Optimal bounds for the tangent and hyperbolic sine means

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


Introduction, definitions and notation
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 We shall show that h increases or -which is equivalent -that z tan z −1 √ 1−z 2 −1 decreases. By Lemma 7.3 it is enough to prove that the function r(z) = (z/ tan z−1) decreases.

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 ≤ 0 and β ≥ 2 3 . Proof. By (4) we shall consider the function We notice immediately that lim z→1 h(z) = 0 and lim z→0 h(z) = 2/3 so the only thing we have to show is that 2/3 is the upper bound for h. Note that the inequality h(z) < 2/3 is equivalent to 3 tan z − z − 2z √ 1−z 2 < 0. Substituting z = sin t transforms this inequality into p(t) = 3 tan(sin t) − sin t − 2 tan t < 0. We have p(0) = 0 and Therefore p is negative in (0, π/2), which completes the proof.
And now it is time for the bound of M sinh .

Theorem 3.2. The inequalities
hold if and only if α ≤ 0 and β ≥ 1 3 . Proof. This time we investigate the function As in the Proof of Theorem 3.1 we notice that lim z→1 h(z) = 0 and lim z→0 h(z) = 1/3. We shall show that h(z) < 1/3 for all 0 < z < 1. This inequality can be written as 3 sinh z − 2z − z (the last inequality is valid by the AG inequality). So p is negative and we are done.

Quadratic bounds
Given three means K < L < M one may try to find the best α, β satisfy- 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 4.1. The inequalities
hold if and only if α ≤ 1 3 and β ≥ 1 tan 2 1 ≈ 0.4123. Proof. By formula (5) we should investigate the function Since h (z) = 2 sin 3 z sin 3 z z 3 − cos z > 0 (by Lemma 7.1), the function h increases. We complete the proof by noting that lim  Proof. The function to be considered here is Its derivative equals h (z) = 2

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 Theorem 5.1. The inequalities hold if and only if α ≤ 0 and β ≥ 2 3 .