Rogosinski’s lemma for univalent functions,
[3mm] hyperbolic Archimedean spirals and the Loewner equation
Oliver Roth and Sebastian Schleißinger
July 12, 2019
[3mm]
Abstract. We describe the region of values of for all normalized bounded univalent functions in the unit disk at a fixed point . The proof is based on the radial Loewner differential equation. We also prove an analogous result for the upper halfplane using the chordal Loewner equation.
1 Results: the unit disk
We denote by the open unit disk in the complex plane and by the set of all holomorphic functions normalized by and . The Schwarz lemma tells us that for any and any . In 1934 Rogosinski [12] (also [4, p. 200]) proved a far reaching sharpening of the Schwarz lemma by giving an explicit description of the region of values of , , at a fixed point . In this note we prove an analogue of Rogosinski’s result for univalent functions in by providing an explicit description of the regions of values
It turns out that the set admits a fairly appealing description in terms of hyperbolic geometry. In order to state the main results, we therefore endow the unit disk with the standard hyperbolic metric
of constant curvature , see [2]. The induced hyperbolic distance between is then given by
(1.1) 
Theorem 1.1
Let . Then
In particular, the boundary of the region of values is composed of parts of the two curves
In complete analogy to the standard euclidean Archimedean spirals
which pass through both the origin and the point , we call the curves and the (standard) hyperbolic Archimedean spirals through the origin and the point .
Now fix and move towards the origin while staying on . Stop either when you reach the point of hyperbolic distance from the origin or when you reach the origin. In the first case, we see from the definition of that , so Let in the second case. In both cases we define
Corollary 1.2
Let . Then the region of values has the following properties.

and is a Jordan domain bounded by the Jordan curve .

The origin is an isolated boundary point of if and only if .

is convex if and only if .^{1}^{1}1The number is also the radius of starlikeness in the class found by Grunsky 1934.

Each boundary curve has hyperbolic length
Theorem 1.1 as well as Corollary 1.2 will be proved in Section 3 by making use of the Loewner differential equation
(1.2) 
with fixed , and its reachable set
Note that by standard results of Loewner’s theory it is clear that . It will turn out that actually the stronger statement holds, so in order to prove Theorem 1.1 it suffices to describe the reachable set . By definition, is made up of the trajectories of the Loewner equation (1.2). Any trajectory of the Loewner equation such that is a boundary point of is therefore of special interest and is called an optimal trajectory. The corresponding “control function” is then called optimal.
Remark 1.3 (The principle of optimality)
The well–known principle of optimality asserts that if a control function ist optimal, then it is also optimal on for any . This means that a trajectory of the Loewner equation which delivers at time a boundary point of the reachable set , then for every . Hence every initial piece of an optimal trajectory is again optimal. This is a general principle which holds for any control system under very general assumptions. It holds in particular for the Loewner equation and has been utilized in this context before e.g. by Friedland and Schiffer [5], Kirwan and Pell [10] and more recently by Graham, Hamada, Kohr and Kohr [6] for the Loewner equation in several complex variables.
The next result shows that for any the Loewner equation (1.2) has exactly two optimal trajectories. These optimal trajectories parametrize the entire boundary of the reachable set . They form exactly the arcs of the hyperbolic Archimedean spirals .
Theorem 1.4
Let . Then there exist two trajectories such that is the connected part of between and .

If , then is optimal for every and .

