Subclasses of uniformly starlike functions

Abstract

ABSTRACT. We study subclasses of the class of uniformly starlike functions which were recently introduced by A.W. Goodman. One new subclass is defined and it is shown that it shares many properties of the parent class.

Uniformly Convex and Uniformly Starlike Functions

“…Ruscheweyh's monograph [57] gives a comprehensive survey on convolutions. To make use of this theory in the investigation of the class UST , Merkes and Salamasi [32] proved the following result.…”

“…Thus, the radius of a property P in the set F is the largest number R such that every function in the set F has the property P in each disk D r = {z ∈ D : |z| < r} for every r < R. For example, a starlike function need not be convex; however, every starlike function maps the disk |z| < 2 − √ 3 onto a convex domain and hence the radius of convexity of the class S * of starlike functions is 2− √ 3. Merkes and Salamasi [32] (using Theorem 2.3) and Rønning [53] (using Theorem 2.4) independently showed that the UST -radius of the class C of convex functions is 1/ √ 2. Merkes and Salamasi [32, Theorem 5, p. 451] also obtained a lower bound for the UST -radius for the class of pre-starlike functions.…”

Growth Theorems for a Subclass of Strongly Spirallike Functions

“…Honda [9] obtained the growth theorems for normalized biholomorphic -symmetric convex mappings on the unit ball in complex Banach spaces. In recent years, there are a lot of new results about the growth and covering theorems for the subclasses of biholomorphic mappings in several complex variables [10][11][12].…”