Generalized Briot–Bouquet differential equation by a quantum difference operator in a complex domain

In the present study, we employ the concept of quantum calculus and the convolution product to impose a generalized symmetric Sàlàgean q-differential operator. By consuming the new operator and the model of the Janowski function, we describe definite new classes of analytic functions in the open unit disk. As a consequence, we deliver some interesting inclusions and inequalities of these classes. Moreover, we apply the q-differential operator to generalize the Briot–Bouquet differential equation (BBDE) and study the existence of the major solution (analytic solution). Finally, we illustrate some examples, including time-space BBDE.