Doctoral Thesis / Dissertation from the year 2022 in the subject Mathematics - Analysis, grade: 4.00, , language: English, abstract: The main subject of this study coincides with the recently adapted methodology in the theory of univalent functions via applications of fractional calculus. The Caputo's definition of fractional derivative of order has not been obviously applied in the field. However, it plays a vital role in other areas like physics and engineering due to its simple restrictions compared to Liouville's integral operator. By modifying the Caputo's derivative operator, a new modified Caputo's operator is defined in this book.

Consequently, new subclasses of analytic functions are introduced, and basic properties, such as distortion theorems, extremal functions, coefficient bounds, the integral transform, superordination and subordination results are discussed. The radii of basic well known properties (Univalency, Starlikeness, Convexity) are also obtained for the new subclasses. Moreover, some results of the modified Hadamard product (convolution) are provided by applying the famous Cauchy-Schwartz inequality.

In order to expand the application of the modified Caputo's operator, two new subclasses of analytic p-valent functions are introduced and investigated through the concepts of neighbourhood and inclusion relations. Further, the definition of Univalency with respect to 1 together with subordination are used to solve a sharp functional of Fekete-Szegö problem. Moreover, some interesting results related to Hankel determinant are obtained.