MATHEMATICS

Yu. Yu. Trokhimchuk

On Conformal Mappings

(Presented by Academician M. A. Lavrent’ev on 17 III 1958)

Let a continuous function \(f(z)\) be given in a domain \(D\) of the complex \(z\)-plane. We shall say that the mapping \(w=f(z)\) preserves angles at a point \(z\in D\) if the limit exists

\[ \lim_{h\to 0}\operatorname{Arg}\frac{f(z+h)-f(z)}{h}. \tag{1} \]

Here values of the argument and their limits that differ by \(2\pi\) must be regarded as identical*.

Theorem 1. If a continuous mapping \(w=f(z)\) preserves angles at every point of the domain \(D\), except possibly for at most a countable set of them, then the function \(f(z)\) is analytic inside \(D\).

This theorem was proved by D. E. Men’shov \((^{1})\) under the additional assumption that the mapping \(w=f(z)\) is single-valued.

Let again \(f(z)\) be a continuous function in the domain \(D\). We shall say that the mapping \(w=f(z)\) has constant stretching at a point \(z\in D\) if there exists the (finite or infinite) limit

\[ \rho(z)=\lim_{h\to 0}\left|\frac{f(z+h)-f(z)}{h}\right|. \tag{2} \]

The following theorem holds, generalizing the well-known theorem of Bohr \((^{2})\):

Theorem 2. If a continuous mapping \(w=f(z)\) is single-valued in the domain \(D\) and has constant stretching at each of its points, except possibly for at most a countable set of them, then inside \(D\) either the function \(f(z)\) itself or its conjugate \(\overline{f(z)}\) is analytic.

This theorem can also be extended to the case of arbitrary noncontinuous mappings, if the property of preserving orientation at a point is defined in a suitable way.

Namely, let \(w=f(z)\) be a continuous mapping of the domain \(D\); consider some point \(z_0\in D\) and its image \(w_0=f(z_0)\) in the \(w\)-plane. We shall call the point \(z_0\) a \(U\)-point of the mapping \(w=f(z)\) if there exists a neighborhood \(V(z_0)\) of it such that for every point \(z'\in V(z_0)\), \(z'\ne z_0\), we have \(f(z')\ne f(z_0)\)**. Take an arbitrary closed Jordan curve \(\lambda\subset V(z_0)\) enclosing the \(U\)-point \(z_0\); it is clear that the continuous curve \(l=f(\lambda)\) does not pass through the point \(w_0=f(z_0)\), and when the point \(z\) traverses the curve \(\lambda\), the point \(w=f(z)\) describes the whole curve \(l\). If now, for positive traversal by the point \(z\) of the closed curve \(\lambda\), the expression \(\arg(w-w_0)=\arg[f(z)-f(z_0)]\) receives a nonnegative increment, and this holds for all possible \(\lambda\subset V(z_0)\), then we shall say,

* We assume that the values of \(\operatorname{Arg}\) in expression (1) depend continuously on \(h\); note also that expression (1) has meaning only in the case when \(f(z+h)\ne f(z)\) for sufficiently small \(|h|\).

** Note that if the mapping \(w=f(z)\) preserves angles at a point \(z_0\in D\), then \(z_0\) is a \(U\)-point, which follows from the very meaning of expression (1).

that at the point $z_0 \in D$ the mapping $w=f(z)$ is direct (or preserves orientation).

One can prove the following generalization of Theorem 2:

Theorem 3. If an arbitrary continuous mapping $w=f(z)$ has constant stretching at each point of the domain $D$, with the exception, possibly, of at most a countable set of such points, and at each $U$-point, if such points exist, is direct, then the function $f(z)$ is analytic inside $D$. Moreover, if $U$-points do not exist, then $f(z)$ is constant.

The proof of all these theorems is based on the notion of a set of uniqueness (in the sense of N. N. Luzin) ($^3$).

For the general case of a mapping with constant stretching one can prove the following result:

Theorem 4. If a continuous mapping $w=f(z)$ has constant stretching at each point of the domain $D$, with the exception, possibly, of at most a countable set of such points, and, moreover, $\rho(z)$ ($^2$) can be equal to zero only on a set of points of category I (in $D$), then there exists a set $O$, open and everywhere dense in $D$, in each component of which either the function $f(z)$ itself or its conjugate $\overline{f(z)}$ is analytic.

An example of such a mapping is given by the function already indicated by Bohr ($^2$):

\[ f(z)= \begin{cases} z, & \text{for } \operatorname{Im} z>0,\\ z, & \text{for } \operatorname{Im} z\le 0. \end{cases} \]

Some generalizations of the results of D. E. Menshov and H. Bohr were formulated by Kuramochi ($^4$) for the case of univalent mappings $w=f(z)$ under a certain additional restriction. In the hypotheses of the theorems of the present note only the continuity of the mapping $w=f(z)$ is assumed.

CITED LITERATURE

$^1$ D. Menshov, Math. Ann., 95, 641 (1926).
$^2$ H. Bohr, Math. Zs., 1 (1918).
$^3$ Yu. Yu. Trokhimchuk, Uspekhi Mat. Nauk, vol. 5, 215 (1956).
$^4$ Z. Kuramochi, Osaka Math. J., 3, No. 1, 21 (1951).