MATHEMATICS

I. V. OSTROVSKII

ON THE CONNECTION BETWEEN THE GROWTH OF A MEROMORPHIC FUNCTION AND THE DISTRIBUTION OF ITS VALUES BY ARGUMENTS

(Presented by Academician S. N. Bernstein on 28 XII 1959)

By investigations of B. Ya. Levin and A. Pfluger (for the main results, as well as references to the original papers, see \((^1)\)) it was established that certain regularity requirements imposed on the moduli of the zeros of an entire function entail regularity of its growth. The purpose of the present note is to show that certain restrictions imposed only on the arguments of the zeros and ones of an entire function entail a rather strong regularity of its growth*. We shall show here that these results carry over also to meromorphic functions.

\(1^\circ\). We shall adhere to the following notation: \(f(z)\) is a function meromorphic in the whole finite plane; \(\alpha\) and \(\beta\) are numbers satisfying the inequality \(0 \le \alpha < \beta \le 2\pi\), \(\gamma=\beta-\alpha\); \(\{\alpha_j\}_{j=1}^n\) are numbers satisfying the inequality \(0 \le \alpha_1 < \alpha_2 < \cdots < \alpha_n < 2\pi\); \((R)\) is the system of rays \(\arg z=\alpha_j\); \(\gamma_j=\alpha_{j+1}-\alpha_j\) \((\alpha_{n+1}=\alpha_1+2\pi)\); \(\theta=\min_{1\le j\le n}\gamma_j\), \(\Theta=\max_{1\le j\le n}\gamma_j\); \(O(1)\) is a quantity remaining bounded as \(r\to\infty\); \(\{a_k\}\) are the poles, \(\{b_l\}\) the zeros, \(\{c_m\}\) the ones of the function \(f(z)\), lying outside the disk \(|z|<1\), considered with multiplicities; \(A_{\alpha\beta}(r,f)\), \(B_{\alpha\beta}(r,f)\), \(C_{\alpha\beta}(r,f)\), \(S_{\alpha\beta}(r,f)=A_{\alpha\beta}(r,f)+B_{\alpha\beta}(r,f)+C_{\alpha\beta}(r,f)\) are the quantities (everywhere \(r\ge 1\)) introduced by Nevanlinna** \((^3)\), characterizing the distribution of the values of the function \(f(z)\) in the angle \(\alpha<\arg z<\beta\). Recall that the quantity \(C_{\alpha\beta}(r,f)\) characterizes the distribution of the poles of \(f(z)\) in this angle, taking their arguments into account in an essential way. In particular, the relation \(C_{\alpha\beta}(r,f)=O(1)\) is equivalent to the convergence of the series

\[ \sum_{\alpha\le \varphi_k\le \beta} \left[\sin\frac{\pi}{\gamma}(\varphi_k-\alpha)\right] r_k^{-\pi/\gamma} \qquad (a_k=r_k e^{i\varphi_k}). \]

We shall use the following results.

Theorem A. If the function \(f(z)\) satisfies the condition

\[ \int_1^\infty \ln^+ T(r,f)\, r^{-\pi/\gamma-1}\,dr<\infty, \]

then for any finite set of \(q\ge 3\) distinct complex numbers \(a^{(1)}, a^{(2)}, \ldots, a^{(q)}\) from the extended plane the following relation holds***

\[ (q-2)S_{\alpha\beta}(r,f) \le \sum_{\nu=1}^{q} C_{\alpha\beta}\bigl(r,(f-a^{(\nu)})^{-1}\bigr)+O(1). \]

* We note that a qualitative assertion close to this was formulated in the paper of A. A. Goldberg \((^2)\).

** Definitions of these quantities can also be found in \((^9)\).

*** By definition, for \(a=\infty\), \(C_{\alpha\beta}(r,(f-a)^{-1})=C_{\alpha\beta}(r,f)\).

Theorem B. If \(S_{0\pi}(r,f)=O(1)\), then there exists a finite limit

\[ \eta=\lim_{r\to\infty} r^{-1}\int_0^\pi \ln |f(re^{i\vartheta})|\sin\vartheta\,d\vartheta \]

and the representation holds

\[ \ln |f(z)|= \frac{1}{\pi}\int_{-\infty}^{\infty} \frac{r\sin\varphi\ln |f(t)|}{r^2+t^2-2rt\cos\varphi}\,dt +\frac{2\eta}{\pi}r\sin\varphi+ \]

\[ +\sum_{\operatorname{Im} a_k>0}\ln\left|\frac{z-\overline{a}_k}{z-a_k}\right| -\sum_{\operatorname{Im} b_l>0}\ln\left|\frac{z-\overline{b}_l}{z-b_l}\right| \qquad (z=re^{i\varphi}). \tag{1} \]

