Mapping the disk to convex subregions

John A. Velling ^{∗}

11 March 1995

§1. Introduction.

Univalent maps f : D → D have been much studied by many authors. See [1], [2], [5], for just a few references. A quick scan of the content of these references shows that such mappings occur in a number of different settings for a variety of reasons. We will consider here whether or not such mappings exist satisfying certain rather restrictive boundary conditions.

To state our problem, and its partial solution, precisely, it will be helpful to summarize several known results. These may be found in [3] and [4].

Theorem A: Given two collections of distinct points, z_{1}, …, z_{n} ∈ S^{1}, and w_{1}, …, w_{n} ∈ S^{1}, there exists a locally univalent f : D → D, continuous up to S^{1}, such that f(z_{i}) = w_{i} and if z ∈ S^{1} \ {z_{1}, … z_{n}} then |f(z)| < 1.

Similar in flavor is

Theorem B: If z_{1}, …, z_{n} and w_{1}, …, w_{n} from theorem A are ordered cyclically then there exists a univalent f satisfying the conclusion of theorem A.

The modulus of the rectangle with vertices at ±a±i in the complex plane is a > 0. Any convex proper subdomain of C with four distinct boundary points z_{1}, …, z_{4} marked cyclically is conformally equivalent to a unique such domain by a conformal map satisfying z_{1} ↦ a − i, z_{2} ↦ a + i, z_{3} ↦ −a + i, z_{4} ↦ −a − i. Thus we refer to a as the modulus of such a marked domain.

∗Partially supported by NSF Grant #4401728.

Given four points z_{1}, …, z_{4} ∈ S^{1} ordered cyclically, we let Ω(z_{1}, …, z_{4}) denote a convex domain contained in D with z_{1}, …, z_{4} in its boundary. Let mod(Ω(z_{1}, …, z_{4})) denote its modulus as a quadrilateral with vertices z_{1}, …, z_{4}. If in this paper the z_{j} are understood, Ω and mod(Ω) will be used. We have

Theorem 1: For fixed z_{1}, …, z_{4} cyclically ordered in S^{1}, inf_{Ω} mod(Ω) is realized uniquely by the convex domain having linear segments from z_{1} to z_{2} and from z_{3} to z_{4}, and circular arcs in S^{1} from z_{2} to z_{3} and from z_{4} to z_{1}. Similarly sup_{Ω} mod(Ω) is realized uniquely by the convex domain having circular arcs in S^{1} from z_{1} to z_{2} and from z_{3} to z_{4}, and linear segments from z_{2} to z_{3} and from z_{4} to z_{1}.

If z_{1}, …, z_{4} ∈ S^{1} are ordered cyclically, let Q(z_{1}, …, z_{4}) denote the convex domain which is the circle with the four points marked. We will show

Theorem 2: Let z_{j}, w_{k} be cyclically ordered 4-tuples in S^{1}. There exists univalent f : D → D, continuous to S^{1}, with f(z_{i}) = w_{i}, |f(z)| < 1 for z ∈ S^{1} \ {z_{1}, …, z_{4}}, and f(D) convex, if and only if the inequalities

inf_{Ω(w_{1},...,w_{4})} mod(Ω(w_{1}, …, w_{4})) < mod(Q(z_{1}, …, z_{4}))

and

mod(Q(z_{1}, …, z_{4})) < sup_{Ω(w_{1},...,w_{4})} mod(Ω(w_{1}, …, w_{4}))

hold, where Q(z_{1}, …, z_{4}) is D marked at the four points z_{1}, …, z_{4}.

We give the proofs of these two theorems in §2. It is easy to see that similar results hold when we consider convexity with respect to the Poincaré hyperbolic metric on D.

In greater generality, suppose we ask whether a result like that of theorem 2 above holds when we are given two cyclically ordered n-tuples of points in S^{1}. To be precise, given z_{1}, …, z_{n} and w_{1}, …, w_{n} cyclically ordered n-tuples of points in S^{1}, does there exist a univalent map f : D → D extending continuously to S^{1} such that f(z_{k}) = w_{k}, f(D) is convex, and |f(z)| < 1 if z ∈ S^{1} \ {z_{1}, …, z_{n}}? By comparing moduli of a finite number of rectangles, we obtain necessary conditions for the existence of such a function. This is explained in §3. Again, similar conditions hold when considering the Poincaré hyperbolic metric on D. Whether or not these conditions are sufficient is not clear to this author.

§2. Proofs of theorems 1 and 2.

Let Ω(z_{1}, …, z_{4}) be a convex quadrilateral, and label the four boundary components z_{1} to z_{2} by R, z_{2} to z_{3} by T, z_{3} to z_{4} by L, and z_{4} to z_{1} by B. Theorems 1 and 2 depend on the following standard lemma. (See the version of Löwner’s theorem in [6], for example.) Lemma: If Ω_{1} ⊂ Ω_{2} are two convex quadrilaterals sharing opposite sides T and B (L and R, respectively), then

mod(Ω_{1}) ≤ mod(Ω_{2})
(mod(Ω_{1}) ≥ mod(Ω_{2})).

Equality holds if and only if Ω_{2} = Ω_{1}.

