Suppose there is an ellipse with a semi-major axis length of $A$ and a semi-minor axis length of $B$. The task is to prove or disprove that the circumference of the ellipse in question is greater than
$$2\pi B \coth{\frac{\pi B}{2\sqrt{A^2 - B^2}}}$$
This is a problem that I received from a friend, and so far I have tried to compare it to the integral expression for the circumference of an ellipse and use Jenson's inequality to find another lower bound to compare the above expression to. This leading me to no avail, I thought I would pose the problem here to see if there is a better approach.