Anna Vershynina - University of Houston It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. The same question has been asked for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable states, when the distance is measured by the quasi-relative entropy. I will present a few recent results for different general scenarios of states and functions. At the end, I will review a relation between quasi-relative entropy and the trace distance. One way this inequality is known as the Pinsker inequality, but the more challenging direction is the reversed Pinsker inequality. I will present a result on this inequality, and apply it to Umegaki, Tsallis and Renyi relative entropies, improving previously known results.