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 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 z 1 + z ≤ f (z) ≤ z 1 − z produces a mean. The correspondence between means and Seiffert functions is given by the formula , where z = |x − y| x + y .
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 [5], and by arithmetic and quadratic means in [6]. For other bounds of Seiffert-like means by the arithmetic and centroidal means, see e.q. [7,2,3,9].
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.  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 [8, Lemma 3.1] and Lemma 7.1.

Linear bounds
Given three means K < L < M , one may try to find the best α, β satisfying the double inequality (1 − α)K + αM < L < (1 − β)K + βM or equivalently α < L−K M −K < β. 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).
Proof. By formula (2) and Remark 1.2, 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.
hold if, and only if, α ≤ 3 tanh 1 − 3 ≈ 0.9391 and β ≥ 1. Proof. We use Remark 1.2 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 7.2), so s is concave and, by Property 7.2, 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 K < L < M of the form We shall use the above to prove two theorems.
hold if, and only if, α ≤ 4 sin 1 − 3 ≈ 0.3659 and β ≥ 1 2 . 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.
Proof. We use Remark 1.2 and formula (3) once more and investigate the function We shall show that h increases. We have 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).
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.
Proof. This time we investigate the function This function increases, because by Lemma 7.4 So the function h assumes values between lim z→0 h(z) = 0 and h(1).

Bounds with varying arguments
If N is a mean, then the formula N {t} (x, y) = N x+y 2 + t x−y 2 , x+y 2 − t x−y In the case of N = Ce we see that ce(z) = 3 z 2 +3 and ce −1 (x) = √ 3x −1 − 3.
Theorem 6.2. The inequalities hold if, and only if, α ≤ 1 2 ≈ 0.7071 and β ≥ 3 sin 1 − 3 ≈ 0.7518. Proof. Using Theorem 6.1 we should find the range of the function The monotonicity of the function h 2 follows from the proof of Theorem 2.1, so evaluation of the values of h at the endpoints completes the proof. Proof. According to Theorem 6.1, we shall consider the function but we found the range of its square in the proof of Theorem 2.2.

Tools and lemmas
In this section, we place all the technical details needed to prove our main results.
Other proofs are similar.