Hypergeometric series approach to the p -generalized Masjed-Jamei type inequality

Abstract In this paper, we establish a counterpart of the Masjed-Jamei inequality involving the inverse hyperbolic tangent and inverse trigonometric sine functions, and extend this Masjed-Jamei type inequality to the setting of p -generalized inverse trigonometric and hyperbolic functions. Specifically, we prove that the p -generalized Masjed-Jamei type inequality a r c t a n h p 2 (x) ≤ x ⁡ arcsin p (x) (1 − | x | p) 1 / p arctanh₏^2 (x) xarcsin₏ (x) { (1- x{ ^p) }^1/p} holds for all x ∈ (−1, 1) if and only if p ∈ (0, 2]. A key feature of our results is the analysis performed in the first proof for the case p ∈ (2, + ∞), which relies on investigating the piecewise monotonicity of ratios of power series whose coefficients satisfy two second-order recurrence relations.