Properties for new classes of analytic functions defined by generalized fractional derivative and integral operators

Abstract

This thesis discusses some classical problems in geometric complex theory related to new generalized fractional differential and integral operators. It includes monitoring geometric properties for several classes of univalent functions introduced using the constructed fractional differential and integral operators. The main idea in constructing these operators was motivated by the fractional operators introduced by Owa in 1978. The new generalized fractional differential and integral operators are formed by utilizing mathematical induction. Then, the new classes of univalent functions are introduced. A class of functions with positive first derivative, uniformly convex, uniformly starlike, close-to-convex, α–close-to-convex, and strongly α-close-to-convex are studied. Some properties are studied, such as the coefficient inequality, growth and distortion theorem, radius of starlike and extreme value. Moreover, we give generalizations for results in the unified class of certain uniformly analytic functions with negative coefficients in the unit disk, showing the relation between the new classes with well-known classes. In addition, a generalized unified class is shown, in which coefficient inequalities for functions in the newly introduced class are presented. Additionally, we study the classical Fekete-Szegö problem in several classes of univalent functions defined by generalized fractional operators and Hadamard products. Here, the results that we investigated generalized well-known previous results. Here, the Hankel matrix also plays an important part in the thesis. The generalized fractional operators are used to introduce new analytic and univalent function classes related to convex and starlike classes. Finally, the upper bounds for the second Hankel determinant are obtained and generalized previous results.