Sharp Arcsine-Type Bounds and Analytic Approximations for the Gauss Lemniscate Sine Function

This paper develops two novel arcsine-type analytic approximation formulas for arcsl(x), the Gauss lemniscate sine function, each equipped with a monotonic and bounded remainder term of order x29 as x→0, which demonstrates the high accuracy of the obtained formulas near the origin. Based on these approximations, we establish several new sharp inequalities valid on the interval (0,1). Numerical evidence confirms that the resulting bounds provide improved accuracy compared with existing estimates in the literature, particularly in a neighborhood of the origin.