MATHEMATICS

V. P. PETRENKO

GROWTH OF A MEROMORPHIC FUNCTION ALONG A RAY

(Presented by Academician M. A. Lavrent′ev, November 25, 1963)

§ 1. In Nevanlinna theory the growth of entire functions is characterized by the quantity

\[ T(r)=\frac{1}{2\pi}\int_{0}^{2\pi}\ln^{+}\left|f\left(re^{i\theta}\right)\right|\,d\theta \]

and the order (lower order) of growth is called

\[ \overline{\lim_{r\to\infty}}[\ln r]^{-1}\ln T(r) \quad \left(\underline{\lim_{r\to\infty}}[\ln r]^{-1}\ln T(r)\right). \tag{1,1} \]

In the classical theory of entire functions, for this purpose the quantity

\[ M(r)=\max_{0\leq\theta<2\pi}\left|f\left(re^{i\theta}\right)\right| \]

is used, and the order (lower order) is called

\[ \overline{\lim_{r\to\infty}}[\ln r]^{-1}\ln\ln M(r) \quad \left(\underline{\lim_{r\to\infty}}[\ln r]^{-1}\ln\ln M(r)\right). \tag{1,2} \]

The relation between the quantities \(T(r)\) and \(\ln M(r)\) was investigated in works \((^{1-3,5,8})\) and at present has not been fully studied.

R. Nevanlinna \((^1)\) proved the inequality

\[ T(r)\leq \ln M(r)\leq \frac{k+1}{k-1}\,T(kr)\quad (k>1), \]

from which it follows that both definitions (1,1), (1,2) of order and lower order are equivalent. (We agree to denote by \(\rho\) the order of a meromorphic function, and by \(\lambda\) its lower order.) Paley \((^2)\) proved that for any \(0\leq \rho\leq \infty\) there exists an entire function of order \(\rho\) for which the equality

\[ \overline{\lim_{r\to\infty}}[T(r)]^{-1}\ln M(r)=+\infty \]

holds, and put forward the hypothesis: for an entire function of order \(\rho\) the inequality

\[ \overline{\lim_{r\to\infty}}\frac{\ln M(r)}{T(r)} \leq \begin{cases} \pi\rho\,\cosec \pi\rho, & 0\leq \rho\leq 0.5,\\ \pi\rho, & \rho>0.5. \end{cases} \tag{1,3} \tag{1,4} \]

This hypothesis was proved by Valiron \((^3)\) only for \(0\leq \rho\leq 0.5\). For \(\rho>0.5\) the question remains open. A. A. Gol′dberg \((^6)\) obtained the following result:

\[ \overline{\lim_{r\to\infty}}[T(r)]^{-1}\ln^{+}\left|f\left(re^{i\varphi}\right)\right| \leq \pi\rho \quad (\rho>0.5,\;0\leq\varphi<2\pi). \tag{1,5} \]

From this result follows the validity of Paley’s hypothesis for functions satisfying the condition
\(\left|f\left(re^{i\varphi}\right)\right|\sim M(r)\) for some fixed \(\varphi\).

A. A. Gol’dberg and I. V. Ostrovskii \((^{4,7})\) strengthened Valiron’s result by proving that, for entire functions of lower order \(\lambda \leqslant 0.5\), in inequality (1.3) one may replace \(\rho\) by \(\lambda\).

In the present paper we prove that, for an entire function of lower order \(\lambda > 0.5\), in inequality (1.5) one may replace \(\rho\) by \(\lambda\). This result is a special case of a more general result obtained in this paper*:

Theorem. Let \(f(z)\) be a meromorphic function of finite lower order \(\lambda > 0.5\). The inequality
\[ \lim_{r\to\infty} [T(r)]^{-1}\ln^+ |f(re^{i\varphi})| \leqslant \pi\lambda \]
holds.

§ 2. We shall carry out the proof of the theorem for the case \(\lambda < \rho\). First we establish several auxiliary relations. Let
\[ G_{\alpha,R}=\{z:0<|z|<R,\ |\arg z|<\alpha\}, \]
where \(0<\alpha<\pi,\ R>0\); let \(g_{\alpha,R}(z,\zeta)\) be the Green function for \(G_{\alpha,R}\).

