Reports of the Academy of Sciences of the USSR

1958, Volume 120, No. 5

MATHEMATICS

I. V. OSTROVSKII

ON MEROMORPHIC FUNCTIONS TAKING CERTAIN VALUES AT POINTS LYING NEAR A FINITE SYSTEM OF RAYS

(Presented by Academician S. N. Bernstein on 6 II 1958)

In papers \((^{1-4})\) there were considered functions \(f(z)\), meromorphic for \(|z|<\infty\), which possess values attained in a certain neighborhood of a finite system of rays

\[ \arg z=\theta_n,\quad n=1,2,\ldots,m,\quad 0\leq \theta_1<\theta_2<\cdots<\theta_m<2\pi . \tag{1} \]

It then turned out that, if any one of these values is assumed by the function \(f(z)\) sufficiently rarely (in the sense that, at least, its Nevanlinna defect is positive), then this entails an estimate of the growth of \(T(r,f)\) depending only on the disposition of the rays (1).

A result of A. Edrei \((^3)\) is as follows:
Let the roots of the three equations

\[ f(z)=0,\quad f^{-1}(z)=0,\quad f^{(l)}(z)=1 \tag{2} \]

(\(l\) is some nonnegative integer), with the possible exception of a finite number, lie on the rays (1). In addition, let

\[ \delta(0,f)+\delta(\infty,f)+\delta(1,f^{(l)})>0 . \tag{3} \]

Then the order of \(f(z)\) does not exceed \(\pi\gamma^{-1}\), where
\(\gamma=\min_{1\leq n\leq m}(\theta_{n+1}-\theta_n)\)
\((\theta_{m+1}=2\pi+\theta_1)\).

In paper \((^4)\), which generalizes results of \((^2)\), an assertion is proved that may be formulated as follows:
Let the function \(f(z)\) be representable in the form

\[ f(z)=\sum A_k(z-h_k)^{-1},\quad \sum |A_kh_k^{-1}|<\infty,\quad \sum |\operatorname{Im}(h_k^{-1})|<\infty, \tag{4} \]

and let \(\delta(b,f)>0\) for some \(b\). Then
\[ \lim_{r\to\infty} r^{-1}T(r,f)<\infty . \]
The connection between this assertion and the result \((^3)\) becomes immediately apparent if one uses the following simple observation. If \(f(z)\) is represented in the form (4), then for any \(a\) (including \(0\) and \(\infty\)) the relation
\[ \sum |\operatorname{Im} z_k^{-1}(a)|<\infty \]
holds, where \(z_k(a)\) are the roots of the equation \(f(z)=a\). Consequently one may say that the roots of the equations

\[ f(z)=0,\quad f^{-1}(z)=0,\quad f(z)=1 \]

are situated “near” the rays \(\arg z=0\) and \(\arg z=\pi\). Moreover, since, without loss of generality, one may take \(b=1\), condition (3) is satisfied with \(l=0\). Meanwhile, Edrei’s theorem is not applicable here.

Comparison of the results \((^3)\) and \((^4)\) led the author to the establishment of a theorem with respect to which both results are particular cases. Before formulating it, we introduce the necessary notation.

Let \(f(z)\) be a function meromorphic for \(|z|<\infty\), and let \(r_k e^{i\varphi_k}\) be its poles. Following (1), put

\[ C(R,\alpha,\beta,f)=2\sum_{\substack{1\leq r_k<R\\ \alpha<\varphi_k<\beta}} \left(\frac{1}{r_k^{\pi/\gamma}}-\frac{r_k^{\pi/\gamma}}{R^{2\pi/\gamma}}\right) \sin\frac{\pi}{\gamma}(\varphi_k-\alpha) \]

\[ (0<\beta-\alpha=\gamma\leq 2\pi). \]

By the symbols \(K_1(t)\) and \(K_2(t)\) we shall denote positive nondecreasing functions of \(t\geq 0\), and by the letters \(k_i\) the quantities
\[ \overline{\lim}_{t\to\infty}\ln K_i(t)(\ln t)^{-1},\quad i=1,2. \]

