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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi mathvariant="normal">a</m:mi> <m:mi mathvariant="normal">r</m:mi> <m:mi mathvariant="normal">c</m:mi> <m:mi mathvariant="normal">t</m:mi> <m:mi mathvariant="normal">a</m:mi> <m:mi mathvariant="normal">n</m:mi> <m:mi mathvariant="normal">h</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mfrac> <m:mrow> <m:mi>x</m:mi> <m:mo>⁡</m:mo> <m:msub> <m:mrow> <m:mi>arcsin</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfrac> </m:math> arctanhₚ²(x)≤ xarcsinₚ(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.