To see theorem 1, if z_{1}, …, z_{4} are cyclically ordered, let Ω_{1} and Ω_{2} be the convex regions described in theorem 1 as yielding the minimum and maximum moduli. If Ω ⊂ D is a convex quadrilateral with corners at z_{1}, …, z_{4} then Ω_{1} ∩ Ω and Ω_{2} ∩ Ω are both convex regions sharing opposite sides with Ω. By the above lemma we have

mod(Ω_{1}) ≤ mod(Ω_{1} ∩ Ω) ≤ mod(Ω) ≤ mod(Ω_{2} ∩ Ω) ≤ mod(Ω_{2}),

and theorem 1 follows.

The necessity of the condition in theorem 2 now follows, as conformal univalent maps preserve modulus. The sufficiency also follows readily, as one may consider regions bounded by circular arcs of decreasing curvature through w_{1}, w_{2}, etc.

We see that if the condition of theorem 2 holds then Q(z_{1}, …, z_{4}) maps to an uncountable number of convex Ω(w_{1}, …, w_{4}).

§3. A necessary condition for the general problem.

The general problem of mapping the disk to a convex region with prescribed boundary points in S^{1} seems to some extent accessable via consideration of moduli of rectangles. If we have an n-tuple z_{1}, …, z_{n} of points in S^{1} ordered cyclically we may consider convex subdomains of D which contain all of the z_{k} in their boundary. For any choice of four of the z_{k} ordered cyclically, we can identify among the convex domains here considered, as in theorem 1, the unique such domains for which the modulus is maximized and minimized. In the following extension of theorem 1, Ω will denote such a convex domain with four specified boundary points.

Theorem 1′: For fixed z_{1}, …, z_{n} cyclically ordered in S^{1}, let z_{k_{1}}, …, z_{k_{4}} denote four of the z_{k} ordered cyclically. In the collection of convex subdomains of S^{1} having all z_{k} in the boundary and marked by the choice of z_{k_{i}} as vertices, inf_{Ω} mod(Ω) is realized uniquely by the convex domain having linear segments from z_{k_{1}} to z_{k_{1}+1}, z_{k_{1}+1} to z_{k_{1}+2}, …, z_{k_{2}−1} to z_{k_{2}}, and from z_{k_{3}} to z_{k_{3}+1}, z_{k_{3}+1} to z_{k_{3}+2}, …, z_{k_{4}−1} to z_{k_{4}}, and circular arcs in S^{1} from z_{k_{2}} to z_{k_{3}} and from z_{k_{4}} to z_{k_{1}}. Similarly sup_{Ω} mod(Ω) is realized uniquely by the convex domain having circular arcs in S^{1} from z_{k_{1}} to z_{k_{2}} and from z_{k_{3}} to z_{k_{4}}, and linear segments from z_{k_{2}} to z_{k_{2}+1}, z_{k_{2}+1} to z_{k_{2}+2}, …, z_{k_{3}−1} to z_{k_{3}}, and from z_{k_{4}} to z_{k_{4}+1}, z_{k_{4}+1} to z_{k_{4}+2}, …, z_{k_{1}−1} to z_{k_{1}}.

The proof of this is the same as that of theorem 1.

Again, since conformal maps preserve modulus, the following necessary conditions on moduli are immediate.

Theorem 2′: Let z_{j}, w_{k} be cyclically ordered n-tuples in S^{1}. For there to exist a univalent f : D → D, continuous to S^{1}, with f(z_{i}) = w_{i}, |f(z)| < 1 for z ∈ S^{1} \ {z_{1}, …, z_{n}}, and f(D) convex, we must have, for every choice of four cyclically ordered points z_{k_{1}}, …, z_{k_{4}}, the satisfaction of the inequalities

inf_{Ω(w_{k_{1}},...,w_{k_{4}})} mod(Ω(w_{k_{1}}, …, w_{k_{4}})) < mod(Q(z_{k_{1}}, …, z_{k_{4}}))

and

mod(Q(z_{k_{1}}, …, z_{k_{4}})) < sup_{Ω(w_{k_{1}},...,w_{k_{4}})} mod(Ω(w_{k_{1}}, …, w_{k_{4}}))

where Q(z_{k_{1}}, …, z_{k_{4}}) is D marked at the z_{k_{i}}.

As mentioned in §1, whether or not these conditions suffice is not known to this author.

References

[1] Ahlfors, L., Conformal Invariants, McGraw–Hill, New York (1973).

[2] Caratheodory, C., Theory of Functions, vol.II, Chelsea, New York (1954).

[3] Clunie, J.; Hallenbeck, D. J. and MacGregor, T. H., A peaking and interpolation problem for univalent functions, J. Math. Anal. Appl. 111 (1985) 559–570.

[4] MacGregor, T. H. and Tepper, D. E., Finite interpolation by univalent functions, J. Approx. Theory 52 (1988) 315–321.

[5] Shapiro, J., Composition Operators, Springer–Verlag, New York (1993).

[6] Velling, J., The uniformization of rectangles: an exercise in Schwarz’s lemma, Amer. Math. Monthly 99 (1992) 112–115.