On Circularly Symmetric Functions
نویسنده
چکیده
1. In addition to the study devoted to the general class of normalized regular univalent functions in the unit circle considerable attention has been given to certain subclasses of these functions such as starlike functions, convex functions, and others. In view of the important role played by symmetrization methods in recent work it seems worth while to study a further subclass—the class of circularly symmetric functions. These functions map the unit circle on a domain of the type obtained by application of the process of circular symmetrization. In the present paper these functions are shown to admit a characterization of the same type as the familiar one for starlike functions. From this it follows that a circularly symmetric function maps the concentric subcircles of the unit circle onto circularly symmetric domains. Further these functions are found to be related to the class of typically real functions introduced by Rogosinski [4] in essentially the same way as starlike functions are related to the class of normalized regular functions with positive real part. From this fact can be drawn some interesting consequences concerning coefficients and function values associated with the class of circularly symmetric functions.
منابع مشابه
On circularly symmetric functions
Let D ⊂ C and 0 ∈ D. A set D is circularly symmetric if for each % ∈ R a set D ∩ {ζ ∈ C : |ζ| = %} is one of three forms: an empty set, a whole circle, a curve symmetric with respect to the real axis containing %. A function f ∈ A is circularly symmetric if f(∆) is a circularly symmetric set. The class of all such functions we denote by X. The above definitions were given by Jenkins in [2]. In ...
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