Examining the behavior of parametric convex operators on a certain set of analytical functions

Mathematical operators that maintain convex functional combinations involving at least one parameter are called parametric convex operators (PCOs) on analytic function spaces. In the context of analytic functions, these operators are often described on spaces of holomorphic (analytic) functions. The most important challenge in this direction is to investigate the boundedness and discovers the upper bound element (in this case, it is the extreme analytic function in the open unit disk). The boundedness of weighted composition operators has been studied extensively on several analytic function spaces. In the present study, we examine the boundedness of two proposed weighted composition operators on different analytic function spaces with differences structure. Examples are illustrated containing the Koebe function (the extreme convex function in the open unit disk). The methodology of this study based on:•Define a new convex operator with a set of parameters.•A special subclass of analytic functions in the open unit disk including the proposed convex operator is suggested.•The boundedness of the convex operator is investigated by illustrating a set of conditions on the parameters.