The derivation of thermodynamic stability conditions for the Renyi and Dar6czy-Tsallis entropies recently given by Ramshaw [Phys. Lett. A 198 (1995) 119] are based on an incorrect assumption about the ''equilibrium'' state for a composite non-interacting system. Moreover, contrary to the claim by Ramshaw, it can be shown that these conditions are satisfied when the parameter q entering the definition of the entropies is less than 1. We also establish a number of inequalities for the entropies that are important in the context of stability considerations.