MATHEMATICS

V. GAVRILOV

ON LIMITS ALONG SEQUENCES OF POINTS OF NORMAL MEROMORPHIC FUNCTIONS

(Presented by Academician I. G. Petrovskii on 2 XII 1960)

1. Let a meromorphic function \(f(z)\) be given in the unit disk \(D: |z|<1\). Following Lehto and Virtanen \((^{1})\), we shall call this function normal in the disk \(D\) if, for an arbitrary family \(\{T(z)\}\) of one-to-one conformal mappings of the disk \(D\) onto itself, the family of functions \(\{f(T(|z|))\}\) is normal in the disk \(D\) in the sense of Montel. Lehto and Virtanen showed that a normal meromorphic function \(f(z)\) will have a definite angular boundary value \(a\) at some point \(e^{i\theta}\) on the boundary of the disk \(D\), if \(a\) is an asymptotic value of the function \(f(z)\) along some Jordan curve lying in the closure \(\overline{D}\) and ending at \(e^{i\theta}\).

An immediate consequence of this result and of the well-known theorem of I. I. Privalov is the following

Theorem 1. If a normal meromorphic function \(f(z)\) in the disk \(D\) has a definite asymptotic value along some Jordan curve lying in \(D\) and having at least two distinct limiting points on the boundary, then \(f(z)=\mathrm{const}\).

Since every regular function in \(D\) has at least one definite asymptotic value along some Jordan curve whose limiting points lie on the boundary \((^{2})\), Theorem 1 immediately implies the following assertion, announced in \((^{3})\):

For any normal regular function in the disk \(D\) one can find on the unit circle a point \(e^{i\theta}\) at which this function has a definite angular boundary value.

1. In the above-mentioned result of Lehto and Virtanen and in Theorem 1, the existence of the limit of the function along all points of some curve was assumed. A natural question arises: is it possible, while preserving the assertions, to weaken this assumption by requiring the existence of the limit of the function only along some, in a certain sense “dense,” sequence of points \(\{z_n\}\) from the disk \(D\), whose limiting points lie on the boundary.

This question was considered in \((^{3})\) and was answered affirmatively for certain classes of normal functions; moreover, as the “measure of density” of the sequence \(\{z_n\}\), depending on the breadth of the classes, the conditions were:

1) \(\rho(z_n,z_{n+1})<M\), where \(M\) is some constant independent of \(n\), or

1′) \(\displaystyle \lim_{n\to\infty}\rho(z_n,z_{n+1})=0\), where
\[ \rho=(a,b)=\frac12\ln\left|\frac{1-a\overline{b}}{1-\overline{a}b}\right|+\left|\frac{a-b}{1-\overline{a}b}\right|,\quad |a|<1,\ |b|<1. \]

Theorem 2. If: 1) the sequence \(\{z_n\}\) of points from the disk \(|z|<1\), \(|z_n|\to 1\) as \(n\to\infty\), whose limiting points cover some arc \(A\) of the unit circle, \(\operatorname{mes} A>0\), is such that for almost all points \(\zeta\in A\) from \(\{z_n\}\) one can choose at least one subsequence \(\{z_n^{(\zeta)}\}\), which converges to \(\zeta\), and for all \(n\)
\[ \rho\bigl(z_n^{(\zeta)},z_{n+1}^{(\zeta)}\bigr)<M(\zeta) \]
with a constant \(M(\zeta)\) depending, perhaps, on the choice of the point \(\zeta\); 2) for

for a normal meromorphic function \(f(z)\) in \(|z|<1\) having at least one exceptional value, there exists \(\lim\limits_{n\to\infty} f(z_n)=a\), then almost everywhere on \(A\) the function \(f(z)\) has angular boundary values equal to \(a\), and, consequently, \(f(z)\equiv a\).

Theorem 2 is a simple consequence of a theorem of Noshiro \({}^{(4)}\) and auxiliary Lemma 1, whose proof, in essence, is contained in \({}^{(3)}\) and is a consequence of known properties of normal families of functions \({}^{(5)}\).

Lemma 1. If: 1) the sequences of points \(\{z_n\}\), \(\{z'_n\}\) from \(D\), \(|z'_n|\to 1\), \(n\to\infty\), are such that \(\rho(z_n,z'_n)<M\) with some constant \(M\) independent of \(n\); 2) for a normal meromorphic function \(f(z)\) in \(D\), omitting the value \(c\), there exists \(\lim\limits_{n\to\infty} f(z_n)=c\), then \(\lim\limits_{n\to\infty} f(z'_n)=c\).

1. Let us note some further consequences of Theorem 1, Lemma 1, and the assertion corresponding to it in \({}^{(3)}\).

Theorem 3. If: 1) a sequence of points \(\{z_n\}\) from \(D\), having two distinct limit points on the boundary, is such that \(\rho(z_n,z_{n+1})<M\) with some \(M\) independent of \(n\); 2) a normal meromorphic function \(f(z)\), omitting the value \(c\), has \(\lim\limits_{n\to\infty} f(z_n)=c\), then \(f(z)\equiv c\).

Theorem 4. If: 1) a sequence of points \(\{z_n\}\) from \(D\), having two distinct limit points on the boundary, is such that \(\lim\limits_{n\to\infty}\rho(z_n,z_{n+1})=0\); 2) a normal meromorphic function \(f(z)\) has \(\lim\limits_{n\to\infty} f(z_n)=a\), finite or infinite, then \(f(z)\equiv a\).

1. If one introduces the concept of a normal quasiconformal mapping \({}^{(6)}\), then, taking into account the estimates \({}^{(7)}\) of the distortion of the hyperbolic metric under one-to-one quasiconformal mappings, the results stated above are automatically carried over to the case of normal quasiconformal mappings of the disk \(D\) onto the corresponding Riemann surfaces.

The author expresses his gratitude to his scientific adviser A. I. Markushevich for his attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
1 XII 1960

CITED LITERATURE

\({}^{1}\) O. Lehto, K. Virtanen, Acta Math., 97, 1—2, 47 (1957).
\({}^{2}\) F. Kierst, E. Szpilrajn, Fund. Math., 21, 276 (1953).
\({}^{3}\) F. Bagemihl, W. Seidel, Suom. tiedeak. toim., Sar. AI Math., No. 280 (1960).
\({}^{4}\) K. Noshiro, Proc. Nat. Acad. Sci. USA, 41, 6 (1955).
\({}^{5}\) P. Montel, Normal Families of Analytic Functions, M.—L., 1936.
\({}^{6}\) J. Väisälä, Suom. tiedeak. toim., Sar. AI Math., No. 266, 1 (1959).
\({}^{7}\) J. Hersch, Comm. Math. Helv., 30, 1 (1956).