We study the fundamental properties of the quantum f-relative entropy, where f(.) is an operator convex function. We give the equality conditions under various properties including monotonicity and joint convexity, and these conditions are more general than, since they hold for a class of operator convex functions, and different for f(t) = -log(t) from, the previously known conditions. The quantum f-entropy is defined in terms of the quantum f-relative entropy and we study its properties giving the equality conditions in some cases. We then show that the f-generalizations of the Holevo information, the entanglement-assisted capacity, and the coherent information also satisfy the data processing inequality, and give the equality conditions for the f-coherent information.