Optimal bounds for the sine and hyperbolic tangent means IV

We provide optimal bounds for the sine and hyperbolic tangent means in terms of various weighted means of the arithmetic and centroidal means


Introduction, definitions and notations
The means   1 It was shown that every symmetric and homogeneous mean of positive arguments can be represented in the form (1) and that every function f : (0, 1) → R (called Seiffert function) satisfying produces a mean. The correspondence between means and Seiffert functions is given by the formula Comparing the means and examining the relationships between them is considered important. A cursory overview of MathSciNet shows over 200 papers on this subject, and the number is constantly growing. The aim of this paper is to determine various optimal bounds for the M tanh and M sin with the arithmetic and centroidal means (denoted here by A and Ce). Similar bounds by the arithmetic and contraharmonic means were obtained in [12], and by arithmetic and quadratic means in [11]. For other bounds of Seiffert-like means by the arithmetic and centroidal means, see e.g. [7,8,17,20]. Similar subjects were considered also in [2][3][4][5][6]10,[13][14][15][16]18,21].
For two means M, N , the symbol M < N denotes that for all positive x = y the inequality M(x, y) < N (x, y) holds.
Our main tool will be the obvious fact that if for two Seiffert functions the inequality f < g holds, then their corresponding means satisfy M f > M g . Thus every inequality between means can be replaced by the inequality between their Seiffert functions.

Remark 1
Throughout this paper all means are defined on (0, ∞) 2 .

Remark 2
Note that the Seiffert function of the centroidal mean Ce(x, y) = 2 3 x 2 +xy+y 2 x+y is ce(z) = 3z 3+z 2 and that of the arithmetic mean A(x, y) = x+y 2 is the identity function a(z) = z. Clearly, the Seiffert functions of M sin and M tanh are the functions sin and tanh, respectively.
For the reader's convenience, in the following sections we place the main results with their proofs, while all lemmas and technical details can be found in the last section of this paper.
The motivation for our research are the inequalities A < M sin < M tanh < Ce proven in [19, Lemma 3.1] and Lemma 1.

Linear bounds
Given three means K < L < M, one may try to find the best α, β satisfying the double If k, l, m are respective Seiffert functions, then the latter can be written as Therefore the problem reduces to finding upper and lower bounds for certain functions defined on the interval (0, 1).

Theorem 1 The inequalities
hold if, and only if, α ≤ 1 2 and β ≥ 3 sin 1 − 3 ≈ 0.5652. Proof By formula (2) and Remark 2, we investigate the function We shall show that h increases. Observe that Using the known inequalities so h (z) > 0. We complete the proof by noting that lim z→0 h(z) = 1/2.

Theorem 2 The inequalities
hold if, and only if, α ≤ 3 tanh 1 − 3 ≈ 0.9391 and β ≥ 1. Proof We use Remark 2 and formula (2) once more and investigate the function The function s satisfies lim z→0 s(z) = 0 and s (z) = 2 sinh 3 z cosh z − sinh 3 z z 3 < 0 (by Lemma 2), so s is concave and, by Property 2, its divided difference (and consequently the function h) decreases. To complete the proof note that lim z→0 h(z) = 1.

Harmonic bounds
In this section, we look for optimal bounds for means We shall use the above to prove two theorems.
Proof According to formula (3), we investigate the function We shall show that h decreases. We have The function s satisfies s(0) = s (0) = s (0) = 0 and Thus s is negative and so is h . We complete the proof by noting that lim z→0 h(z) = 1/2.

Theorem 4 The inequalities
Proof We use Remark 2 and formula (3) once more and investigate the function We shall show that h increases. We have By Lemma 4 we get Therefore h increases from lim z→0 h(z) = 0 to h(1).

Quadratic bounds
Given three means K < L < M, one may try to find the best α, β satisfying the If k, l, m are respective Seiffert functions, then the latter can be written as Thus, the problem reduces to finding upper and lower bounds for certain functions defined on the interval (0, 1).

Theorem 5 The inequalities
hold if, and only if, α ≤ 1 2 and β ≥ 9 7 cot 2 1 ≈ 0.5301. Proof Using formula (4) we investigate the function To show that h increases we use Lemma 3. A simple calculation shows that Thus r is positive and both r and h increase. We complete the proof by noting that lim z→0 h(z) = 1/2.
And here comes the hyperbolic tangent version of the previous theorem.

Bounds with the weighted power mean of order −2
In this section, we look for optimal bounds for means K < L < M of the form 1

Bounds with varying arguments
If N is a mean, then the formula N {t} (x, y) = N x+y hold if, and only if, p ≤ p 0 and q ≥ q 0 .
Other proofs are similar.

Conflict of interest
The authors declare that they have no conflict of interest.
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