MATHEMATICS

V. I. GAVRILOV

THE SET OF LIMIT VALUES OF PSEUDOANALYTIC FUNCTIONS IN THE UNIT DISK*

(Presented by Academician I. G. Petrovskii on 28 VI 1962)

Let \(w=f(z)\) be a pseudoanalytic function defined in the disk \(D:\ |z|<1\) (a many-sheeted quasiconformal mapping of the disk \(D\)). Let \(A\) be some arc of the circle \(\Gamma:\ |z|=1\), and let \(E\) be a closed set on \(A\). To each point \(e^{i\theta}\in A\setminus E=\mathscr C E\) we assign a certain (arbitrary) Jordan curve \(\Lambda_\theta\), lying in \(D\) and ending at \(e^{i\theta}\), and denote by \(C_{\Lambda_\theta}(f,e^{i\theta})\) the set of limit values of the function \(f(z)\) at the point \(e^{i\theta}\) along the curve \(\Lambda_\theta\). As usual, \(C_D(f,e^{i\theta})\) is the set of all limit values of the function \(f(z)\) at the point \(e^{i\theta}\); \(R_D(f,e^{i\theta})\) is the aggregate of values assumed by the function \(f(z)\) infinitely often in every neighborhood of the point \(e^{i\theta}\). At an arbitrary point \(z_0=e^{i\theta_0}\in E\) we define the set \(C_{\Gamma\setminus E}(f,z_0)\) as follows:

\[ C_{\Gamma\setminus E}(f,z_0)=\bigcap_{\eta>0} M_\eta, \]

where \(M_\eta\) is the closure of the union

\[ \bigcup_\theta C_{\Lambda_\theta}(f,e^{i\theta}) \]

over all points \(e^{i\theta}\in \mathscr C E\cap\{|e^{i\theta}-z_0|<\eta\}\).

Consider the set \(\Omega=C_D(f,z_0)\setminus C_{\Gamma\setminus E}(f,z_0)\).

The aim of the present note is to prove the following theorem.

Theorem. a) The set \(\Omega\) is open;

b) if the set \(\Omega\) is nonempty, and \(\Omega_n\) is any connected component of it, then \(\Omega_n\setminus R_D(f,z_0)\) consists of at most two points;

c) if for some \(n_0\) the set \(\Omega_{n_0}\setminus R_D(f,z_0)=\{w_0,w_1\}\), \(w_0\ne w_1\), then the set \(R_D(f,z_0)\) coincides with the extended \(w\)-plane from which the points \(w_0,w_1\) have been removed;

d) every value \(\alpha\in \Omega\setminus R_D(f,z_0)\) is an asymptotic value of the function \(f(z)\) either at the point \(z_0\), or at each point of some sequence of points \(\{z_n\}\), \(z_n\in\Gamma\), converging to \(z_0\).

Assertion c) of the theorem refines the theorem of Størvick \((^1)\) and is a generalization, to the case of pseudoanalytic functions, of a result of Wolff \((^2)\), established for meromorphic functions when the curves \(\Lambda_\theta\) are radii of the disk \(D\). Assertion d) of the theorem was proved by Størvick \((^1)\) in the case when the \(\Lambda_\theta\) are radii of the disk \(D\). The method of our proof differs from the methods of \((^{1,2})\) and is close to the methods of Noshiro \((^3)\), who established an analogous theorem for meromorphic functions under a weaker restriction on the set \(E\): Noshiro assumed \(\operatorname{mes} E=0\). However, the proofs in \((^3)\) do not go through in the case of pseudoanalytic functions. Therefore a stronger restriction had to be imposed on the set \(E\) (\(\operatorname{cap} E=0\)), and in addition to the methods of \((^3)\) several modernized methods from other works of Noshiro \((^{4,5})\) had to be used.

We establish the validity of assertion a) of the theorem. Let \(w_0\in\Omega\); choose \(\eta>0\) so that the point \(w_0\) lies outside \(M_\eta\) at a distance \(\rho\). In the neighborhood \(U(z_0,\eta):\ |z-z_0|<\eta\), consider two points \(e^{i\theta_1}, e^{i\theta_2}\in \mathscr C E\) \((\theta_1<\theta_0<\theta_2)\), and join by a straight-line segment \(s\) the final parts of the curves \(\Lambda_{\theta_1}, \Lambda_{\theta_2}\) that have fallen into \(U(z_0,\eta)\). Denote the resulting curve by \(l\), and the domain bounded by the curve \(l\) and the arc

* The result of the note was reported at the VI All-Union Conference on the Theory of Functions of a Complex Variable in Moscow in May–June 1962.

