I found this formula in Brian McCartin's interesting book "Mysteries of the equilateral triangle" http://www.kettering.edu/news/mysteries-equilateral-triangle and it looks as follows:

Suppose that $f(z)$ is analytic on the equilateral triangle, $T$ , with vertices at 1, $\omega$, $\omega^2$, where $$\omega=\exp{\frac{2\pi i}{3}}.$$ Then $$\iint \limits_T f^{\prime \prime}(z)\,dx\,dy=\frac{\sqrt{3}}{2}[f(1)+ \omega f(\omega)+\omega^2 f(\omega^2)].$$

A variant of the proof, as well as the generalization to other polygons, can be found in http://www.jstor.org/discover/10.2307/2002943?uid=3738936&uid=2&uid=4&sid=21104022760067 (Triangle Formulas in the Complex Plane, by Philip J. Davis).

Can the Grunsky-Motzkin-Schoenberg formula be generalized to other hypercomplex numbers (especially to quaternions and octonions)?