The Inner Mapping Radius of Harmonic Mappings of the Unit Disk
نویسنده
چکیده
The class SH consists of univalent, harmonic, and sense-preserving functions f in the unit disk, ∆, such that f = h + g where h(z) = z + ∑∞ 2 akz , g(z) = ∑∞ 1 bkz . Using a technique from Clunie and Sheil-Small, we construct a family of 1-slit mappings in SH by varying ω(z) = g ′(z)/f ′(z). As ω(z) changes, the tip of the slit slides along the negative real axis from the point 0 to −1. In doing so, we establish that the inner mapping radius, ρ(f) can be as large as 4. In addition, we show that the inner mapping radius for functions in S H can be as small as 1/2 and as large as 2.
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SOME HARMONIC n - SLIT MAPPINGS
The class SH consists of univalent, harmonic, and sense-preserving functions f in the unit disk, , such that f = h+g where h(z) = z+ P 1 2 akz , g(z) = P 1 1 bkz . S H will denote the subclass with b1 = 0. We present a collection of n-slit mappings (n 2) and prove that the 2-slit mappings are in SH while for n 3 the mappings are in S H . Finally we show that these mappings establish the sharpne...
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