Optimal bounds for arithmetic-geometric and Toader means in terms of generalized logarithmic mean

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Introduction
(.) It is well known that L p (a, b) is continuous and strictly increasing with respect to p ∈ R for fixed a, b >  with a = b. Many remarkable inequalities for the generalized logarithmic mean can be found in the literature [-].
The classical arithmetic-geometric mean AG(a, b) of two positive numbers a and b is defined by starting with a  = a, b  = b and then iterating a n+ = a n + b n  , b n+ = a n b n (.) for n ∈ N until two sequences {a n } and {b n } converge to the same number. The well-known Gauss identity [] shows that for r ∈ (, ), where K(r) = π /  (r  sin  t) -/ dt, r ∈ [, ), is the complete elliptic integral of the first kind.
In [], the Toader mean T(a, b) of two positive numbers a and b was given by where E(r) = π /  (r  sin  θ ) / dθ , r ∈ [, ] is the complete elliptic integral of the second kind.
Recently, the bounds for the arithmetic-geometric mean AG(a, b) and Toader mean T(a, b) have attracted the attention of many mathematicians. The double inequality ab is the power mean of order p. This conjecture was proved by Qiu and Shen [] and Barnard et al. [].
In [], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows: In [-], the authors proved that denotes the pth Lehmer mean and S p (a, b) is the generalized Seiffert mean given by for all a, b >  with a = b if and only if p  ≤ /, q  ≥  and p  ≤ , q  ≥ /. Here the pth Gini mean of two positive numbers a and b is defined by The main purpose of this paper is to find the greatest values α  , α  and the smallest val- ) hold for all a, b >  with a = b and give some new bounds for the complete elliptic integrals.

Basic knowledge and lemmas
In order to prove our main results, we need several formulas and lemmas, which we present in this section.
For r ∈ (, ) and r = √ r  , the well-known complete elliptic integrals of the first and second kinds are defined by respectively, and the following formulas were presented in [], Appendix E, pp.-: In what follows, four special values E( √ /), K( √ /) and E(.), K(.) will be used. By numerical computations, these are given by .
is strictly monotone, then the monotonicity in the conclusion is also strict.
A simple calculation yields Following from Lemma .() and (.) together with the monotonicity of /r  , we clearly see that f  (r)/ f  (r) is strictly increasing on (, ). Equations (.)-(.) and Lemma . lead to the conclusion that f (r) is strictly increasing on (, ).
The following double inequalities can be obtained from Lemma . immediately.

Lemma . Let
, then φ p (r) >  for  < r <  if and only if p ≤ /; φ p (r) <  for  < r <  if and only if p ≥ p  .
Proof It is well known that L p (a, b) is strictly increasing with respect to p ∈ R for fixed a, b >  with a = b, then φ p (r) is strictly decreasing with respect to p ∈ R. In order to prove Lemma ., we divide it into three cases. Case  p = /. From Corollary . and Lemma ., we clearly see that We divide it into two subcases. Subcase A φ p  (r) <  for r ∈ (, .).
Since φ p (r) is strictly decreasing with respect to p ∈ R, we clearly see that φ p  (r) < φ  (r). It suffices to prove that φ  (r) <  for r ∈ (, .).

Main results
Theorem . Inequality L - (a, b) < AG(a, b) < L -/ (a, b) holds for all a, b >  with a = b, where L - (a, b) and L -/ (a, b) are the best possible lower and upper generalized logarithmic mean bounds for the arithmetic-geometric mean AG(a, b), respectively.
Proof Firstly, from (.) we clearly see that L - (a, b) < AG(a, b) for all a, b >  with a = b.
Next, we prove that AG(a, b) < L -/ (a, b) for all a, b >  with a = b. Since AG(a, b) and L p (a, b) are symmetric and homogeneous of degree , without loss of generality, it suffices to give an assumption that a =  > b. Let t = b ∈ (, ), r = (t)/( + t), then (.) and (.) lead to We can rewrite h(r) as where λ(r ) = ( + r ) log(e  /r ). A simple calculation yields Equations (.)-(.) lead to the conclusion that λ(r ) is strictly decreasing on (, ) with respect to r . Moreover, the function r = √ r  is strictly decreasing on (, ). Hence the function λ(r ) is strictly increasing on (, ) with respect to r. It follows from (.) and Lemma .() that h(r) is strictly decreasing on (, ). This implies that h(r) <  for  < r <  together with h() = .
Therefore, AG(a, b) < L -/ (a, b) for all a, b >  with a = b follows from (.) and h(r) < . Finally, we prove that L - (a, b) and L -/ (a, b) are the best possible lower and upper generalized logarithmic mean bounds for the arithmetic-geometric mean AG(a, b).
For any  < ε < / and  < x < , it follows from (.) and (.) that and making use of the Taylor expansion as x → , one has Equations (.) and (.) imply that for any  < ε < / there exist δ  = δ  (ε) ∈ (, ) and where p  is defined as in Lemma . and L / (a, b), L p  (a, b) are the best possible lower and upper generalized logarithmic mean bounds for the Toader mean T(a, b), respectively.
Proof From (.) and (.) we clearly see that both T(a, b) and L p (a, b) are symmetric and homogeneous of degree . Without loss of generality, we assume that a =  > b. Let t = b ∈ (, ), r = (t)/( + t), then from (.) and (.) together with (.) we have where φ p (r) is defined as in Lemma .. Therefore, Theorem . follows from (.) and Lemma ..

Corollaries and remarks
From Theorem . we get a lower bound for the complete elliptic integral of the first kind K(r) as follows.

Remark .
We define H(r) = π[ + √ r  -(r  ) / ]/( -√ r  )  . Computational and numerical experiments show that the lower bound in (.) can be regarded as an approximation of K(r) for some r ∈ (, ), refer to Table  for numerical values. Table 1 Comparison of K(r) with H(r) for some r ∈ (0, 1) r