Abstract
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Reports of the Academy of Sciences of the USSR
- Volume 152, No. 5
MATHEMATICS
A. A. GOL’DBERG
ON THE GROWTH OF AN ENTIRE FUNCTION ALONG A RAY
(Presented by Academician M. A. Lavrent’ev on 9 V 1963)
Let \(f(z)\) be an entire function of order \(\rho\), and let \(T(r)\) be its Nevanlinna characteristic:
\[ T(r)=\frac{1}{2\pi}\int_{0}^{2\pi}\ln^{+}|f(re^{i\varphi})|\,d\varphi . \]
In the present note the growth of \(\ln |f(re^{i\varphi})|\) for fixed \(\varphi\) is compared with the growth of \(T(r)\). Without loss of generality we shall assume that the fixed ray is \(\varphi=0\), i.e. the positive real axis. Pólya \({}^{(1)}\) constructed examples of entire functions of arbitrary order \(\rho\), \(0\leq \rho\leq \infty\), for which
\[ \limsup_{r\to\infty}\ln |f(r)|/T(r)=\infty . \]
We shall supplement this result by showing that the following estimate always holds:
\[ \liminf_{r\to\infty}\frac{\ln |f(r)|}{T(r)} \leq \begin{cases} \pi\rho\cosec \pi\rho, & 0\leq \rho\leq {1}/{2},\\ \pi\rho, & \rho>{1}/{2}. \end{cases} \tag{1} \]
The fact that estimate (1) cannot be improved follows from consideration of the well-known Mittag-Leffler function \(E_{\rho}(z)\). Estimate (1) for \(\rho=\infty\) is trivial, and for \(0\leq \rho\leq {1}/{2}\) follows at once from an inequality obtained almost simultaneously by Valiron and Valiron \({}^{(2-4)}\); the new result is estimate (1) for \({1}/{2}<\rho<\infty\).
The proof of (1) (for \({1}/{2}<\rho<\infty\)) uses several lemmas.
Lemma 1. Let the functions \(h(r)\) and \(g(r)\) be continuous for \(a\leq r<\infty\), with \(g(r)>0\) for \(a\leq r<\infty\). Let \(K(\eta,r)\geq 0\) be continuous for \(0\leq \eta<\eta_{0}\) and \(a\leq r<\infty\), and let the integrals
\[ \int_{a}^{\infty} h(r)K(\eta,r)\,dr,\qquad \int_{a}^{\infty} g(r)K(\eta,r)\,dr>0 \]
converge for \(0<\eta<\eta_{0}\), while
\[ \int_{a}^{\infty} g(r)K(0,r)\,dr=\infty . \]
Then
\[ \limsup_{r\to\infty}\frac{h(r)}{g(r)} \geq \limsup_{\eta\to 0+} \frac{\displaystyle\int_{a}^{\infty} h(r)K(\eta,r)\,dr} {\displaystyle\int_{a}^{\infty} g(r)K(\eta,r)\,dr}, \]
\[ \liminf_{r\to\infty}\frac{h(r)}{g(r)} \leq \liminf_{\eta\to 0+} \frac{\displaystyle\int_{a}^{\infty} h(r)K(\eta,r)\,dr} {\displaystyle\int_{a}^{\infty} g(r)K(\eta,r)\,dr}. \]
A function \(\rho(r)\), differentiable on \([1,\infty)\), is called a refined order \({}^{(5)}\) for an entire function \(f(z)\) of order \(\rho\), if the following conditions are satis-
the following conditions are fulfilled: a) \(\lim_{r\to\infty}\rho(r)=\rho\); b) \(\lim_{r\to\infty} r\rho'(r)\ln r=0\); c) \(\limsup_{r\to\infty} T(r)r^{-\rho(r)}=1\).
Lemma 2. If \(\rho(r)\) is a proximate order for an entire function \(f(z)\) of order \(\rho>0\), then
\[ \int_1^\infty T(r)r^{-\rho(r)-1}\,dr=\infty . \]
Lemma 3. For every entire function \(f(z)\) of order \(\rho\), \(1/2<\rho<\infty\), one can find a proximate order \(\rho(r)\) such that for \(1\le R<\infty\) the following holds:
\[ \limsup_{\eta\to 0+} \frac{ \displaystyle \int_R^\infty r^{-\rho(r)-\eta-1}\ln^+|f(r)|\,dr }{ \displaystyle \int_R^\infty r^{-\rho(r)-\eta-1}T(r)\,dr } \le \pi\rho+\omega(R), \tag{2} \]
where \(\lim_{R\to\infty}\omega(R)=0\).
From Lemma 1, putting \(h(r)=\ln^+|f(r)|\), \(g(r)=T(r)\), \(K(\eta,r)=r^{-\rho(r)-\eta-1}\), \(a=R\), we have
\[ \liminf_{r\to\infty}\ln^+|f(r)|/T(r)\le \pi\rho+\omega(R), \]
whence, letting \(R\to\infty\), we obtain (1).
In the case when \(f(z)\) is an entire function of order \(1/2<\rho<\infty\) of the divergence class \((^5,\text{ p. }495)\), inequality (2) was obtained by Rauch \((^6)\) with \(\rho(r)\equiv \rho\), \(\omega(R)\equiv 0\)*, and the results of Valiron \((^8)\) were used essentially. Our proof of Lemma 3 follows basically the same path, but the consideration of proximate order complicates all the calculations.
In the case \(0\le \rho\le 1/2\), the inequality of Wiman and Valiron \((^{2-4})\) implies an inequality stronger than (1):
\[ \liminf_{r\to\infty}\ln M(r)/T(r)\le \pi\rho\cosec\pi\rho,\qquad M(r)=\max_{|z|=r}|f(z)|. \]
Pólya \((^1)\) conjectured that for \(\rho>1/2\) the inequality
\[ \liminf_{r\to\infty}\ln M(r)/T(r)\le \pi\rho, \tag{3} \]
is valid, but we have not been able to prove it. If, however, one additionally assumes that for some fixed \(\varphi_0\) one has
\(\ln|f(re^{i\varphi_0})|\sim \ln M(r)\), then (3) follows from (1) for \(\rho>1/2\) and, as a special case, a result of I. V. Ostrovskii \((^9,\text{ p. }31)\).
Uzhgorod State University
Received
18 IV 1963
References
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- Wahlund, Ark. Math., Astr. och Fys., 21A, No. 23, (1929).
- G. Valiron, Opuscula math. A. Wiman dedicata, 1 1930.
- G. Valiron, Mathematica, 11, 264 (1935).
- B. Ya. Levin, Distribution of zeros of entire functions, Moscow, 1956.
- A. Rauch, C. R., 206, 1076 (1938).
- A. Rauch, Bull. sci. math., 63, 66 (1939).
- G. Valiron, J. math. pures et appl., 10, 457 (1931).
- I. V. Ostrovskii, Zap. Mekh.-Mat. Fak. Kharkov State Univ. and Kharkov Math. Soc., 28, 23 (1961).
* In fact, A. Rauch asserts in \((^6)\) the existence of the limit on the left-hand side of (2) (under the assumptions he made), but this assertion is not justified by anything. In \((^7)\) Rauch attempts to prove an even stronger inequality than in \((^6)\), but his arguments contain gross errors.