Sharp Bounds for the Arc Lemniscate Sine Function
arXiv:1903.03897
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 They proved $$ \frac{1}{8} \, x \, Φ(x^4, 3/2, 1/4) < \mbox{arcsl}(x)< \frac{1}{4} \, x \, Φ(x^4,3/2,1/4)\qquad{(0<x<1)}. $$ We %use the monotone form of l'Hopital's rule to show that the factor can be replaced by $\mbox{arcsl}(1)/Φ(1,3/2,1/4)=0.12836...$. This constant is best possible.
To appear in Applied Mathematics E-Notes