UDC 517.544

MATHEMATICS

N. V. GOVOROV

ON THE INDICATOR OF FUNCTIONS OF INTEGER ORDER, ANALYTIC AND OF COMPLETELY REGULAR GROWTH IN A HALF-PLANE

(Presented by Academician S. N. Bernstein on 8 IV 1966)

B. Ya. Levin and A. Pfluger showed \((({}^{1}), \text{ Chs. I, III}; ({}^{2,3}))\) that an entire function of order \(\rho>0\) has completely regular growth if and only if the set of its zeros \(\{z_n\}\) is regularly distributed, i.e.:

1. For all \(\theta_k \in [0,2\pi]\setminus N\), where \(N\) is at most countable, there exists the limit

\[ \lim_{r\to\infty}\frac{n(r,\theta_1,\theta_2)}{r^\rho} = \Delta(\theta_1,\theta_2), \tag{1} \]

called the angular density (see \(({}^{1})\), p. 118).

1. In the case of integer \(\rho\), the zeros \(z_n\) are arranged with a special symmetry, namely, there exists the limit

\[ \lim_{r\to\infty}\sum_{|z_n|<r} z_n^{-\rho}=\sigma\ne\infty. \tag{2} \]

In the works of the authors cited, a formula for the indicator of these entire functions was also obtained. In article \(({}^{4})\) these results were extended to functions of noninteger order, regular in a half-plane. For them, the concept of argument density, which is an analogue of angular density, is introduced in a special way, and a formula for the indicator is obtained. In the present note we consider functions of integer order in a half-plane. Definitions of the order of a function and of functions of completely regular growth (in an open or closed angle) are given in article \(({}^{4})\).

Theorem 1. Every function, regular and of integer order \(\rho\ge 0\) in \(\operatorname{Im} z>0\), is uniquely representable in the form

\[ \begin{aligned} f(z)=\exp\Biggl\{& i\sum_{k=0}^{\rho} a_k z^k +\frac{1}{\pi i}\biggl[ \int_{-1}^{1}\frac{\ln |f(t)|}{t-z}\,dt +\int_{-1}^{1}\frac{i\,d\varphi(t)}{t-z} \\ &\quad +z^{\rho+1}\int_{|t|>1}\frac{\ln |f(t)|\,dt}{t^{\rho+1}(t-z)} +z^{\rho+1}\int_{|t|>1}\frac{d\varphi(t)}{t^{\rho+1}(t-z)} \biggr]\Biggr\} \prod_{|z_n|\le 1}\frac{z-z_n}{z-\overline{z}_n}\times \end{aligned} \]

\[ \times \prod_{|z_n|>1} \frac{ \left(1-\dfrac{z}{z_n}\right) \exp\left[ \dfrac{z}{z_n} +\dfrac{1}{2}\left(\dfrac{z}{z_n}\right)^2 +\cdots+ \dfrac{1}{\rho}\left(\dfrac{z}{z_n}\right)^\rho \right] }{ \left(1-\dfrac{z}{\overline{z}_n}\right) \exp\left[ \dfrac{z}{\overline{z}_n} +\dfrac{1}{2}\left(\dfrac{z}{\overline{z}_n}\right)^2 +\cdots+ \dfrac{1}{\rho}\left(\dfrac{z}{\overline{z}_n}\right)^\rho \right] }, \tag{3} \]

where \(a_k\) are real constants; \(z_n\) are the interior (lying in \(\operatorname{Im} z>0\)) zeros of \(f(z)\); \(\varphi(t)\) is a real nondecreasing function for which \(\varphi'(t)=0\) almost everywhere and the integral

\[ \int_{|t|\ge 1}\frac{d\varphi(t)}{t^{\rho+2}} \]

converges.

The theorem is readily obtained from the analogous theorem given in \(({}^{4})\).*

* For the case of integer order, Theorem 1 is in a certain sense more convenient than the one mentioned (\(\varphi(t)\) and \(a_k\) in these theorems are different).

