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).