Hi,

I asked some time ago the following [question][1] on math.stackexchange, but I ask it here too since it remains unanswered.

The question concerns a function I encountered during research :

$$f(k):= k K(k) \sinh \left(\frac{\pi}{2} \frac{K(\sqrt{1-k^2})}{K(k)}\right)$$
for $k \in (0,1)$. 

Here $K$ is the [Complete elliptic integral of the first kind][2], defined by
$$K(k):= \int_{0}^{1} \frac{dt}{\sqrt{1-t^2} \sqrt{1-k^2t^2}}.$$

More specifically, my question is the following :

**Is $f$ decreasing on $(0,1)$?**

This seems to be true, as the graph below suggests (obtained with Maple) :

![graph of $f$][3]

In fact, as remarked by Henry Cohn, much more seems to be true : all the derivatives of $f$ seem to be negative. This can be seen by looking at the Taylor series expansion of $f$ (see the link to math.stackexchange). The Taylor series expansion seems to have all negative coefficients (except the constant term), and the coefficient of $k^{2j}$ seems to be $\pi$ times a rational number with denominator dividing $16^j$...

Any comment or relevant reference is welcome.

Thank you,
Malik

**EDIT (20-07-2012)**
It was remarked by J.M. on M.SE that $f$ can be written as
$$f(k)=kK(k)\frac{1-q(k)}{2\sqrt{q(k)}},$$
where $q(k)$ is the [Elliptic nome][4]. Maybe this is useful...


  [1]: https://math.stackexchange.com/questions/167498/is-this-function-decreasing-on-0-1
  [2]: http://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind
  [3]: https://i.sstatic.net/cOtoK.jpg
  [4]: http://mathworld.wolfram.com/Nome.html