Optimal bounds for the tangent and hyperbolic sine means
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Aequationes Mathematicae
Optimal bounds for the tangent and hyperbolic sine means Monika Nowicka
and Alfred Witkowski
Abstract. We provide optimal bounds for the tangent and hyperbolic sine means in terms of various weighted means of the arithmetic and geometric means. Mathematics Subject Classification. Primary 26D15. Keywords. Seiffert-like mean, Seiffert function, Convex function.
1. Introduction, definitions and notation The means
and
⎧ x−y ⎨ x−y Mtan (x, y) = 2 tan x+y ⎩ x
⎧ ⎨
x−y x−y Msinh (x, y) = 2 sinh x+y ⎩ x
x = y
(tangent mean)
x=y
x = y
,
(hyperbolic sine mean)
x=y
defined for all positive x, y, were introduced in [4], where one of the authors investigates means of the form ⎧ |x − y| ⎪ ⎨ x= y |x−y| Mf (x, y) = 2f x+y . (1) ⎪ ⎩ x x=y 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
AEM
M. Nowicka, A. Witkowski
(called Seiffert function) satisfying z z ≤ f (z) ≤ 1+z 1−z produces a mean. The correspondence between means and Seiffert functions is given by the formula f (z) =
z , M (1 − z, 1 + z)
where z =
|x − y| . x+y
The aim of this paper is to determine various optimal bounds for the Mtan and Msinh with the arithmetic and geometric means (denoted here by A and G). 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 means the inequality f < g holds, then their corresponding means satisfy Mf > Mg . Thus every inequality between means can be expressed in terms of their Seiffert functions. Remark 1.1. Note that the Seiffert function of the geometric mean G(x, y) = √ x+y z xy is g(z) = √1−z is the 2 and that of the arithmetic mean A(x, y) = 2 identity function a(z) = z. Clearly, the Seiffert functions of Mtan and Msinh are the functions tan and sinh, respectively. Remark 1.2. Throughout this paper all means are defined on (0, ∞)2 . 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 G < L < Mtan < Msinh < A proven in [4, Lemma 3.2]. The results obtained in this paper show what the distance is between the new and the classical means measured in different ways.
2. 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 L−K α< M −K < β. If k, l, m are respective Seiffert functions, then the latter can be written as 1 − k1 (2) α < 1l 1 < β. m − k Thus the problem reduces to finding upper and lower bounds for certain functions defined on the interval (0, 1).
Tangent and hyperbolic sine means
Theorem 2.1. The inequalities (1 − α) G + αA < Mtan < (1 − β) G + βA hold if and only if α ≤
1 3
and β ≥ cot 1 ≈ 0.6421.
Proof. Taking Remark 1.1 and the formula (2) into account we should
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