BOUNDS FOR A TOADER-TYPE MEAN BY ARITHMETIC AND CONTRAHARMONIC MEANS

In this paper, we present the best possible parameters αi and βi with i = 1, 2, 3, 4 such that the double inequalities α1A(a, b) + (1− α1)C(a, b) < T [A(a, b), C(a, b)] < β1A(a, b) + (1− β1)C(a, b), A2(a, b)C2(a, b) < T [A(a, b), C(a, b)] < A2(a, b)C2(a, b), α3 A(a, b) + 1− α3 C(a, b) < 1 T [A(a, b), C(a, b)] < β3 A(a, b) + 1− β3 C(a, b) , C[α4a+ (1− α4)b, α4b+ (1− α4)a] < T [A(a, b), C(a, b)] < C[β4a+ (1− β4)b, β4b+ (1− β4)a] hold for all a, b > 0 with a 6= b, as consequences, we provide several new bounds for the complete elliptic integral E(r) = ∫ π/2 0 (1 − r2 sin 2 θ)1/2dθ (r ∈ (0, √ 3/2) of the second kind, where T (a, b) = 2 π ∫ π/2 0 √ a2 cos2 θ + b2 sin θdθ, A(a, b) = (a + b)/2 and C(a, b) = (a2 + b2)/(a + b) are the Toader, arithmetic and contraharmonic means of a and b, respectively. AMS Subject Classification: 26E60


Introduction
For a, b > 0 with a = b, let p ∈ [0, 1] and q ∈ R, then the p-th generalized Seiffert mean S p (a, b), q-th Gini mean G q (a, b), q-th power mean M q (a, b), q-th Lehmer mean L q (a, b), harmonic mean H , q = 2, a a b b 1/(a+b) , q = 2, M q (a, b) = a q +b q 2 1/q , q = 0, √ ab, q = 0, It is well known that S p (a, b), G q (a, b), M q (a, b) and L q (a, b) are continuous and strictly increasing with respect to p ∈ [0, 1] and q ∈ R for fixed a, b > 0 with a = b, and the inequality chain hold for all a, b > 0 with a = b.
In [1], Toader introduced the Toader mean T (a, b) of two positive numbers a and b as follows: where ) is the complete elliptic integral of the second kind.
Recently, the Toader mean T (a, b) has been the subject of intensive research.Vuorinen [2] conjectured that the inequality holds for all a, b > 0 with a = b.This conjecture was proved by Qiu and Shen [3], and Barnard et al. [4], respectively.
Alzer and Qiu [5] presented a best possible upper power mean bound for the Toader mean as follows: for all a, b > 0 with a = b.
Neuman [6], and Kazi and Neuman [7] proved that the inequalities hold for all a, b > 0 with a = b, where AGM (a, b) is the arithmetic-geometric mean of a and b.
In [8][9][10], the best possible parameters ) hold for all a, b > 0 with a = b were presented.

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 ∈ (0, 1) and r ′ = √ 1 − r 2 , the well-known complete elliptic integrals of the first and second kinds are defined by respectively, and the following formulas were presented in [18, Appendix E, pp.474-475]: In what follows, two special values E(1/3) and K(1/3) will be used.By numerical computations, these are given by ) is strictly monotone, then the monotonicity in the conclusion is also strict.
As an application, Corollary 3.5 follows immediately from Theorems 3.1, 3.2, 3.3.We remark that the optimal bounds for the complete elliptic integral of second kind obtained from Theorem 3.4 is the same as those from Theorem 3.1 since λ 1 + λ 4 = 1.
(a, b), geometric mean G(a, b), quadratic mean Q(a, b), centroidal mean C(a, b), arithmetic mean A(a, b) and contraharmonic mean C(a, b) are respectively defined by S p (a, b) =