Theorems A and B are due to Nevanlinna \((^3)\), who in Theorem A assumed, instead of our condition, the finite order of \(f(z)\). In order to obtain the result in the form in which we have formulated it, one must use the estimate

\[ A_{\alpha\beta}(r,f'f^{-1}) = O\left(\int_1^{2r}\ln^+T(r,f)r^{-\pi/\gamma-1}\,dr\right). \]

Let the function \(u(r,\varphi)\) be defined for \(1\le r<\infty\), \(\alpha\le\varphi\le\beta\). If it is possible to specify a set \(E\subset[1,\infty)\) of finite logarithmic length \(\left(\text{i.e. } \int_E d\ln r<\infty\right)\) such that \(\lim_{\substack{r\to\infty\\ r\notin E}}u(r,\varphi)=h(\varphi)\) exists uniformly in \(\varphi\), \(\alpha\le\varphi\le\beta\), then we shall agree to say that for \(\alpha\le\varphi\le\beta\) there exists

\[ \lim_{r\to\infty}^{(l)}u(r,\varphi)=h(\varphi). \]

Theorem V. If the function \(f(z)\) is representable in the form (1), then for \(0\le\varphi\le\pi\) there exists

\[ \lim_{r\to\infty}^{(l)} r^{-1}\ln |f(re^{i\varphi})| = 2\eta\pi^{-1}\sin\varphi. \]

This theorem is a simple consequence of a result of Heiman \((^4)\). We note that Theorems B and V are easily generalized to the case of an angle of arbitrary opening.

Theorem G. If the function \(f(z)\) satisfies the conditions:

a) for at least two distinct values \(a\) from the extended plane and for some system of rays \((R)\)

\[ \sum_{j=1}^{n} C_{\alpha_j\alpha_{j+1}}(r,(f-a)^{-1})=O(1); \]

b) for some value \(a\), different from those which occur in condition a), the positive quantity

\[ \Delta^*(a)=\sup_{\mathfrak B\in K}\lim_{\substack{r\to\infty\\ r\in\mathfrak B}} \frac{m(r,a)}{T(r,f)} \]

(here \(K\) denotes the class of sets lying on the positive half-axis with upper density\(^*\) less than 1),

then the growth of \(f(z)\) is not higher than order \(\pi\delta^{-1}\) and of normal type.

This theorem is a special case of the main result of the note \((^5)\).

\(2^\circ\). From a comparison of Theorems A, B, and V the following follows immediately:

Theorem 1. If the function \(f(z)\) satisfies the conditions:

1) there exist \(\alpha\) and \(\beta\) such that, for at least three distinct values \(a\) from the extended plane,

\[ C_{\alpha\beta}(r,(f-a)^{-1})=O(1); \]

2)

\[ \int_1^\infty \ln^+T(r,f)r^{-\pi/\gamma-1}\,dr<\infty, \]

\(^*\) The upper density of a set \(E\subset[1,\infty)\) is the quantity

\[ \lim_{r\to\infty} r^{-1}\operatorname{mes}\{E\cap[1,r]\}. \]

then

3) for \(\alpha \leqslant \varphi \leqslant \beta\) there exists
\[ \lim_{r\to\infty}^{(l)} r^{-\pi/\gamma}\ln |f(re^{i\varphi})| = c\sin \frac{\pi}{\gamma}(\varphi-\alpha); \]

4) the integrals
\[ \int_1^\infty |\ln |f(te^{i\alpha})||\,t^{-\pi/\gamma-1}\,dt \quad\text{and}\quad \int_1^\infty |\ln |f(te^{i\beta})||\,t^{-\pi/\gamma-1}\,dt \]
converge.

Corollary 1. Let there be a certain system of rays \((R)\). If the function \(f(z)\) satisfies the conditions:

1) for at least three distinct values \(a\) from the extended plane
\[ C_{\alpha_j\alpha_{j+1}}(r,(f-a)^{-1})=O(1)\quad (j=1,2,\ldots,n); \]

2)
\[ \int_1^\infty \ln^+ T(r,f)r^{-\pi/\Theta-1}\,dr<\infty, \]
then

3) for \(\alpha_j \leqslant \varphi \leqslant \alpha_{j+1}\) there exists
\[ \lim_{r\to\infty}^{(l)} r^{-\pi/\gamma_j}\ln |f(re^{i\varphi})| = c_j\sin \frac{\pi}{\gamma_j}(\varphi-\alpha_j) \]
\[ (j=1,2,\ldots,n); \]

4) the integrals
\[ \int_1^\infty |\ln |f(te^{i\alpha_j})||\,t^{-\pi/\delta_j-1}\,dt \quad (\delta_j=\max(\gamma_j,\gamma_{j-1}), \]
\[ \gamma_0=\gamma_n,\ j=1,\ldots,n) \]
converge.