Definition 1. The set of \(a\)-points of the function \(f(z)\) is called “close” to the system of rays (1) if the inequality

\[ \sum_{n=1}^{m} C(R,\theta_n,\theta_{n+1},(f-a)^{-1}) \leq K_1(R)K_2(T(R,f)), \]

where \(k_1\) is finite and \(k_2<1\), holds for all \(R\geq 0\), except perhaps for some set \(\mathfrak A\subset[0,\infty)\) such that
\[ \lim_{R\to\infty}R^{-1}\operatorname{mes}\{\mathfrak A\cap[0,R]\}=0. \]

Definition 2. A number \(a\) is called a \(*\)-deficient value of the function \(f(z)\) if there exists a set \(\mathfrak B\subset[0,\infty)\) such that:
\[ 1)\quad \overline{\lim}_{R\to\infty}R^{-1}\operatorname{mes}\{\mathfrak B\cap[0,R]\}<1;\qquad 2)\quad \lim_{\substack{R\to\infty\\ R\in\mathfrak B}} m(R,a,f)(T(R,f))^{-1}>0. \]

Obviously, a Nevanlinna deficient value is a fortiori \(*\)-deficient.

Theorem 1. Let \(f(z)\) be a function meromorphic for \(|z|<\infty\), and suppose that at least one of the conditions A, B, C is satisfied:

A. 1) The zeros and poles of \(f(z)\) are “close” to the system of rays (1); 2) at least one of the functions \(f^{(l)}(z)\), \(l\geq 0\), has at least one \(*\)-deficient value distinct from \(0\) and \(\infty\).

B. 1) The poles of \(f(z)\) and the \(a\)-points of \(f^{(l)}(z)\), for some \(a\ne 0,\infty\) and some integer \(l\geq 0\), are “close” to the system of rays (1); 2) zero is a \(*\)-deficient value of \(f(z)\).

C. 1) The zeros and poles of \(f(z)\) and the \(a\)-points of \(f^{(l)}(z)\), for some \(a\ne 0,\infty\) and some integer \(l\geq 0\), are “close” to the system of rays (1); 2) \(\infty\) is a \(*\)-deficient value of \(f(z)\).

Then the order of the function \(f(z)\) is finite and does not exceed the quantity

\[ \chi=\chi(\gamma,k_1,k_2)=(\pi+\gamma k_1)\gamma^{-1}(1-k_2)^{-1}, \]

where

\[ \gamma=\min_{1\leq n\leq m}(\theta_{n+1}-\theta_n),\quad \theta_{m+1}=2\pi+\theta_1. \]

Moreover, if both quantities
\[ \sigma_i=\overline{\lim}_{t\to\infty}K_i(t)t^{-k_i}\quad (i=1,2) \]
are finite, then the growth of \(T(R,f)\) does not exceed the normal type of order \(\chi\), and if in addition one of them is equal to zero, then it does not exceed the minimal type of order \(\chi\).*

For \(k_1=k_2=0\) we obtain that the order of \(f(z)\) does not exceed \(\pi\gamma^{-1}\), and if, moreover, \(K_i(t)=O(1)\) \((i=1,2)\) (which is always fulfilled if, apart from a finite number, the roots of equations (2) are situated on the rays (1)), then the type is not above normal. This assertion is stronger than Edrei’s theorem \((^3)\) and stronger than the result \((^4)\).

For the proof of Theorem 1 the following are used: a) the estimate
\[ m(R,a,f^{(l)})\leq m(R,f^{(l)}/f^{(l+1)})+C_l\ln\{RT(R,f)\};\quad l=0,1,\ldots;\quad a\ne 0,\infty, \]
valid

* One can also estimate the refined order of \(T(R,f)\) through the refined orders \(K_1(t)\) and \(K_2(t)\).

for all \(R \geq 0\), except perhaps for a set of finite length; b) an estimate of the quantity \(m(R, f^{(l)}/f^{(l+1)})\), following from the following theorem, which lies at the basis of our investigation:

Theorem 2. Whatever may be given: a function \(f(z)\) meromorphic for \(|z|<\infty\); \(l=0,1,2,\ldots\); \(\alpha\) and \(\beta\) \((0<\beta-\alpha=\gamma\leq 2\pi)\); \(\varepsilon>0\), there exists a set \(\mathfrak A=\mathfrak A_{f,l,\alpha,\beta,\varepsilon}\subset[0,\infty)\) such that:

