  We study the fundamental properties of the quantum f-relative entropy, where
f(.) is an operator convex function. We give the equality conditions under
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) = -ln(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.
