Several sharp inequalities about the first Seiffert mean

In this paper, we deal with the problem of finding the best possible bounds for the first Seiffert mean in terms of the geometric combination of logarithmic and the Neuman–Sándor means, and in terms of the geometric combination of logarithmic and the second Seiffert means.


Introduction
A mean is a function f : R 2 + → R + which satisfies Each mean is reflexive, namely f (a, a) = a, ∀a > 0. (1.2) That is also used as the definition of f (a, a). A mean is symmetric if f (a, b) = f (b, a), ∀a, b > 0; (1.3) it is homogeneous (of degree 1) if f (ta, tb) = tf (a, b), ∀a, b, t > 0. (1.4) We shall refer here to some symmetric and homogeneous means as follows.
For a, b > 0 with a = b, the Neuman-Sándor mean M(a, b) [16], the first Seiffert mean P(a, b) [18], the second Seiffert mean T(a, b) [19]  , (1.7) and L(a, b) = ab log a -log b , (1.8) respectively. Let M p (a, b) = ((a p + b p )/2) 1/p (p = 0) stand for the pth power means. The mean M 1 = A is the arithmetic mean, and the mean M 2 = Q is the root-square mean. The geometric mean is given by G(a, b) = √ ab, but verifying also the property As Carlson remarked in [2], the logarithmic mean can be rewritten as , (1.9) thus the means M, P, T and L are very similar. In [16] it is also proven that these means can be defined using the non-symmetric Schwab-Borchardt mean SB given by (1.10) see [1]. It has been established in [16] that It is well known that the inequalities (1.12) Recently, the inequalities for means have been the subject of intensive research. Many remarkable inequalities can be found in the literature [5,7,9,14,15,17,20].
In [6], Costin and Toader presented holding for all a, b > 0 with a = b.
The following sharp power mean bounds for the first Seiffert mean P(a, b) are given by Jagers in [13]: for all a, b > 0 with a = b. Hästö [11] improved the results of [13] and the sharp result was found that In [8,12], the authors proved that the double inequalities hold for all a, b > 0 with a = b if and only if α 1 ≥ 1/3, β 1 ≤ (1/2)π(4/π -1/ sinh -1 (1)), α 2 ≥ 1/3, β 2 ≤ log(4 log(1 + √ 2)/π)/ log 2. In [3,4,10], the authors proved that the double inequalities The main purpose of this paper is to find the least values α and β such that the inequalities hold for all a, b > 0 with a = b. Moreover, we find that both upper bounds for P(a, b) are trivial cases. That is to say, the inequalities P(a, b) if and only if λ 1 = 0 and λ 2 = 0, which we will address at the end of this paper.

Main results
Note that L(a, b), P(a, b), T(a, b) and M(a, b) are symmetric and homogeneous of degree 1.
In this section, without loss of generality, we can assume that a > b, then x := (ab)/(a + b) ∈ (0, 1). We have the following two theorems. Proof Noting that we have Direct computations lead to Next, we take the logarithm of (3.1) and consider the difference between the convex combination of log L(a, b), log P(a, b) and log T(a, b) as follows: It follows that Therefore, it follows from (3.15)-(3.17) that for any x ∈ (0, 1). According to (3.14) and (3.18), we conclude that P(a, b) > L  L(a, b) < P(a, b) < M(a, b) holds for all a, b > 0 with a = b, we can see that (3.10) holds for all a, b > 0 with a = b and β ≥ 2 3 . If β < 2 3 , then Eqs. (3.12) and (3.13) imply that there exists 0 < σ 2 < 1 such that P(a, b) < L β (a, b)M 1-β (a, b) for all a, b, with (ab)/(a + b) ∈ (0, σ 2 ). The proof is completed.

Conclusion
In the article, we give the best possible bounds for the first Seiffert mean in terms of the geometric combination of logarithmic and the Neuman-Sándor means, and in terms of the geometric combination of logarithmic and the second Seiffert means.