Properties of close-to-convex functions and special functions / Jonathan Aaron Azlan Mosiun - Jonathan Aaron Azlan, Mosiun (2019)
Let S be the class of functions of the form f(z) = z + 1P n=2 anzn that are univalent and analytic in the unit disc U = fz 2 C : jzj < 1g. Study on functions derived via geometric properties such as S_, C and K, which are subclasses of S, has been ongoing for many decades and has been done extensively and exhaustively. Among the many subclasses of K, Sakaguchi introduced the class of starlike functions with respect to symmetric point, denoted by S_ s . Since its introduction in 1959, many authors have introduced generalizations of S_ s or classes resembling it. Inspired by this, Gao & Zhou introduced another subclass of K which was denoted as Ks which was further generalized by Wang, Gao, & Yuan. Following their inspirations, this dissertation introduces a subclass of close-to-convex functions, denoted by Kk;N s , where k;N 2 N, that combines the concepts of S_ s and Ks and investigates them for their properties which include, but not limited to, coefficient estimates, distortion and growth theorems, and radius of convexity. Moreover, we also introduce the class of p-valent functions, denoted by Kk;N s;p , in this dissertation which further generalizes the class Kk;N s and investigate it for its properties. In addition to investigating properties of geometric functions, many other mathematicians have also expressed interest in finding sufficient conditions such that certain special functions has certain geometric properties, such as univalency, starlikeness or convexity. Examples of special functions that have undergone this investigation include Bessel and Struve functions. Motivated by this, this dissertation also investigates sufficient conditions for the function Tp;b;c(z) = (f _ gp;b;c)(z), a convolution between f(z) = z + 1P n=2 anzn and gp;b;c(z) = z + 1P n=2 (