$e^{i\theta}$, $\theta_1 \leqslant \theta \leqslant \theta_2$), through $D_1$. We may assume that the image $l$ under $w=f(z)$ is at a positive distance $\rho$ ($\rho \leqslant \rho'$) from $M_\eta$. Since $w_0 \in C_D(f,z_0)$, there exists a sequence of points $z_\mu \in D_1$, $z_\mu \to z_0$, $f(z_\mu)\to w_0$, $\mu\to\infty$. We shall regard the sequence $\{z_\mu\}$ as fixed throughout the proof. The preimage of the disk $|w-w_0|<\rho$ in the domain $D_1$ consists of at most a countable number of connected components; denote by $\Delta_\mu$ the component of the preimage containing $z_\mu$.

First consider the case when the number of distinct components $\Delta_\mu$ is infinite. We may assume that $\Delta_\mu \ne \Delta_\nu$, $\mu\ne\nu$. Then the sequence $\{\Delta_\mu\}$ converges to the point $z$. Otherwise one could choose points $z'_1=e^{i\theta'_1}$, $z'_2=e^{i\theta'_2}$, $z'_1,z'_2\in \mathscr{C}E$ ($\theta_1<\theta'_1<\theta_0<\theta'_2<\theta_2$) and construct for them, by the method indicated above, a curve $l'$ such that the intersection $l'\cap \Delta_{\mu_n}$, $n=1,2,\ldots$, would be nonempty. Let $\zeta_n$ be a point of intersection of the curve $l'$ and the boundary of $\Delta_{\mu_n}$, and let $\zeta_0$ be a limit point of the sequence $\{\zeta_n\}$; then $\zeta_0\in \mathscr{C}E$ or $\zeta_0\in D$. The case $\zeta_0\in \mathscr{C}E$ is impossible, because then the set $M_\eta$ would intersect $l^*$: $|w-w_0|=\rho$ ($f(\zeta_n)\in\{|w-w_0|=\rho\}$). The case $\zeta_0\in D$ is also impossible, since then an arbitrarily small neighborhood of the point $\zeta_0$ would intersect infinitely many level lines $|f(z)-w_0|=\rho$.

If $\Delta_\mu$ is compact in $D$, then $w=f(z)$ assumes in $\Delta_\mu$ every value from the disk $|w-w_0|\leqslant \rho$. Let $\Delta_\mu$ be noncompact in $D$. We shall show that the domain $\Delta_\mu$ is locally connected on the boundary; otherwise there exists a sequence $\{\Gamma_n\}$ of boundary contours of the domain $\Delta_\mu$, converging to some arc $C$ on $\Gamma$ in the neighborhood $U(z_0,\eta)$, and, consequently, the sets $C_{\Delta\theta}(f,e^{i\theta})$ at the points $e^{i\theta}\in \mathscr{C}E\cap C$ would intersect the curve $l^*$, i.e. $M_\eta\cap l^*$ would be nonempty, which contradicts the choice of $M_\eta$. We shall show that the part of the boundary of the domain $\Delta_\mu$ lying on $\Gamma$—the set $E_\mu$—has capacity zero. For this, write $E_\mu=(E_\mu\cap E)\cup(E_\mu\cap \mathscr{C}E)$. By the preceding, through every point $e^{i\theta}\in E_\mu\cap \mathscr{C}E$ one can draw a continuous curve $L_\theta$, lying entirely inside $\Delta_\mu$. Therefore the sets $C_{L_\theta}(f,e^{i\theta})$ and $C_{\Delta\theta}(f,e^{i\theta})$ do not intersect. According to Beurling’s theorem $(^6)$, there are at most countably many such points $e^{i\theta}$. Consequently, $E_\mu$ has capacity zero.

Now we have the right to apply the following lemma $(^4)$.

Lemma $(^4)$. Let $w=f(z)$ be a pseudoanalytic function in a bounded domain $D$, and let $E$ be a closed set of capacity zero on the boundary $\Gamma$. If $\overline{\lim}_{z\to\zeta}|f(z)|\leqslant M$ at every point $\zeta\in\Gamma\setminus E$, and $f(z)$ is bounded in a neighborhood of every point of the set $E$, then $|f(z)|\leqslant M$ at all points of the domain $D$.

