Some Linear Subordination Results for Classes of Univalent Functions

Abstract

In this note we determine necessary and sufficient conditions on complex numbers A and p. such that Az/(1-pa z) is subordinate 2 ... to f(z)-z + a z + ... for all functions / in certain classes of univalent functions. 1. Introduction. Let S denote the class of functions fiz)-z + a.z +... analytic and univalent in the unit disk F = \z : \z\ < 1 \. If / is in S and g is analytic in F with g(0) = 0 and g(F) is a subset of /(F), then g is said to be subordinate to /, and / is called a univalent majorant oí g. We express this by writing giz)-< f(z). Let K and S denote the convex and starlike subclasses, respectively, of S. The well-known result that for each / in K, f(E) contains the disk {z : \z\ < Y2\ is equivalent to the statement z/2 ■< f(z). Similarly, the Koebe "]4 theorem" implies z/A < f(z) for all / in S. T. Basgoze, J. L. Frank, and F. R. Keogh [1] examined necessary and sufficient conditions on complex numbers À and p such that z/2 < Xz + ¡ia z-< f(z) for f(z) = z + a z +...