Publication: Functions Starlike With Respect To A Boundary Point
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Date
1996-11
Authors
Abdullah, Siti Aishah Sheikh
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Abstract
Let u = i z : izi < l} be the unit disk and s be the class of analytic univalent functions f normalized such that i(o) = f'(d) -l = o. Let s* denote the subclass of functions f in s which satisfy the condition rei zf'(z) / fez) } > o, z e u. Functions f e s* map u univalently onto domains starlike with respect to the origin. The class s* has been extensively studied over the last fi fly years. However not much seems to be known about the class of analytic functions that map u onto domains starlike with respect to a boundary point. M.S.Robei1son [23] was the first to initiate a systematic study of this class. 00 let g denote the class of functions g(z) = l + i, d n z n analytic and non-vanishing in u n=1 and satisfying r {2zg'(z) 1+ e +--z} > o ,ze u. G(z) l-z robertson [231 had shown that nonconstant functions in the class g map u univalently onto domains starlike with respect to the boundary point. By a rotation we may assume that this point is g( i) = o. A close relation between the class g and the class s* was given such that g e g if and only if g2(z) = (l-z)2 fez) / z , f e s*. Further, g is either close-to-convex with respect to f or g is the constant function 1. Leverenz theorem r 17] is introduced in chapter 2. An equivalent finite form of the theorem is then obtained. Examples are given to illustrate its application. In chapter 3, the important role of the koebe function k(z) = z 1(1- z)2 and its rotations as related to the class g is examine.