Further Refinements of Shafer–Fink Type Inequalities for Lemniscatic Functions

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Abstract
The lemniscatic functions and their inverses play a vital role in various fields of applied science. In classical mechanics, they appear in the exact solutions of the nonlinear pendulum equation and describe the trajectories of particles in central force fields. In electromagnetism, they are involved in the modeling of electric and magnetic field distributions with elliptic symmetry. Moreover, in engineering mathematics and optical system design, lemniscatic functions arise in problems involving wave propagation, signal transmission, and conformal mappings. Their mathematical properties make them valuable tools in both theoretical analysis and computational modeling. Lemniscatic functions, such as Gauss’ arc lemniscate sine and the hyperbolic arc lemniscate sine, arise from the incompletely symmetric elliptic integral of the first kind. In 2007, Neuman introduced additional lemniscatic functions including the arc lemniscate tangent and its hyperbolic counterpart. These functions have attracted considerable attention due to their deep connections with elliptic integrals and their applications in inequality theory. In this paper, we build upon recent work by Wei, He, and Wang (2020), who established Shafer–Fink type inequalities for these lemniscatic functions. Employing methods from real analysis, we present new and sharper refinements of these inequalities. Specifically, we establish double inequalities for Gauss’ arc lemniscate sine, the hyperbolic arc lemniscate sine, arc lemniscate tangent and its hyperbolic counterpart bounded by rational expressions involving radical terms. These results not only improve the existing Shafer–Fink type inequalities but also provide further insight into the monotonicity and convexity properties of the involved lemniscatic functions. Our approach offers a unified framework for deriving tight bounds for all four Neuman lemniscatic functions, contributing to the broader understanding of special function inequalities. These results represent significant refinements over previous known estimates and open avenues for further generalizations.

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