1)
\[ \varlimsup_{R\to\infty} R^{-1}\operatorname{mes}\{\mathfrak A\cap[0,R]\}<\varepsilon; \]

2)
\[ \int_{\alpha}^{\beta}\ln^+\left|\frac{f^{(l)}(Re^{i\theta})}{f^{(l+1)}(Re^{i\theta})}\right|\,d\theta \leq A_{f,l,\alpha,\beta,\varepsilon}\ln^4\{RT(R,f)\}\times \]
\[ \times R^{4\pi/\gamma}\{C(\widetilde R,\alpha,\beta,f)+C(\widetilde R,\alpha,\beta,f^{-1})+q(\gamma,R)\ln\{RT(R,f)\}\} \tag{5} \]
for \(R\in C\mathfrak A\), where
\[ \widetilde R=R+\frac{R}{\ln T(R,f)};\qquad q(\gamma,R)= \begin{cases} 1, & (0<\gamma\leq\pi),\\ R^{1-\pi/\gamma}, & (\pi<\gamma\leq 2\pi). \end{cases} \]

If the order of \(f(z)\) is finite, then inequality (5) may be replaced by the more precise one:
\[ 2')\qquad \int_{\alpha}^{\beta}\ln^+\left|\frac{f^{(l)}(Re^{i\theta})}{f^{(l+1)}(Re^{i\theta})}\right| \leq A_{f,l,\alpha,\beta,\varepsilon}R^{\pi/\gamma} \{C(4^{\gamma/\pi}R,\alpha,\beta,f)+ \]
\[ +C(4^{\gamma/\pi}R,\alpha,\beta,f^{-1})+1\}. \]

The known theorem of Nevanlinna on the logarithmic derivative makes it possible to estimate from above the quantity \(m(r,f')\) in terms of \(m(r,f)\). Theorem 2 may be applied in some cases to obtain the converse estimate.

In the proof we start from Nevanlinna’s formula \((^1)\) and proceed by methods that constitute a refinement of the methods of \((^4)\). Our methods also make it possible to obtain the following assertion:

Theorem 3. Let \(f(z)\) be meromorphic in the disk \(|z|<1\), and suppose that at least one of the conditions A, B, C is satisfied:

A. 1) The zeros and poles of \(f(z)\) lie on the radii (1). 2) \(\delta(a,f^{(l)})>0\) for some \(a\ne0,\infty\) and some integer \(l\geq0\).

B. 1) The poles of \(f(z)\) and the \(a\)-points of \(f^{(l)}(z)\), for some \(a\ne0,\infty\) and some integer \(l\geq0\), lie on the radii (1); 2) \(\delta(0,f)>0\).

C. 1) The zeros and poles of \(f(z)\) and the \(a\)-points of \(f^{(l)}(z)\), for some \(a\ne0,\infty\) and some integer \(l\geq0\), lie on the radii (1); 2) \(\delta(\infty,f)>0\).

Then, independently of the number (provided only that it is finite) and the arrangement of the radii (1), the order of \(f(z)\), i.e.
\[ \lim_{r\to1}\frac{\ln T(r,f)}{\ln(1/(1-r))} \]
does not exceed 4.

We note that there is also a stronger theorem, in whose hypothesis there occur (adapted to the hyperbolic case) the notions of “proximity” of the set of \(a\)-points of \(f(z)\) to the radii (1) and of a \(*\)-defective value.

I express my gratitude to A. A. Goldberg, to whom I owe the idea of comparing the results \((^3)\) and \((^4)\), and to B. Ya. Levin for a number of valuable suggestions and attention to the work.

Kharkov State University
named after A. M. Gorky

Received
6 II 1958

CITED LITERATURE

1. R. Nevanlinna, Acta Soc. Sci. Fenn., 50, No. 12 (1925).
2. M. G. Krein, Izv. AN SSSR, ser. matem., 11, No. 4, 309 (1947).
3. A. Edrei, Trans. Am. Math. Soc., 78, No. 2, 276 (1955).
4. I. V. Ostrovskii, DAN, 116, No. 5, 742 (1957).