New inequalities for hyperbolic functions based on reparameterization
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New inequalities for hyperbolic functions based on reparameterization Wangkang Huang1,2 · Xiao-Diao Chen1 · Linqiang Chen3 · Xiaoyang Mao1,2 Received: 8 June 2020 / Accepted: 28 September 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In this paper, we present new inequalities about hyperbolic functions with much better approximation effect. It firstly provides two-sided bounds of (sinh(x)/x) p for the case p ∈ (0, 1], and lower bound for the case p ≥ 75 as well. It also provides inequalities about mixed hyperbolic functions consisting of tanh(x) and sinh(x). Numerical examples show that the new inequalities can achieve much better approximation effect than those of prevailing methods. Keywords Inequalities · Inverse tangent function · Inverse hyperbolic sine function · Inverse hyperbolic tangent function · Sine function Mathematics Subject Classification 26D05 · 26D07
1 Introduction It is well known that (see [1,6,7,9,10,13,15,16]) 1
a2 x 4 x x x 2 x tanh(x) 4/7 x + , (7) − 2 < b2 x 4 sinh(x) tanh(x) x 8 2 and b2 ≥ 45 . holds if and only if a2 ≤ 45 This paper discusses the Cusa-type and other types of inequalities on hyperbolic functions, where the
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