Starting from representation (3), for an arbitrary function \(f(z)\), regular and of integer order \(\rho>0\) in \(\operatorname{Im} z>0\), we introduce the following notation:

\[ \tau(t)= \begin{cases} \dfrac{1}{2\pi}\displaystyle\int_{1}^{t}\dfrac{\ln |f(x)|}{x}\,dx+ \dfrac{1}{2\pi}\displaystyle\int_{1}^{t}\dfrac{d\varphi(x)}{x}, & t>1,\\[1.2em] \dfrac{1}{2\pi}\displaystyle\int_{t}^{-1}\dfrac{\ln |f(x)|}{|x|}\,dx+ \dfrac{1}{2\pi}\displaystyle\int_{t}^{-1}\dfrac{d\varphi(x)}{|x|}, & t<-1, \end{cases} \tag{4} \]

\[ c(r,\eta_1,\eta_2)= \sum_{\eta_1<\theta_n\leq \eta_2,\; 1\leq r_n\leq r} \sin\theta_n,\qquad r>1\quad (z_n=r_n e^{i\theta_n}), \tag{5} \]

\[ a(r,\eta_1,\eta_2)= \begin{cases} c(r,\eta_1,\eta_2), & 0<\eta_1<\eta_2<\pi,\\ c(r,0,\eta_2)-\tau(r), & 0=\eta_1<\eta_2<\pi,\\ c(r,\eta_1,\pi)-\tau(-r), & 0<\eta_1<\eta_2=\pi,\\ c(r,0,\pi)-\tau(r)-\tau(-r), & \eta_1=0,\ \eta_2=\pi, \end{cases} \tag{6} \]

\[ a(r,\eta_1,\eta_2)=-a(r,\eta_2,\eta_1),\quad \eta_1>\eta_2;\qquad a(r,\eta,\eta)\equiv 0. \]

Definition 1. If for a function \(f(z)\), regular and of order \(\rho>0\) in \(\operatorname{Im} z>0\), for all \(\eta_1,\eta_2\in[0\leq\theta\leq\pi]\setminus N\), where \(N\) is at most countable and contains no points \(\eta=0\), \(\eta=\pi\), there exists the finite limit

\[ \lim_{r\to\infty}\frac{a(r,\eta_1,\eta_2)}{r^\rho} =\lambda(\eta_1,\eta_2), \]

then we shall say that the set of zeros of the function \(f(z)\) has an argument-boundary density in the half-plane \(\operatorname{Im} z>0\).

Definition 2. A function \(f(z)\) of order \(\rho>0\) in \(\operatorname{Im} z>0\) is called a function of finite type if, asymptotically,

\[ \sup_{|z|\leq r,\ \operatorname{Im} z>0}|f(z)|<\exp(Kr^\rho),\qquad K=\mathrm{const}. \tag{7} \]

Denote by \(A_\rho\) (by \(\overline{A}_\rho\)) the class of functions regular of order \(\rho\) and of finite type in \(\operatorname{Im} z>0\), having completely regular growth in the open angle \(0<\arg z<\pi\) (in the closed angle \(0\leq\arg z\leq\pi\)).

Theorem 2. Let the function \(f(z)\) be regular of integer order \(\rho>0\) and of finite type in \(\operatorname{Im} z>0\). Then, in order that \(f(z)\) belong to the class \(A_\rho\), it is necessary and sufficient that the set of its zeros have an argument-boundary density and that there exist the finite limit

\[ \sigma=\lim_{r\to\infty}\left[ \frac{1}{\rho}\sum_{1\leq r_n\leq r}\frac{\sin \rho\theta_n}{r_n^\rho} -\frac{1}{2\pi}\int_{1\leq |x|\leq r}\frac{\ln |f(x)|}{x^{\rho+1}}\,dx -\frac{1}{2\pi}\int_{1\leq |x|\leq r}\frac{d\varphi(x)}{x^{\rho+1}} \right]. \tag{8} \]

Theorem 3. If \(f(z)\in A_\rho\) and \(\rho>0\) is an integer, then the indicator of the function \(f(z)\) is expressed by the formula

\[ h_f(\theta)=(2\sigma-a_\rho)\sin\rho\theta +2\cos\rho\theta\int_{0}^{\pi}\psi\,\frac{\sin\rho\psi}{\sin\psi}\,d\lambda(\psi) + \]

\[ +2\int_{0}^{\theta}\frac{\psi}{\sin\psi}\sin\rho(\theta-\psi)\,d\lambda(\psi) +2\int_{\theta}^{\pi}\frac{\psi-\pi}{\sin\psi}\sin\rho(\theta-\psi)\,d\lambda(\psi), \tag{9} \]

where \(\lambda(\psi)\equiv\lambda(0,\psi)\) is the argument-boundary density of the zeros of \(f(z)\), and \(\sigma\) and \(a_\rho\) are defined respectively by relations (8) and (3).