According to this lemma, in our case the set of values $\mathfrak{D}_\mu=f(\Delta_\mu)$ is everywhere dense in $|w-w_0|<\rho$ and, consequently, its closure $\overline{\mathfrak{D}}_\mu$ coincides with $|w-w_0|\leqslant \rho$. Since $\{\Delta_\mu\}$ converges to $z_0$, $C_D(f,z_0)$ contains the closed disk $|w-w_0|\leqslant \rho$.

Now consider two monotonically decreasing sequences of positive numbers $\{\eta_n\}$, $\{\rho_n\}$ and the corresponding sequence of curves $\{l_n\}$; for fixed $n$ the number $\rho_n$ and the curve $l_n$ from the neighborhood $|z-z_0|<\eta_n$ are chosen by the method indicated above. Denote by $\Delta_\mu^{(n)}$ the connected component containing $z_\mu$ of the preimage of the disk $|w-w_0|<\rho_n$. Suppose that there is at least one index $n$ for which the sequence $\{\Delta_\mu^{(n)}\}$ ($\mu\geqslant N(n)$) consists of infinitely many members. Then, by the preceding, $C_D(f,z_0)$ contains the closed disk $|w-w_0|\leqslant \rho_n$.

It remains to consider the case when, for every $n$, $\{\Delta_\mu^{(n)}\}$ consists of a finite number of distinct domains. Denote by $\Delta^{(1)}$ some

a domain \(\Delta_\mu^{(1)}\) containing an infinite subsequence \(\{z_\mu^{(1)}\}\) of the sequence \(\{z_\mu\}\); by \(\Delta^{(2)}\), some domain \(\Delta_\mu^{(2)}\) containing an infinite subsequence \(\{z_\mu^{(2)}\}\) of the sequence \(\{z_\mu^{(1)}\}\), etc. We obtain a new sequence of domains \(\{\Delta^{(n)}\}\), \(\Delta^{(1)} \supset \Delta^{(2)} \supset \cdots \supset \Delta^{(n)} \supset \cdots\); all \(\Delta^{(n)}\) have on their boundary a common point \(z_0\). Since the set of values of the function \(w=f(z)\) in \(\Delta^{(n)}\) lies inside \(|w-w_0|<\rho_n\), and the diameters of the domains \(\Delta^{(n)}\) tend to zero as \(n\to\infty\), in \(D_1\) there exists a curve \(\Lambda\), ending at \(z_0\), along which \(w=f(z)\) tends to \(w_0\).*

Denote by \(\Delta\) the connected component of the preimage of the disk \(|w-w_0|<\rho\) that contains the last part of the curve \(\Lambda\). Just as above, one can show that the boundary of \(\Delta\) consists of a closed set \(E_0\) of capacity zero on \(\Gamma\) and at most a countable number of curves in \(D\). We are now in the conditions of applicability of the methods used by Noshiro in [7] to prove part (ii) of Theorem 3. By means of these methods it is established that \(C_D(f,z_0)\setminus R_\Delta(f,z_0)\) has capacity zero.

To prove assertion c) of the theorem, in the preceding arguments one must replace the point \(w_0\) by the point \(\alpha\).

We shall prove assertion b) of the theorem. We shall give two proofs of this part: one for meromorphic functions, the other for pseudoanalytic functions.

Let the component \(\Omega_{n_0}\) contain three distinct values \(w_0,w_1,w_2\in {\mathcal C}R_D(f,z_0)\). Through the point \(w_2\) draw a closed analytic curve \(\mathcal L\) so that its interior \(G\) consists entirely of points of the domain \(\Omega_{n_0}\), and \(w_0,w_1\in G\). The domain \(G\) plays the same role as the disk \(|w-w_0|<\rho\) in the preceding proof. As above, we construct the domain \(D_1\) and denote by \(\Delta\) the component of the preimage of the domain \(G\) containing an asymptotic curve \(\Lambda\), which ends at some point \(z_0'=e^{i\theta_0}\), \(\theta_1<\theta_0'<\theta_2\) (possibly \(z_0'=z_0\)), and along which \(f(z)\to w_0\) (the existence of the curve \(\Lambda\) is guaranteed by part c) of the theorem). One may assume that the image of the curve \(\Lambda\) under \(w=f(z)\) lies entirely inside \(G\). The component \(\Delta\) is simply connected. Indeed, the boundary of \(\Delta\) cannot contain any closed curve in \(D_1\), since otherwise the image of such a curve would lie on the curve \(\mathcal L\), passing through the exceptional value \(w_2\) of the function \(w=f(z)\). The boundary of the domain \(\Delta\) consists of a finite number of segments \(q_i\) on \(s\), at most a countable number of analytic arcs \(\{\Gamma_n\}\), \(\Gamma_n\in D_1\), and a closed set \(E_0\) on \(\Gamma\). As above, one can show that \(E_0\) has capacity zero.

