Estimation of Radii of Regions of Starlikeness and Spirallikeness for Analytic Mappings

Abstract

Many researchers in complex analysis have invested time particularly in investigating geometric properties like starlikeness and spirallikeness of analytic mappings on the unit disk. Finding the exact lengths of the radii in the starlike and or spirallike regions for these functions is very difficult since these shapes are not regular. This therefore requires that an estimation of the radii of these regions be carried out. This research seeks to estimate the radii within which analytic mappings in the unit disk remain starlike or spirallike. This note derives sharp radii estimates and constructs extremal functions achieving these estimates. The methodology involved using techniques of differential subordination, coefficient estimates, distortion theorems, and subordination principles. Moreover, algorithm development techniques were used as well as numerical methods to come up with pictorial representation of starlikeness and spirallikeness regions. Advancement of theoretical development of geometric function theory and also in providing sharp radii bounds useful in modeling involving conformal maps which enhances applications in engineering, fluid dynamics, and signal theory requires the results generated in this work.