We examine the evolution of H-functions during the process of violent relaxation of collisionless self-gravitating systems. Our arguments follow a few, relatively recent, works which dealt with this subject. We point out that the functional that is extremized through violent relaxation does not necessarily increase at all times and that it may decrease if the system has specific, far from equilibrium, conditions. We propose to distinguish between two phases of violent relaxation. In one phase the H-function may decrease with time, while in the other it is always a nondecreasing function of time. The phase during which the H-function may decrease is characterized by the conversion of kinetic to gravitational energy, and it happens before the central-dense and outer-dilute equilibrium structure is formed. We propose initial conditions to be imposed on N-body numerical simulations, which can be performed in order to test our claims.
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