**András Horváth, Sándor Rács, and Miklós Telek**

The class of order 3 phase type distributions (PH(3)) is
known to be a proper subset of the class of order 3 matrix exponential
distributions (ME(3)). In this paper we investigate the relation of
these two sets for what concerns their moment bounds. To this end we
developed a procedure to check if a matrix exponential function of order
3 defines a ME(3) distribution or not. This procedure is based on the
time domain analysis of the density function. The proposed procedure
requires the numerical solution of a transcendent equation in some cases.

The presented moment bounds are based on some unproved conjectures which are verified only by numerical investigations