Remark. For writing formula (9) in other forms, Theorem 4 may be useful.

Theorem 4. If \(\lambda(\psi)=\lambda(0,\psi)\) is the argument-boundary density of the function \(f(z)\in A_\rho\), \(\rho>0\) an integer, then the equality

\[ \int_0^\pi \frac{\sin \rho\psi}{\sin \psi}\,d\lambda(\psi)=0 \tag{10} \]

holds.

Definition 3. Let the function \(f(z)\) be regular and of order \(\rho>0\) in \(\operatorname{Im} z>0\), and let \(\tau(r)\) be defined by equality (4). Then the limits

\[ \lim_{r\to+\infty}\frac{\tau(r)}{r^\rho}=l_1,\qquad \lim_{r\to+\infty}\frac{\tau(-r)}{r^\rho}=l_2, \tag{11} \]

if they exist and are finite, are called, respectively, the right- and left-hand boundary density of the set of zeros of the function \(f(z)\). If in (11) upper limits are taken, then we shall call them upper boundary densities \((l_1^*\) and \(l_2^*)\).

Definition 4. Let a set of points \(\{z_n\}\), \(n=1,2,\ldots\), be given in \(\operatorname{Im} z>0\), all of whose limit points lie on the real axis. Then, if for all \(\eta_1,\eta_2\in(0\leq \eta\leq \pi)\setminus N\), where \(N\) is at most countable and does not contain \(\eta=0,\eta=\pi\), there exists the finite limit

\[ \lim_{r\to\infty}\frac{c(r,\eta_1,\eta_2)}{r^\rho}=\mu(\eta_1,\eta_2), \tag{12} \]

then we shall say that the set \(\{z_n\}\) has, in the domain \(\operatorname{Im} z>0\), an argument density with exponent \(\rho\) (or an upper argument density \(\mu^*(\eta_1,\eta_2)\), if in (12) an upper limit is taken).

Theorem 5. If \(f(z)\) is a function of order \(\rho\geq 1\) and of finite type in \(\operatorname{Im} z>0\), then its upper boundary densities are finite, and its upper argument density is bounded.

Theorem 6. If \(f(z)\) is a function of class \(A_\rho\) and \(\rho>0\) is an integer, then the indicator of \(f(z)\) can be expressed by the formula \((0<\theta<\pi)\)

\[ \begin{aligned} h_f(\theta)=2\Biggl\{& \sin \rho\theta\cdot \left[\sigma-l_1^*+(-1)^\rho l_2^*-\frac{a_\rho}{2}\right] +\cos \rho\theta\cdot \left[\int_0^\pi \psi\,\frac{\sin \rho\psi}{\sin \psi}\,d\mu^*(\psi) +\pi\rho(-1)^\rho l_2^*\right]\\ &+\int_0^\theta \frac{\psi}{\sin \psi}\,\sin \rho(\theta-\psi)\,d\mu^*(\psi) +\int_\theta^\pi \frac{\psi-\pi}{\sin \psi}\,\sin \rho(\theta-\psi)\,d\mu^*(\psi) \Biggr\}. \end{aligned} \]

Theorem 7. Let \(f(z)\) be regular in \(\operatorname{Im} z>0\) and continuous in \(\operatorname{Im} z\geq 0\), and have integer order \(\rho>0\) and finite type. Then \(f(z)\in A_\rho\) if and only if:

1. The zeros of \(f(z)\) have boundary and argument density, the latter being continuous at \(\psi=0\) and \(\psi=\pi\).
2. The function \(\varphi(t)\), defined from (3), satisfies the equality

\[ \lim_{r\to\infty}\frac{1}{r^\rho}\int_{1\leq |t|\leq r}\frac{d\varphi(t)}{|t|}=0. \tag{13} \]

