On the Set of Angular Boundary Values of Normal Meromorphic Functions

V. I. Gavrilov

(Presented by Academician I. G. Petrovskii on 30 VI 1961)

Mathematics

In the present note we consider the question of the structure of the set of angular boundary values of meromorphic functions in the unit disk \(|z|<1\) that are normal there in the sense of Lehto and Virtanen \((^1)\). In the proofs of our assertions we shall rely essentially on the results of Collingwood and Cartwright \((^2)\).

1. In the terminology of \((^2)\) one can formulate the following theorem:

Theorem 1. If the function \(w=f(z)\) is meromorphic and normal in the disk \(|z|<1\), then:

a) \(\Gamma(f,\theta_1<\theta<\theta_2)=\Gamma_P(f,\theta_1<\theta<\theta_2)\) on every arc \((\theta_1,\theta_2)\) of the unit circle \(|z|=1\) \((^3)\);

b) if \(a\in\Gamma_P(f,e^{i\theta})\), then \(\Gamma_P(f,e^{i\theta})\) contains no other values and \(e^{i\theta}\in F(f)\) \((^1)\).

Theorem 1, establishing a one-to-one correspondence between the sets \(\Gamma(f)\) and \(F(f)\) considered for a normal meromorphic function \(f(z)\), says nothing, however, about whether these sets contain even a single point. Moreover, an example is known \((^1)\) of a normal meromorphic function \(\varphi(z)\) in \(|z|<1\) for which the sets \(\Gamma(\varphi)\) and \(F(\varphi)\) are empty. In this case the set \(CR(\varphi)\) was also empty for the function \(\varphi(z)\).

If, however, for a normal meromorphic function \(f(z)\) in \(|z|<1\) the set \(CR(f)\) is nonempty, then the sets \(\Gamma(f)\) and \(F(f)\) are also nonempty, as is indicated by Corollary 9.3 of \((^2)\).*

1. Theorem 2. If the meromorphic function \(w=f(z)\not\equiv \mathrm{const}\) is normal in the disk \(|z|<1\) and the set \(CR(f)\) is nonempty: \(a\in CR(f)\), then:

a) on every arc \((\theta_1,\theta_2)\), \(0\leq\theta_1<\theta_2\leq 2\pi\), of the circle \(|z|=1\), the set \(F(f)\) is nonempty;

b) if, moreover, on \((\theta_1,\theta_2)\) \(\operatorname{mes} F(f)=0\), then there exists a point \(e^{i\theta_0}\), \(\theta_1<\theta_0<\theta_2\), \(e^{i\theta_0}\in F(f)\), at which the function \(w=f(z)\) has the angular boundary value equal to \(a\), and \(F(f)\) will be a set of first category on \((\theta_1,\theta_2)\).

Proof. Suppose the contrary, and let the arc \((|\theta|<\eta)\), \(\eta>0\), contain no points of the set \(F(f)\). Then, by Lemma 11 of \((^1)\): 1) either \(a\in CC(f,1)\), 2) or \(a\in R(f,1)\), 3) or \(a\in\Phi(f,1)\).

If \(a\in CC(f,1)\), then, by Theorem 14 of \((^2)\), the arc \((|\theta|<\eta)\) would contain points of the set \(F(f)\). The second case is also impossible, by the hypothesis.

If \(a\in\Phi(f,1)\), then there exists a sequence of mutually nonintersecting continuous arcs \(\{\gamma_n\}\), whose endpoints tend to the points \(e^{\pm i\delta}\), \(0<\delta<\eta\), on which \(f(z)\) tends uniformly to \(a\) as \(|z|\to 1\).

Draw through the point \(z=1\) a diameter \(L_0\) of the disk \(|z|<1\), which intersects the arcs \(\{\gamma_n\}\) at the points \(\{z_n^0\}\). If the non-Euclidean distances \(\rho(z_n^0,z_{n+1}^0)\)

* This assertion was announced in \((^4)\); for a proof by a method different from the present one, see also \((^3)\).

between neighboring points of the sequence are uniformly bounded by some constant \(M\), then, by the lemma from \((^3)\), \(a\in \Gamma_p(f,1)\).

