THE ONE-QUARTER THEOREM FOR MEAN UNIVALENT FUNCTIONS
STANFORD UNIV CA APPLIED MATHEMATICS AND STATISTICS LABS
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The class of mapping functions of Spencer Ann. Math. 42614-63 2, 1941 i w - fz z sub p a sub p1 z p1 sub a sub p2 sub z p2 ..., regular in the unit circle z 1, is considered which transform the unit circle into a Riemann surface R over the w-plane so that, for each r 0, the area of the sheets of R covering the circle w r does not exceed p pi r squared. These functions are called mean p-valent, and mean univalent when p 1. The analytic functions w fz are considered of the form i in the unit circle, with p 1, which map the unit circle onto a Riemann surface R over the w-plane satisfying the condition integral from r to 0 integral alpha phi - 2 pi 1p dp or equal to 0 for each r , where the integration with respect to phi is extended over all sheets of R covering the circle w pho. For this class of weakly mean univalent functions, any omitted value d is shown to satisfy the sharp inequality ii d or 14, where d is any value which fz does not assume in the unit circle. The first part of the proof of ii is based on the work of Hayman J. dAnalyse Mathematique 1155-179, 1951 who gave elegant sharp estimates for the distortion of p-valent mappings by using the concept of circular symmetrization due to Polya Compt. rend. 23025-27, 1950. The later part of the proof depends on the polygonal Hadamard variations of Garabedian and Royden Proc. Nat. Acad. Sci. 3857-61, 1952 and closes with an inequality from the theory of free streamline flows.
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