Complete Continuity of Composition-Differentiation Operators on the Hardy Space $H^{}$

We study composition-differentiation operators on the Hardy space $H^1$ on the unit disk. We prove that if $\phi$ is an analytic self-map of the unit disk such that the composition-differentiation operator induced by $\phi$ is bounded on the Hardy space $H^1$ , then it is completely continuous. This result is stronger than the similar result for composition operators which says that the composition operator induced by $\phi$ is completely continuous if and only if $\left|\phi \left(e^{i\theta }\right)\right|<1$ almost everywhere on the unit circle.