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


Introduction
For p ∈ 0, 1 , the generalized Seiffert mean of two positive numbers a and b is defined by

1.1
It is well known that S p a, b is continuous and strictly increasing with respect to p ∈ 0, 1 for fixed a, b > 0 with a / b.In particular, if p 1/2, then the generalized Seiffert mean

1.2
Recently, the Seiffert mean and its generalization have been the subject of intensive research, many remarkable inequalities for these means can be found in the literature 1-5 .
In 6 , Toader introduced the Toader mean T a, b of two positive numbers a and b as follows: The main purpose of this paper is to find the greatest value α and least value β such that the double inequality S α a, b < T a, b < S β a, b holds for all a, b > 0 with a / b and give new bounds for the complete elliptic integrals of the second kind.

Lemmas
In order to establish our main result, we need several formulas and lemmas, which we present in this section.
The following formulas were presented in 10, Appendix E, pages 474-475 : Let r ∈ 0, 1 , then If f x /g x is strictly monotone, then the monotonicity in the conclusion is also strict.
It follows from 2.3 and part 1 together with Lemma 2.1 that F r is strictly increasing in 0, 1 and F 0 π/16.
From 2.4 , parts 2 and 5 together with Lemma 2.1, we know that G r is strictly increasing in 0, 1 , and f 0 3π/16.
Proof.For part 1 , clearly g 0 0 and g where Making use of Lemma 2.2 1 , 2 , and 6 , we get

2.6
Therefore, part 1 follows from 2.5 and 2.6 together with the limiting values of g r at r 0 and r 1.
For part 2 , simple computations yield that lim Abstract and Applied Analysis 5 where From Lemma 2.2 1 , 3 , and 4 together with the monotonicity of E r we know that f 2 r is strictly increasing in 0, 1 .Moreover, lim

2.13
Equations 2.11 -2.13 and the monotonicity of f 2 r lead to the conclusion that there exists r 0 ∈ 0, 1 such that f 1 r is strictly decreasing in 0, r 0 and strictly increasing in r 0 , 1 .
It follows from 2.8 -2.10 and the piecewise monotonicity of f 1 r that there exists r 1 ∈ 0, 1 such that f r is strictly decreasing in 0, r 1 and strictly increasing in r 1 , 1 .
Therefore, part 2 follows from 2.7 and the piecewise monotonicity of f r .

2 Abstract
1 is the complete elliptic integral of the second kind.Vuorinen 7 conjectured that M 3/2 a, b < T a, b 1.4 for all a, b > 0 with a / b, where mean of order p of two positive numbers a and b.This conjecture was proved by Barnard et al. 8 .In 9 , Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows: T a, b < M log 2/ log π/2 a, b 1.6 for all a, b > 0 with a / b.

Theorem 3 . 1 .
Inequality S √ 3/4 a, b < T a, b < S 1/2 a, b holds for all a, b > 0 with a / b, and S √ 3/4 a, b and S 1/2 a, b are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean T a, b , respectively.Proof.Firstly, we prove that S √ 3/4 a, b < T a, b < S 1/2 a, b 3.1 for all a, b > 0 with a / b. 6 Abstract and Applied Analysis Without loss of generality, we assume that a > b.Let t b/a < 1, r 1 − t / 1 t .Then 1.1 and 1.3 lead to T a, b − S √ 3/4 a, b