Abstract
In this paper, we obtain some new inequalities which reveal the further relationship between the inverse tangent function arctanx and the inverse hyperbolic sine function sinh−1x. At the same time, we give the analogue for inverse hyperbolic tangent and inverse sine.
Highlights
In 2010, Masjed-Jamei [1] obtained the following inequality: √(arctan x)2 ≤ x ln(x√+ 1 + x2), |x| < 1. (1.1) 1 + x2[1] reminded us that the above inequality is established in a larger interval (–∞, ∞) because it was detected by Maple software
Inequality (1.1) gives the upper bound for the square of the√inverse tangent function arctan x by the inverse hyperbolic sine function sinh–1 x = ln(x + 1 + x2)
We show the analogue for inverse hyperbolic tangent function arctanh x = (1/2) ln((1 + x)/(1 – x)) and inverse sine function arcsin x
Summary
In 2010, Masjed-Jamei [1] obtained the following inequality:. |x| < 1. In 2010, Masjed-Jamei [1] obtained the following inequality:. [1] reminded us that the above inequality is established in a larger interval (–∞, ∞) because it was detected by Maple software. Inequality (1.1) gives the upper bound for the square of the√inverse tangent function arctan x by the inverse hyperbolic sine function sinh–1 x = ln(x + 1 + x2). We first affirm Masjed-Jamei’s quest, conclude that the scope of the inequality is the large interval (–∞, ∞), and give a simple proof of this result. We obtain some natural generalizations of this inequality. We show the analogue for inverse hyperbolic tangent function arctanh x = (1/2) ln((1 + x)/(1 – x)) and inverse sine function arcsin x. 1 + x2 holds for all x ∈ (–∞, ∞), and the power number 2 is the best in (1.2)
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