Abstract
In 1958, E. Heinz obtained a lower bound for \(|\partial _x F|^2+|\partial _y F|^2\), where \(F\) is a one-to-one harmonic mapping of the unit disk onto itself keeping the origin fixed. We show various variants of Heinz’s inequality in the case where \(F\) is the Poisson integral of a function of bounded variation in the unit circle. In particular, we obtain such inequalities for \(F\) when it is a locally injective quasiregular mapping or an injective mapping of the unit disk onto a bounded convex domain in the complex plane.
Highlights
Let D(a, r ) := {z ∈ C : |z − a| < r }, D(a, r ) := {z ∈ C : |z − a| ≤ r } andT(a, r ) := {z ∈ C : |z − a| = r } for a ∈ C and r > 0
The Poisson integral P[ f ] is the unique solution to the Dirichlet problem for the unit disk D provided the boundary function f is continuous; cf. e.g. [6, Thm. 2.11]. This means that P[ f ] is a harmonic mapping in D which has a continuous extension to the closed disk cl(D) and its boundary limiting valued function is identical with f
We focus our attention to Poisson integrals mapping the unit disk onto bounded convex domains
Summary
Theorem 1.1 ([13, Thm. 2.1]) Given an injective harmonic mapping F of D onto a bounded convex domain including 0, assume that F(0) = 0, |∂ F(0)|−|∂ ̄ F(0)| > 0 and that F has a continuous extension to D(0, 1). This means that P[ f ] is a harmonic mapping in D which has a continuous extension to the closed disk cl(D) and its boundary limiting valued function is identical with f. Theorem 1.3 Given a Dini smooth function f : T → C assume that F := P[ f ] is an injective mapping of D onto a convex domain including 0, F(0) = 0 and |∂ F(0)| − |∂ ̄ F(0)| > 0. We present a few applications of the earlier results
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