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

This paper develops two novel arcsine-type analytic approximation formulas for $\mathrm{arcsl}\left(x\right)$, the Gauss lemniscate sine function, each equipped with a monotonic and bounded remainder term of order $x^{29}$ as $x\to 0$, 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.