1. There exists the finite limit (8).
2. The indicator \(h_f(\theta)\) is continuous at \(\theta=0\) and \(\theta=\pi\).

The indicator of the function \(f(z)\in \overline{A}_\rho\) has the following form \((0\leq \theta\leq \pi)\):

\[ \begin{aligned} h_f(\theta)=2\Biggl\{& \sin \rho\theta\cdot \left[\sigma-l_1+(-1)^\rho l_2-\frac{a_\rho}{2}\right] +\cos \rho\theta\cdot \left[\int_0^\pi \psi\,\frac{\sin \rho\psi}{\sin \psi}\,d\mu(\psi) +\pi\rho(-1)^\rho l_2\right]\\ &+\int_0^\theta \frac{\psi}{\sin \psi}\,\sin \rho(\theta-\psi)\,d\mu(\psi) +\int_\theta^\pi \frac{\psi-\pi}{\sin \psi}\,\sin \rho(\theta-\psi)\,d\mu(\psi) \Biggr\}. \end{aligned} \]

Theorem 8. If the assumptions of Theorem 7 are satisfied and \(\rho\) is odd, then \(f(z)\in \overline{A}_\rho\) if and only if:

1. The set of zeros of the function \(f(z)\) has, in the open angle \(0<\theta<\pi\), an angular density (see (1), where it is assumed that \(0<\theta_k<\pi\)).

2. The limit (8) exists.

3. The indicator \(h_f(\theta)\) satisfies the condition
   \[ h_f(0)+h_f(\pi)=2\pi\int_{+0}^{\pi-0}\sin\rho\psi\,d\Delta(\psi). \]

We note that the last improper integral certainly converges for every function of the class \(A_\rho\) ((\(^{4}\), p. 498)).

In conclusion we give two theorems concerning functions of finite degree, i.e., functions of order not exceeding one and of finite type.

Definition 5. A function analytic in \(\operatorname{Im} z\geqslant 0\), whose zeros \(z_n=r_n e^{i\theta_n}\) satisfy the condition (see (\(^{1}\), p. 289))
\[ \sum_{n=1}^{\infty}\frac{\sin\theta_n}{r_n}<\infty, \]
is called a function of the class \(A\).

Theorem 9. In order that a function \(f(z)\), regular and of finite degree in \(\operatorname{Im} z\geqslant 0\), belong simultaneously to the classes \(A\) and \(A_1\), it is necessary and sufficient that the limits
\[ \lim_{r\to\infty}\frac{1}{r}\int_{1}^{r}\frac{\ln|f(-x)|}{x}\,dx=l_2,\qquad \lim_{r\to\infty}\frac{1}{r}\int_{1}^{r}\frac{\ln|f(x)|}{x}\,dx=l_1 \]
(the left and right boundary densities) exist, and that the integral
\[ \int_{1}^{\infty}\frac{\ln|f(x)f(-x)|}{x^2}\,dx \tag{14} \]
converge.

Theorem 10. In order that a function, regular and of finite degree in \(\operatorname{Im} z\geqslant 0\), belong simultaneously to the classes \(A\) and \(\overline{A}_1\), it is necessary and sufficient that the integral (14) converge and that the condition
\[ h_f(0)+h_f(\pi)=0. \]
be satisfied.

It must be noted that, for the case of sufficiency, Theorem 10 was established earlier (see (\(^{1}\), p. 317)).

In conclusion I express my deep gratitude to Prof. F. D. Gakhov, who supervised the present work.

Novocherkassk Polytechnic Institute

Received
1 IV 1966

REFERENCES

\(^{1}\) B. Ya. Levin, Distribution of Zeros of Entire Functions, Moscow, 1956.
\(^{2}\) B. Ya. Levin, Matem. sborn., 2 (44), 6, 1097 (1937).
\(^{3}\) A. Pfluger, Comm. Math. Helv., 12, 25 (1939).
\(^{4}\) N. V. Govorov, DAN, 162, No. 3, 495 (1965).