New classes of analytic functions determined by a modified differential-difference operator in a complex domain

In the recent study, we utilize a differential-difference operator in a complex domain to define some new classes of analytic functions. A modification of the differential-difference operator in the open unit disk is formulated in order to impose some geometric properties. The modified operator is studied in the Hardy space. Additionally, we define a new subspace of the Hardy space comprising the normalized analytic functions with the set of all bounded functions. We shall validate that the modified operator is closed in the subspace of normalized functions with the bounded first derivative. Our proofs are based on the Jack Lemma.