Lemma. For a function meromorphic in \(G_{\alpha,R}\) the following inequality holds
\[ \ln^+ |f(r)| \leqslant \frac{1}{2\pi}\int_0^R \{\ln^+ |f(te^{i\alpha})|+\ln^+ |f(te^{-i\alpha})|\}\,P(t,r,\alpha)\,dt+ \]
\[ +\frac{1}{2\pi}\int_{-\alpha}^{\alpha}\ln^+ |f(Re^{i\theta})|\,P(R,r,\alpha,\theta)\,d\theta+ \sum_{b_k\in G_{\alpha,R}}\ln\left|\frac{r^\sigma+\overline{b_k^{\sigma}}}{r^\sigma-b_k^{\sigma}}\right|, \tag{2.1} \]
where \(\sigma=0.5\,\pi\alpha^{-1}\),
\[ P(t,r,\alpha)=\pi\alpha^{-1}t^{\sigma-1}r^\sigma(t^{2\sigma}+r^{2\sigma})^{-1}, \]
\[ P(R,r,\alpha,\theta)=\sigma(R^{2\sigma}-r^{2\sigma}) \frac{4R^\sigma r^\sigma\cos\sigma\theta} {R^{4\sigma}-2R^{2\sigma}r^{2\sigma}\cos 2\sigma\theta+r^{4\sigma}}, \]
and \(b_k\) are the poles of \(f(z)\).

This lemma follows from the known representation for \(\ln |f(z)|\) in the sector \(G_{\alpha,R}\) and from the following relations:
\[ \left|\frac{\partial}{t\partial\varphi}\, g_{\alpha,R}(te^{i\varphi},r)\right|_{\varphi=\pm\alpha} \leqslant P(t,r,\alpha), \]
\[ \frac{\partial}{\partial R}\,g_{\alpha,R}(Re^{i\theta},r) =R^{-1}P(R,r,\alpha,\theta), \]
\[ g_{\alpha,R}(r,a)\leqslant \ln\left|(r^\sigma+\overline{a^\sigma})(r^\sigma-a^\sigma)^{-1}\right|. \]

We now proceed to the proof of the theorem. Without loss of generality, we shall assume \(\varphi=0,\ |f(0)|<1\). Choose \(\gamma\) so that \(\lambda<\gamma<\rho\), and take \(\alpha<\pi(2\gamma)^{-1}\); then from inequality (2.1) we find:
\[ \int_{r_0}^{0.5R} r^{-\gamma-1}\ln^+ |f(r)|\,dr \leqslant \frac{1}{2\pi}\int_0^R\int_{r_0}^{0.5R} \{\ln^+ |f(te^{i\alpha})|+\ln^+ |f(te^{-i\alpha})|\}\times \]
\[ \times r^{-\gamma-1}P(t,r,\alpha)\,dt\,dr +\frac{1}{2\pi}\int_{r_0}^{0.5R}\int_{-\alpha}^{\alpha} \ln^+ |f(Re^{i\theta})|\,r^{-\gamma-1}P(R,r,\alpha,\theta)\,dr\,d\theta+ \]
\[ +\sum_{b_k\in G_{\alpha,R}}\int_{r_0}^{0.5R} r^{-\gamma-1}\ln\left|(r^\sigma+\overline{b_k^{\sigma}})(r-b_k^\sigma)^{-1}\right|\,dr \tag{2.2} \]
where \(0<r_0<0.5R\).

\[ \text{* We use the standard notation of Nevanlinna theory.} \]

Noting that*

\[ \int_{r_0}^{0.5R} r^{-\gamma-1} P(t,r,\alpha)\,dr = t_{-\gamma}^{-1}\int_{r_0t^{-1}}^{0.5Rt^{-1}} s^{-\gamma-1}P(1,s,\alpha)\,ds \leq \]

\[ \leq t^{-\gamma-1}\int_{0}^{\infty}s^{-\gamma-1}P(1,s,\alpha)\,ds = \pi t^{-\gamma-1}\sec\alpha\gamma, \]

\[ \int_{r_0}^{0.5R} r^{-\gamma-1}\ln\left|\frac{r^\sigma+\overline{b}_k^{\sigma}}{r^\sigma-b_k^\sigma}\right|\,dr \leq \int_{r_0}^{0.5R} r^{-\gamma-1}\ln\left|\frac{r^\sigma+|b_k|^\sigma}{r^\sigma-|b_k|^\sigma}\right|\,dr = \]

\[ = |b_k|^{-\gamma} \int_{r_0|b_k|^{-1}}^{0.5R|b_k|^{-1}} s^{-\gamma-1}\ln\left|(s^\sigma+1)(s^\sigma-1)^{-1}\right|\,ds \leq \]

\[ \leq |b_k|^{-\gamma}\int_{0}^{\infty} s^{-\gamma-1}\{\ln(s^\sigma+1)-\ln|s^\sigma-1|\}\,ds = \pi\gamma^{-1}|b_k|^{-\gamma}\tg\alpha\gamma, \]

\[ P(R,r,\alpha,\theta)\leq C_1\alpha^{-1}r^\sigma R^{-\sigma} \quad \text{for } \quad 0<r<0.5R, \]

from (2.2) we find

\[ 2\cos\alpha\gamma\int_{r_0}^{0.5R} r^{-\gamma-1}\ln^+|f(r)|\,dr \leq \int_{r_0}^{0.5R} r^{-\gamma-1} \{\ln^+|f(re^{i\alpha})|+\ln^+|f(re^{-i\alpha})|\}\,dr + \]

