  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.