If , then is optimal for every , but not optimal for any . Here
Furthermore, .
Remark 1.5
Theorem 1.1 has a very well–known classical counterpart for the class
established by Grunsky 1932, see [8], who proved the remarkable fact that the set
is exactly the disk
Grunsky’s result was extended e.g. by Gorjainov and Gutljanski [7], who obtained a precise description of the sets
for any . The results in [8, 7] are much more difficult to prove than the results of the present paper since the sets and have in fact a much more complicated structure than the set . In principle, the set can certainly be described using the results in [7] about the sets . The purpose of the present paper is to give a simple and direct proof of the simple structure of the set emphasizing some of its remarkable hyperbolic geometric properties without making appeal to the deeper results about the sets and due to Grunsky [8] and Gorjainov and Gutljanski [7].
2 Results: The upper halfplane
We now replace the interior normalization and , which was used throughout section 1, by an appropriate boundary condition. For this purpose, it is convenient to use the upper halfplane with its distinguished boundary point instead of the unit disk . We consider the set of all holomorphic functions such that the so–called hydrodynamic normalization,
is satisfied.
Remark 2.6
The hydrodynamic normalization for functions in corresponds to the interior normalization for functions in in the following way. First note that the interior condition and for every enforces that the only conformal automorphism of contained in is the identity. Now, every function has an angular derivative at ,
and hence the angular limit
Note that the two boundary conditions and alone allow for infinitely many conformal automorphisms , , of , but the sharper hydrodynamic condition for guarantees that the only conformal automorphism of contained in is the identity.
The following result is the analogue of Theorem 1.1 for the upper halfplane.
Theorem 2.7
Let . Then
Remark 2.8
Using the well–known Nevanlinna representation for holomorphic functions in with positive imaginary part (see [13, Theorem 5.3]), it is immediate that
Hence, Theorem 2.7 tells us that the set of values for all univalent functions is the same as the set of values for all . This is a significant difference to the unit disk case, where a comparison of Theorem 1.1 with Rogosinski’s Lemma shows that the set of values for all univalent functions is strictly smaller than the set of values for all functions .
As in the unit disk case, the proof of Theorem 2.7 relies on the Loewner differential equation, but now we have to use the chordal version
(2.1) 
with fixed , and its reachable set
It is well–known (see [11, Chapter 4] and [3, 9]) that every solution of the chordal Loewner equation
generated by a continuous driving function is a univalent function on such that
In particular, for every . Hence Remark 2.8 implies that Theorem 2.7 is an immediate consequence of the following result, which describes the reachable set for any .
Theorem 2.9
Let . Then .
3 Proofs: the unit disk
The proofs are based on an extremely simple differential inequality for the “hyperbolic polar coordinates” of the solutions of Loewner’s equation, which follows immediately from the particular form of the Loewner equation.
3.1 The basic differential inequality
3.2 Integrating the basic differential inequalities and proof of Theorem 1.1
Integrating (3.5) yields
(3.6) 
with equality if
(Case I)  
or if  
(Case II) 
This shows that
(3.7) 
We now show that this inclusion is sharp. We first consider Case I. Then equality holds in (3.6) in this case, i.e.,
and equality holds in (3.4) with for every . This means that
(3.8) 
In a similar way, (3.2) reduces to
and another integration gives
Using (3.8), we see that for every . We can now define
This gives a (real–analytic) control function . The corresponding solution of the Loewner equation (1.2) satisfies
(3.9) 
by construction. Case II can be handled in a similar way. In fact, if we set
where
then the corresponding solution of the Loewner equation (1.2) satisfies
(3.10) 
This shows that the inclusion (3.7) is sharp. In order to conclude
(3.11) 
we just note that is a compact subset of the complex plane, which is starlike with respect to the boundary point (i.e., if , then for any ) and that by Loewner’s theory. This proves (3.11) and . The proof of Theorem 1.1 is complete.
3.3 Proof of Theorem 1.4
We note that in view of (3.9) and (3.10) the two trajectories form the connected parts of the hyperbolic Archimedean spirals between and . These two trajectories never meet if and only if and for every , which happens if and only if . In this case, we thus have . If , then the two trajectories first meet at time , where , that is,
i.e.,
Clearly, is not optimal on for any and . This proves Theorem 1.4.
3.4 Proof of Corollary 1.2
(a) is clear from the above discussion. (b) The origin is an isolated boundary point of if and only if , that is, . (c) is convex if and only if and for every . This occurs if and only if , that is, .
(d) In hyperbolic polar coordinates the curve has the parametrization , where and
Now, a computation shows
so
4 Proofs: the upper halfplane
Fix and let . In order to prove Theorem 2.9, we first note that for every
(4.1) 
This follows immediately from the Poisson representation
where is a nonnegative Borel measure on such that
(see [13]) and
i.e., for . Clearly, (4.1) shows that .
Now let . We need to find a driving function such that the solution of (2.1) passes through . We separate into real and imaginary parts and write and . Now, we claim that can be chosen such that connects and by a straight line segment, i.e.
where
In order to prove this, we separate equation (2.1) into real and imaginary parts and obtain
with initial conditions and . We now assume that and are related by
Then we get the following initial value problem:
which can be solved directly:
Hence if we now define
then by construction the solution of (2.1) satisfies In particular, the trajectory is the halfline starting at through the point , so . This completes the proof of Theorem 2.9.
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 [11] G. Lawler, Conformally invariant Processes in the Plane, Amer. Math. Soc. 2005.
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Oliver Roth
Sebastian Schleißinger
Department of Mathematics
University of Würzburg
97074 Würzburg
Germany
roth@mathematik.uniwuerzburg.de
sebastian.schleissinger@mathematik.uniwuerzburg.de