Condition 1) can be weakened somewhat, replacing it by the requirement that for each \(j\) there is such a triple of numbers \(a\) from the extended plane that
\[ C_{\alpha_j\alpha_{j+1}}(r,(f-a)^{-1})=O(1). \]

\(3^\circ\). In this item we shall consider functions having the representation
\[ f(z)=\sum_{k=1}^{\infty}\frac{A_k}{z-a_k},\qquad \sum_{a_k\ne 0}\left|\frac{A_k}{a_k}\right|<\infty . \tag{2} \]

M. V. Keldysh proved \((^6)\) that for such functions
\[ m(r,f)=O(1), \tag{3} \]
whence it follows easily that for any \(\alpha\) and \(\beta\) one has
\[ B_{\alpha\beta}(r,f)=O(1). \]
By a method close to that which was used in \((^6)\) to prove (3), one can establish that for functions of the form (2) one has
\[ A_{\alpha\beta}(r,f)=O(1), \]
if \(0<\gamma\leqslant \pi\). Consequently, for functions of the form (2) always
\[ S_{\alpha\beta}(r,f)=C_{\alpha\beta}(r,f)+O(1)\quad (0<\gamma\leqslant\pi). \]

From this relation and Theorems B and C it follows:

Theorem 2. If the function \(f(z)\) is represented in the form (2) and, for some \(\alpha\) and \(\beta\), \(0<\gamma\leqslant\pi\),
\[ C_{\alpha\beta}(r,f)=O(1), \]
then for \(f(z)\) the assertions of Theorem 1 hold; moreover, in 3) we shall have \(c\leqslant 0\).

Corollary 2. If the function \(f(z)\) is represented in the form (2) and for some system of rays \((R)\) with \(\Theta\leqslant\pi\) one has
\[ \sum_{j=1}^{n} C_{\alpha_j\alpha_{j+1}}(r,f)=O(1), \]
then for \(f(z)\) the assertions of Corollary 1 hold, and in 3) we shall have
\[ c_j\leqslant 0\quad (j=1,2,\ldots,n). \]
This result is a generalization of a theorem of M. G. Krein \((^7)\).

\(4^\circ\). From Theorem C it follows that condition 2) of Corollary 1 can be replaced by the following condition:

\(2^{**})\) There is at least one value \(a\) (not necessarily distinct from those which occur in condition \(1)\)) for which the quantity \(\Delta^*(a)\) is positive.

Since for every entire function \(\Delta^*(\infty)=1\), it follows from this that

Theorem 3. If \(f(z)\) is an entire function such that for at least two distinct values \(a\ne\infty\)

\[ \sum_{j=1}^{n} C_{\alpha_j\alpha_{j+1}}\left(r,\left(f-a^{-1}\right)=O(1)\right), \]

then the assertions of Corollary 1 are valid for this function.

Corollary 3. If \(f(z)\) is an entire function for which \(\sum |\operatorname{Im}(b_l^{-1})|<\infty\), \(\sum |\operatorname{Im}(c_m^{-1})|<\infty\), then \(f(z)\) is a function of exponential type satisfying the condition

\[ \int_{-\infty}^{\infty}\frac{|\ln|f(t)||}{1+t^2}\,dt<\infty . \]

\(5^\circ\). Using some results of Mii\({}^{8}\), one can somewhat strengthen the results of \(2^\circ\), \(3^\circ\), and \(4^\circ\), by considering, in addition to the \(a\)-points of the function, also the \(a\)-points of its derivatives. In doing so it is also necessary to use some results from \({}^{5}\).

\(6^\circ\). We regard as highly probable the supposition that condition 2) in Theorem 1 is superfluous.

It seems to us that it would be interesting to prove or refute the possibility of replacing, in Corollary 3, the condition \(\sum |\operatorname{Im}(c_m^{-1})|<\infty\) by the condition \(\delta(1)>0\) or by the condition \(\Delta^*(1)>0\). We note that in \({}^{5}\) it was established (in particular) that if \(\sum |\operatorname{Im}(b_l^{-1})|<\infty\) and \(\Delta^*(1)>0\), then \(f(z)\) is of exponential type.

Kharkov State University
named after A. M. Gorky

Received
21 XII 1959

REFERENCES

\({}^{1}\) B. Ya. Levin, Distribution of zeros of entire functions, Moscow, 1956.
\({}^{2}\) A. A. Gol’dberg, Izv. Vyssh. Uchebn. Zaved., Mathematics, No. 4, 50 (1959).
\({}^{3}\) R. Nevanlinna, Acta Soc. Sci. Fenn., 50, No. 12 (1925).
\({}^{4}\) W. K. Hayman, J. de math. pures et appl., 35, 115 (1956).
\({}^{5}\) I. V. Ostrovskii, DAN, 120, No. 5, 970 (1958).
\({}^{6}\) M. V. Keldysh, DAN, 94, No. 3, 377 (1954).
\({}^{7}\) M. G. Krein, Izv. AN SSSR, ser. matem., 11, No. 4, 309 (1947).
\({}^{8}\) H. Milloux, Les fonctions méromorphes et leurs dérivées, Paris, 1940.
\({}^{9}\) I. V. Ostrovskii, DAN, 130, No. 5 (1960).