\[ + 2\pi\gamma^{-1}\sin\alpha\gamma \sum_{r_0<|b_k|<0.5R}|b_k|^{-\gamma} + C_2\sum_{b_k\leq r_0}|b_k|^{-\gamma} + \]

\[ + C_3\sum_{0.5R<|b_k|<R}|b_k|^{-\gamma} + r_0\int_{0}^{r_0} r^{-\gamma-1} \{\ln^+|f(re^{i\alpha})|+\ln^+|f(re^{-i\alpha})|\}\,dr + \]

\[ + C_4T(R)R^{-\gamma} + \int_{0.5R}^{R} r^{-\gamma-1} \{\ln^+|f(re^{i\alpha})|+\ln^+|f(re^{-i\alpha})|\}\,dr. \]

Integrating the resulting inequality with respect to \(\alpha\) over the interval from \(0\) to \(\pi(2\gamma)^{-1}\), we find

\[ (\pi\gamma)^{-1}\int_{r_0}^{0.5R} r^{-\gamma-1}\ln^+|f(r)|\,dr \leq \int_{r_0}^{0.5R} r^{-\gamma-1}m(r,\infty)\,dr + \]

\[ + \gamma^{-2}\int_{r_0}^{0.5R} r^{-\gamma}\,dn(r,\infty) + C_5T(2R)R^{-\gamma}+C_6 = \]

\[ = \int_{r_0}^{0.5R} r^{-\gamma-1}m(r,\infty)\,dr + \int_{r_0}^{0.5R} r^{-\gamma-1}N(r,\infty)\,dr + C_7T(2R)R^{-\gamma}+C_8. \tag{2.3} \]

Since \(\gamma<\rho\), as \(R\to\infty\),

\[ \int_{r_0}^{0.5R} r^{-\gamma-1}T(r)\,dr \to \infty, \]

* Throughout the article, by the letter \(C\) with subscripts we shall denote positive constants independent of \(R\), \(r\), and \(\alpha\).

on the other hand, since \(\gamma>\lambda\), we have

\[ \lim_{R\to\infty} R^{-\gamma} T(R)=0. \]

Therefore from (2.3) we find

\[ \lim_{r\to\infty} \int_{r_0}^{0.5R} r^{-\gamma-1}\ln^{+}|f(r)|\,dr \left\{ \int_{r_0}^{0.5R} r^{-\gamma-1}T(r)\,dr \right\}^{-1} \leqslant \pi\gamma, \]

whence

\[ \lim_{r\to\infty} [T(r)]^{-1}\ln^{+}|f(r)| \leqslant \pi\gamma. \]

Letting now \(\gamma\) tend to \(\lambda\), we obtain the assertion of the theorem for the case \(\lambda<\rho\).

For \(\lambda=\rho\) one should apply the method of work \({}^{6}\).

Remark. From the results of A. A. Gol’dberg and I. V. Ostrovskii \({}^{4,7}\) there follows the following assertion, generalizing inequalities (1, 3) to meromorphic functions:

Let \(f(z)\) be a meromorphic function of finite lower order \(\lambda\leqslant 0.5\). The inequality holds

\[ \lim_{r\to\infty} [T(r)]^{-1}\ln M(r) \leqslant \pi\lambda \operatorname{cosec}\pi\lambda\,(1-\cos\pi\lambda(1-\Delta(\infty))), \]

where

\[ \Delta(\infty)=1-\lim_{r\to\infty} N(r,\infty)[T(r)]^{-1}. \]

I express my deep gratitude to I. V. Ostrovskii for supervising the work.

Kharkov State University
named after A. M. Gorky

Received
20 XI 1963

REFERENCES

\({}^{1}\) R. Nevanlinna, Single-Valued Analytic Functions, Moscow–Leningrad, 1941.
\({}^{2}\) R. E. A. C. Paley, Proc. Cambridge Phil. Soc., 28, 262 (1932).
\({}^{3}\) G. Valiron, Mathematica, 11, 264 (1935).
\({}^{4}\) A. A. Gol’dberg, I. V. Ostrovskii, Notes of the Mathematical Department of Kharkov State University and the Kharkov Mathematical Society, 27, ser. 4, 3 (1961).
\({}^{5}\) A. A. Gol’dberg, Dokl. AN URSR, 4, 443 (1963).
\({}^{6}\) A. A. Gol’dberg, DAN, 152, No. 5 (1963).
\({}^{7}\) I. V. Ostrovskii, DAN, 150, 32 (1963).
\({}^{8}\) I. V. Ostrovskii, Notes of the Mathematical-Mechanical Faculty of Kharkov State University and the Kharkov Mathematical Society, 28, ser. 4, 23 (1961).