Analytic Studies of a Class of Langevin Differential Equations Dominated by a Class of Julia Fractal Functions

In this investigation, we study a class of analytic functions of type Carathéodory style in the open unit disk connected with some fractal domains. This class of analytic functions is formulated based on a kind of Langevin differential equations (LDEs). We aim to study the analytic solvability of LDEs in the advantage of geometric function theory consuming the geometric properties of the Julia fractal (JF) and other fractal connected with the logarithmic function. The analytic solutions of the LDEs are obtainable by employing the subordination theory.