Normalized Symmetric Differential Operators in the Open Unit Disk

The symmetric differential operator SDO is a simplification functioning of the recognized ordinary derivative. The purpose of this effort is to provide a study of SDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to deliver two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function, bounded turning function subclass and convolution structures. Consequently, we define a linear combination differential operator involving the Sàlàgean differential operator and the Ruscheweyh derivative. The new operator is a generalization of the Lupus differential operator. Moreover, we aim to solve some special complex boundary problems for differential equations, spatially the class of Briot-Bouquet differential equations. All solutions are symmetric under the suggested SDOs. Additionally, by using the SDOs, we introduce a generalized class of Briot-Bouquet differential equations to deliver, what is called the symmetric Briot-Bouquet differential equations. We shall show that the upper solution is symmetric in the open unit disk by considering a set of examples of univalent functions.