# Sharp Bounds for the Arc Lemniscate Sine Function

## TL;DR

This paper refines bounds for the arc lemniscate sine function by replacing a constant factor with a sharper, optimal value derived using monotone l'Hopital's rule, improving previous inequalities.

## Contribution

It introduces a tighter bound for arcsl(x) in terms of the Lerch zeta function, establishing the best possible constant factor.

## Key findings

- The constant factor 1/4 is replaced by approximately 0.12836.
- The new bound is proven to be optimal.
- The method uses monotone l'Hopital's rule for the proof.

## Abstract

The arc lemniscate sine function is given by $$ \mbox{arcsl}(x)=\int_0^x \frac{1}{\sqrt{1-t^4}}dt. $$ In 2017, Mahmoud and Agarwal presented bounds for $\mbox{arcsl}$ in terms of the Lerch zeta function $$ \Phi(z,s,a)=\sum_{k=0}^\infty \frac {z^k}{(k+a)^s}. $$ They proved $$ \frac{1}{8} \, x \, \Phi(x^4, 3/2, 1/4) < \mbox{arcsl}(x)< \frac{1}{4} \, x \, \Phi(x^4,3/2,1/4)\qquad{(0<x<1)}. $$ We %use the monotone form of l'Hopital's rule to show that the factor $1/4$ can be replaced by $\mbox{arcsl}(1)/\Phi(1,3/2,1/4)=0.12836...$. This constant is best possible.