Rogosinski's lemma for univalent functions, hyperbolic Archimedean spirals and the Loewner equation

Abstract

We describe the region V(z0) of values of f(z0) for all normalized bounded univalent functions f in the unit disk double-struckD at a fixed point z0∈D. The proof is based on the radial Loewner differential equation. We also prove an analogous result for the upper half‐plane using the chordal Loewner equation.

Value range of solutions to the chordal Loewner equation

“…Corollary 1 corresponds to the result due to Roth and Schleissinger [11]. It follows from the optimality principles that every boundary point w ∈ ∂D(T ) \{i}, Im w > 0, is delivered by a unique function f (z) ∈ H(T ), where f is a regular solution to the Loewner differential equation (1).…”

“…Grunsky's result was extended by Goryainov and Gutlyanski [2] who gave a description of the same set for the subclass of bounded functions f ∈ S. Remind also a far-reaching sharpening of the Schwarz lemma due to Rogosinski [10] where the value range {f (z 0 )}, z 0 ∈ D, is precisely described for the class of all holomorphic functions f (z) in D, f (0) = 0, f (0) ≥ 0 and |f (z)| < 1 for |z| < 1. An analogue of Rogosinski's result for univalent functions was obtained by Roth and Schleissinger [11] in terms of hyperbolic geometry. They proved a similar result for the class of univalent holomorphic functions g : H → H with the hydrodynamic normalization.…”

“…It is reduced to the boundary value problem for Eqs. (11) and (13) since they do not contain the phase variable u.…”

“…The first integral relation (14) is equivalent to changing variables t = t(v) in (13) according to (11) and going to…”

“…It is possible to interpret this statement for the class ∪ T >0 H(T ). The proofs in [11] are based on the radial and chordal Loewner differential equations, see the original radial Loewner version [7].…”