Up to this point all the arguments have been valid both for meromorphic functions and for pseudoanalytic functions. Now suppose that \(w=f(z)\) is a meromorphic function in \(D\).

By means of the function \(z=z(\zeta)\), map the domain \(\Delta\) conformally onto the unit disk \(|\zeta|<1\). The image of the curve \(\Lambda\) under \(z=z(\zeta)\) will be some curve ending at the point \(\zeta_0\), \(|\zeta_0|=1\). On \(|\zeta|=1\) consider a sufficiently small arc \(A_\zeta\ni \zeta_0\) having no common points with the images, under \(z=z(\zeta)\), of the segments \(q_i\). On the arc \(A_\zeta\) consider the set \(E_\zeta\), at each point \(\zeta=e^{i\varphi}\) of which both functions \(z=z(\zeta)\) and \(w=w(\zeta)\equiv f(z(\zeta))\) have definite angular boundary values \(z(e^{i\varphi})\), \(w(e^{i\varphi})\), and \(w(e^{i\varphi})\in G\). Suppose that there exists a point \(e^{i\varphi}\in E_\zeta\) at which \(z(e^{i\varphi})=e^{i\theta}\notin E\). According to Beurling’s theorem, such points \(e^{i\varphi}\) are countable—

* Relying on the results from (4) and on the fact that, by definition, the set \({\mathcal C}R_D(f,z_0)\) contains three distinct points \(w_0,w_1,w_2\), one can show that the point \(z_0'\in E\).

set. Consequently, the set \(E_\zeta\) has capacity zero, and at all points \(e^{i\varphi} \in A_\zeta \setminus E_\zeta\) the angular boundary values \(w(e^{i\varphi})\) of the function \(w(\zeta)\) lie on \(\mathcal L\). If by \(W=\Phi(w)\) we denote a conformal mapping of the domain \(G\) onto the disk \(|W|<1\), then the function
\[ W = W(\zeta)=\Phi(f(z(\zeta))) \]
will be a function of class \((U)\) in Seidel’s sense. Indeed, the function \(W(\zeta)\) is regular and bounded, \(|W(\zeta)|<1\), \(|\zeta|<1\), and \(|W(e^{i\varphi})|=1\) for \(e^{i\varphi}\in A_\zeta\setminus E_\zeta\), where \(W(e^{i\varphi})\) is the radial boundary value of the function \(W(\zeta)\) at the point \(e^{i\varphi}\). Further, the function \(W=W(\zeta)\) has at the point \(\zeta_0=e^{i\varphi_0}\) the radial boundary value \(W(e^{i\varphi_0})=\Phi(w_0)\), lying in \(|W|<1\). According to Løvatør’s theorem \((^8)\), the function \(W(\zeta)\) assumes every value from \(|W|<1\) infinitely often in any neighborhood of the point \(\zeta_0\), with the exception, perhaps, of one. On the other hand, the function \(W(\zeta)\) has two exceptional values (corresponding to the values \(w_0,w_1\)). We have arrived at a contradiction. The proof of part b) cannot be transferred to the case when \(w=f(z)\) is a pseudoanalytic function. In this case the arguments of \((^4)\) apply, based on the properties of the Evans–Zelberg function associated with a set \(E_0\) of capacity zero (the set \(E_0\) is the part of the boundary of the component \(\Delta\) lying on \(\Gamma\)), and the proof ends in the same way as in \((^4)\).

By analogous methods one establishes the validity of assertion c) of the theorem.

Moscow State University
named after M. V. Lomonosov

Received
27 V 1962

CITED LITERATURE

\(^1\) D. Storvick, Nagoya Math. J., 18, 43 (1961).
\(^2\) W. Woolf, Ann. Acad. Sci. Fenn., Ser. AI, 305 (1961).
\(^3\) K. Noshiro, Proc. Nat. Acad. Sci. Wash., 41, 398 (1955).
\(^4\) K. Noshiro, Nagoya Math. J., 1, 83 (1950).
\(^5\) K. Noshiro, J. Fac. Sci. Hokk. Univ., 6, 217 (1937).
\(^6\) F. Bagemihl, Proc. Nat. Acad. Sci. Wash., 41, 379 (1955).
\(^7\) K. Noshiro, Japan. J. Math., 29, 83 (1959).
\(^8\) A. Løvatør, DAN, 126, No. 4, 707 (1959).