In the general case, to each pair of neighboring points \(z_k^0, z_{k+1}^0\) we adjoin a finite number of points
\[ z_k^\nu,\quad \nu=1,\ldots,p_k,\quad z_k^1=z_k^0,\ldots,z_k^{p_k}=z_{k+1}^0, \]
lying on \(L_0\), so that
\[ \rho\left(z_k^\nu,z_k^{\nu+1}\right)=1,\quad \nu=1,2,\ldots,p_k. \]
Through the points \(z_k^\nu\) draw curves \(\gamma_k^\nu\) so that
\[ \gamma_k^0=\gamma_k^1,\ldots,\gamma_k^{p_k}=\gamma_{k+1}^0,\quad \rho\left(\gamma_k^\nu,\gamma_k^{\nu+1}\right)=1. \]
To the radius \(L_0\) draw equidistants \(\Lambda_s\), \(s=1,2,\ldots,p_k\), for which
\[ \rho(L_0,\Lambda_s)=s. \]
Let the points of intersection of the curves \(\Lambda_s\) and \(\gamma_k^\nu\) be
\[ z_{k,s}^\nu,\quad \nu=1,\ldots,p_k;\quad s=1,\ldots,p_k;\quad k=1,2,\ldots . \]

Denote the aggregate of points
\[ \{z_k^0;\ z_{k,2}^2;\ z_{k,3}^3;\ldots;\ z_{k+1,p_k}^0;\ z_{k+1,p_k-1}^0;\ldots;\ z_{k+1}^0\} \]
by \(E_k\). Let \(E=\bigcup_k E_k\); by construction, \(E\) is a sequence of points \(\{z_m\}\), whose limit points lie on \(|z|=1\), and
\[ \rho(z_m,z_{m+1})\le M \]
for all \(m\). Since, by the lemma from \((^3)\),
\[ \lim_{\nu\to\infty} f\left(z_{k,s}^\nu\right)=a \]
for every \(s\), it follows, in view of the choice of \(E_k\), that
\[ \lim_{m\to\infty} f(z_m)=a. \]
If now the set of limit points of the sequence \(\{z_m\}\) consists of a single point \(e^{i\theta_0}\), then \(a\in \Gamma_p(f,e^{i\theta_0})\), \(|\theta_0|<\eta\); if, however, this set contains more than one point, then, by Theorem 3 from \((^3)\), \(f(z)\equiv a\).

The assertion of part b) of Theorem 2 is an immediate consequence of Theorem 16 of the work \((^2)\), of the constructions carried out in the proof of part a), and of Theorem 4 of the work \((^5)\).

Remark 1. The construction used in proving the impossibility of case 3), in essence, shows that for a normal meromorphic function in \(|z|<1\) the set of Plessner points cannot fill any arc on \(|z|=1\) completely.

3. We give some consequences of the theorems obtained above.

Theorem 2 and the uniqueness theorem of N. N. Luzin— I. I. Privalov immediately lead to the following assertion:

Theorem 3. If, for a normal meromorphic function \(w=f(z)\not\equiv \mathrm{const}\) in the disk \(|z|<1\), the set \(CR(f)\) contains at least one point, then the set \(\Phi(f)\) is empty.

If one slightly modifies the construction carried out for the proof of the impossibility of case 3) in Theorem 2, and, instead of the lemma from \((^3)\), uses Lemma 1 from \((^4)\), instead of Theorem 3 from \((^3)\) and Theorem 1 from \((^4)\)—Theorems 4 from \((^3)\) and 2 from \((^4)\), then one can show that the following theorem is true:

Theorem 4. If \(w=f(z)\not\equiv \mathrm{const}\) is a meromorphic normal function in the disk \(|z|<1\), then the set \(\Phi(f)\) is empty.

Theorems 3 and 4 may be regarded as generalizations of the Koebe—Gross theorem \((^2)\).

Remark 2. Just as in \((^3)\), one can show that the theorems given above remain valid also in the general case of normal quasiconformal functions.

In conclusion I express my gratitude to my scientific adviser A. I. Markushevich for his attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
29 VI 1961

REFERENCES

1. O. Lehto, K. Virtanen, Acta Math., 97, 1—2, 47 (1957).
2. E. Collingwood, M. Cartwright, Acta Math., 87, 1—2, 83 (1952).
3. V. Gavrilov, DAN, 138, No. 1, 16 (1961).
4. F. Bagemihl, W. Seidel, Ann. Acad. Sci. Fennicae, Ser. A I, 280 (1960).
5. E. Collingwood, Acta Math., 91, 3—4, 165 (1954).