Value range of solutions to the chordal Loewner equation

Abstract

We describe a value range {f (z 0 )} over the class of holomorphic univalent mappings from H \ K(T ) onto the upper half-plane H with the hydrodynamic normalization at infinity where K(T ) is an arbitrary hull of half-plane capacity T > 0. Without loss of generality choose z 0 = i. A similar value range over the class of inverse functions f −1 is also described. This is a specification of the result by Roth and Schleissinger.

Value ranges of univalent self-mappings of the unit disc

“…Our results are analogous to the results of Prokhorov and Samsonova [PS15], who study univalent self-mappings of the upper half-plane having the so called hydrodynamical normalization at the boundary point ∞. Finally we note that in [GG76], the authors consider the set {log(f (z 0 )/z 0 ) : f : D → C univalent, f (0) = 0, |f (z)| ≤ M } for M > 0.…”

Value Regions in Classes of Conformal Mappings

“…Remark that Zherdev [17] developed the results and methods in [16] and described the value region for solutions to the chordal Loewner ODE (2) with T 1 4 under the restriction |λ(t)| c, c 0, for the driving function λ in (2). Besides, he managed to write down explicitly the parametric representation of the boundary of the domain D(T ) in the W -plane, W = X + iY , as follows:…”

Value regions of univalent self-maps with two boundary fixed points

“…Recently, the sharp value regions of f → f (z 0 ) have been determined for other classes of univalent self-maps [22,33,35]. The main instrument is the classical parametric representation of univalent functions, going back to the seminal work by Loewner [27].…”