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Register a new userQuasiconformal Extensions of Harmonic Univalent Mappings of the Interior and Exterior Unit Disk
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Authors
Xiu-Shuang Ma
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In this note, we consider the sufficient coefficient condition for some harmonic mappings in the unit disk which can be extended to the whole complex plane. As an application of this result, we will prove that a harmonic strongly starlike mapping has a quasiconformal extension to the whole plane and will give an explicit form of its extension function. We also investigate the quasiconformal extension of harmonic mappings in the exterior unit disk.
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The conformal mapping $f(z)=(z+1)^2 $ from $mathbb{D}$ onto the standard cardioid has a homeomorphic extension of finite distortion to entire $mathbb{R}^2 .$ We study the optimal regularity of such extensions, in terms of the integrability degree of the distortion and of the derivatives, and these for the inverse. We generalize all outcomes to the case of conformal mappings from $mathbb{D}$ onto cardioid-type domains.
In the present paper, we obtain a more general conditions for univalence of analytic functions in the open unit disk U. Also, we obtain a refinement to a quasiconformal extension criterion of the main result.
Let $f$ be a univalent self-map of the unit disc. We introduce a technique, that we call {sl semigroup-fication}, which allows to construct a continuous semigroup $(phi_t)$ of holomorphic self-maps of the unit disc whose time one map $phi_1$ is, in a sense, very close to $f$. The semigrup-fication of $f$ is of the same type as $f$ (elliptic, hyperbolic, parabolic of positive step or parabolic of zero step) and there is a one-to-one correspondence between the set of boundary regular fixed points of $f$ with a given multiplier and the corresponding set for $phi_1$. Moreover, in case $f$ (and hence $phi_1$) has no interior fixed points, the slope of the orbits converging to the Denjoy-Wolff point is the same. The construction is based on holomorphic models, localization techniques and Gromov hyperbolicity. As an application of this construction, we prove that in the non-elliptic case, the orbits of $f$ converge non-tangentially to the Denjoy-Wolff point if and only if the Koenigs domain of $f$ is almost symmetric with respect to vertical lines.
By using the method of Loewner chains, we establish some sufficient conditions for the analyticity and univalency of functions defined by an integral operator. Also, we refine the result to a quasiconformal extension criterion with the help of Beckerss method.
In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.
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