New inequalities between the inverse hyperbolic tangent and the analogue for corresponding functions

Xiao-Diao Chen,Long Nie,Wangkang Huang

doi:10.1186/s13660-020-02396-8

Xiao-Diao Chen, Long Nie + Show 1 more

Open Access

https://doi.org/10.1186/s13660-020-02396-8

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Abstract

In this paper, we present new inequalities which reveal further relationship for both the inverse tangent function arctan (x) and the inverse hyperbolic function operatorname{arctanh}(x). At the same time, we give the analogue for inverse hyperbolic tangent and other corresponding functions.

Highlights

Masjed-Jamei [1] obtained the following inequality for |x| < 1: √(arctan x)2 ≤ x ln(x√+ 1 + x2) . (1) 1 + x2Many similar or relative inequalities are discussed in references [2,3,4,5,6,7,8,9,10,11,12,13,14]
We prove that κ1(arctan x)2 ≤ F(x)
Note that Gi(–x) = Gi(x), i = 1, 2, combining with Eq (34), we have proved both Eq (13) and Theorem 9

Summary

Introduction

Masjed-Jamei [1] obtained the following inequality for |x| < 1:. 1 + x2). Zhu and Malesevic [15√] affirmed inequality (1) for the large interval (–∞, ∞), pointed out that sinh–1(x) = ln(x + 1 + x2), and provided the following Theorems 1–6, which (or relative results) can be found in [11, 12]. In 2020, Zhu and Malešević [13] proposed natural approximation of Masjed-Jame√i’s inequality and provided two-sided bounds in a polynomial form of (arctan x). Holds for all x ∈ (–1, 1), where κ3 and κ4 are the best constants in (14)

Proofs of Theorems 6–10

Proof of Theorem 8