THE ONE-QUARTER THEOREM FOR MEAN UNIVALENT FUNCTIONS
Abstract
The class of mapping functions of Spencer (Ann. Math. 42:614-63 2, 1941) (i) w - f(z) = z sub p + a sub p+1 z p+1 + sub a sub p+2 sub z p+2 ..., 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 = f(z) 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) 1/p 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 = 1/4, where d is any value which f(z) does not assume in the unit circle. The first part of the proof of (ii) is based on the work of Hayman (J. d'Analyse Mathematique 1:155-179, 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. 230:25-27, 1950). The later part of the proof depends on the polygonal Hadamard variations of Garabedian and Royden (Proc. Nat. Acad. Sci. 38:57-61, 1952) and closes with an inequality from the theory of free streamline flows.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 26, 1953
- Accession Number
- AD0003388
Entities
People
- H. L. Royden
- P. R. Garabedian
Organizations
- Stanford University