RACSAM (2020) 114:25 https://doi.org/10.1007/s13398-019-00735-z ORIGINAL PAPER Natural approximation of Masjed-Jamei’s inequality Ling Zhu 1 · Branko Maleševi´ c 2 Received: 7 August 2019 / Accepted: 25 October 2019 © The Royal Academy of Sciences, Madrid 2019 Abstract In this paper, we obtain a general result on the natural approximation of the function (arctan x ) 2 − (x arcsinh x )/ √ 1 + x 2 , and prove a conjecture raised by Zhu and Maleševi´ c (J Inequal Appl 2019:93, 2019). Keywords Conjecture · Inverse tangent function · Inverse hyperbolic sine function Mathematics Subject Classification Primary 26D05 · 26D07 · 26D15; Secondary 33B10 · 41A58 1 Introduction Masjed-Jamei [1] obtained the following inequality (arctan x ) 2 ≤ x ln  x + √ 1 + x 2  √ 1 + x 2 , |x | < 1, (1.1) where ln(x + √ 1 + x 2 ) is the inverse hyperbolic sinefunction arcsinh x = sinh −1 x . In [1] the author conjectured that the above inequality is established in a larger interval (−∞, ∞). Recently, the authors of this paper [2] first affirmed Masjed-Jamei’s conjecture, obtained some natural generalizations of this inequality, and pose a conjecture about a natural approach of Masjed-Jamei’s inequality inspired by [3–9]. B Ling Zhu zhuling0571@163.com Branko Maleševi´ c branko.malesevic@etf.rs 1 Department of Mathematics, Zhejiang Gongshang University, Hangzhou City 310018, Zhejiang Province, China 2 Faculty of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, Belgrade 11000, Serbia 0123456789().: V,-vol 123