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.
Categories: math.CA