We present the best possible parametersα1,β1,α2,β2∈Randα3,β3∈(1/2,1)such that the double inequalitiesQα1(a,b)A1-α1(a,b)<AG[A(a,b),Q(a,b)]<Qβ1(a,b)A1-β1(a,b),α2Q(a,b)+(1-α2)A(a,b)<AG[A(a,b),Q(a,b)]<β2Q(a,b)+(1-β2)A(a,b),Q[α3a+(1-α3)b,α3b+(1-α3)a]<AG[A(a,b),Q(a,b)]<Q[β3a+(1-β3)b,β3b+(1-β3)a]hold for alla,b>0witha≠b, whereA(a,b),Q(a,b), andAG(a,b)are the arithmetic, quadratic, and Gauss arithmetic-geometric means ofaandb, respectively. As applications, we find several new bounds for the complete elliptic integrals of the first and second kind.

Highlights

We present the best possible parameters α1, β1, α2, β2 ∈ R and α3, β3 ∈ (1/2, 1) such that the double inequalities Qα1 (a, b)A1−α1 (a, b) < AG[A(a, b), Q(a, b)] < Qβ1 (a, b)A1−β1 (a, b), α2Q(a, b) + (1 − α2)A(a, b) < AG[A(a, b), Q(a, b)] < β2Q(a, b) + (1 − β2)A(a, b), Q[α3a + (1 − α3)b, α3b + (1 − α3)a] < AG[A(a, b), Q(a, b)] < Q[β3a + (1 − β3)b, β3b + (1 − β3)a] hold for all a, b > 0 with a ≠ b, where A(a, b), Q(a, b), and AG(a, b) are the arithmetic, quadratic, and Gauss arithmetic-geometric means of a and b, respectively
We find several new bounds for the complete elliptic integrals of the first and second kind
In order to prove our main results we need several derivative formulas and particular values for K(r) and E(r), which we present

Summary

Introduction

E(r) and the Gaussian arithmetic-geometric mean AG(a, b) have many applications in mathematics, physics, mechanics, and engineering [4,5,6,7,8,9]. The bounds for the Gaussian arithmetic-geometric mean AG(a, b) have attracted the attention of many researchers. By using a variant of L’Hospital’s rule and representation theorems with elliptic integrals, Vamanamurthy and Vuorinen [14] proved, among other results, the inequalities. Inequalities (9) and (12) show that AG lies between the arithmetic and geometric means of A and G. Alzer and Qiu [20] proved that λ = 3/4 and μ = 2/π are the best possible parameters such that the double inequality λ/L (a, b). < Q [β3a + (1 − β3) b, β3b + (1 − β3) a] hold for all a, b > 0 with a ≠ b

Lemmas

Main Results

Applications