In this paper, we use the double-time Green’s function method to study the magnetic properties of the mixed spin-1/2 and spin-1 Heisenberg ferrimagnets with magnetic fields on a simple cubic lattice. We derive the equations of motion of the Green’s function by a standard procedure. In the course of this, the higher order Green’s functions have to be decoupled. A Tyablikov or random phase approximation decoupling is used to decouple the higher order Green’s functions. Based on the above procedures, the effects of the magnetic fields on the critical and compensation temperatures are investigated. The cause of compensation temperature appearance of a spin-1/2 and spin-1 ferrimagnetic system is discussed in detail. Our results show that the magnetizations of the two sublattices approach to zero at the phase transition point when h a= h b=0. When h a=0, the dropping speed of the sublattice magnetization | m b| with the increase of temperature is slower than that of sublattice magnetization | m a| of small spin when the value of h b increases. Under general condition, S b is larger than S a so that | m a| is always smaller than | m b|. It means that the relation of m a=– m b≠0 cannot be satisfied. It indicates that when h a=0, no matter what value of other parameters in the Hamiltonian is, the compensation temperature cannot appear. On the contrary, in the case of h b=0, as the value of h a increases, the dropping speed of | m b| with the increase of temperature is faster than that of | m a|. It indicates that the difference between | m a| and | m b| decreases with the increasing of h a below the critical temperature. As h a increases to a certain value h amin, we obtain m a=– m b≠0 below the critical temperature. It means that the compensation point appears. The compensation temperature increases with increasing h a. And the value of h amin depends on h b=0. Therefore the condition for T com to appear is that h a/ h b should be beyond some value that depends on h b. By comparing and analyzing the previous research, we obtain the general condition for the appearance of compensation temperature of the ferrimagnetic system, that is, there is at least one additional force. This additional force causes that the decreasing speed of the sublattice magnetization of the large spin with increasing temperature is greater than that of the sublattice magnetization of the small spin. Only in this way, the relation m a=– m b≠0 can be satisfied, i.e., the compensation temperature appears. Otherwise the compensation point will not occur. Meanwhile, it should be noted that our Hamiltonian will recover an ordinary three-dimensional Heisenberg antiferromagnetic model with the nearest-neighbor interactions at S a= S b. Our results agree with the high temperature series, the linked-cluster series approach and ratio method results.