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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 apply to a class of operator convex functions that we define, and this class is different from the ones previously studied. 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.