Abstract Generated abstract
This paper extends results of Levin and Pfluger on completely regular growth from entire functions of noninteger order to functions analytic in a half-plane. It introduces a notion of order and indicator for functions in an angle, defines complete regularity in open and closed half-planes, and develops corresponding density notions for interior zeros and boundary contributions. Using a canonical representation for half-plane analytic functions, the paper proves necessary and sufficient conditions for membership in classes of completely regular growth in terms of argument-boundary density, angular density, and two-sided boundary density. It also derives integral formulas expressing the indicator through these densities, including separate contributions from interior zeros and boundary behavior.
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MATHEMATICS
N. V. GOVOROV
ON THE INDICATOR OF FUNCTIONS OF NON-INTEGER ORDER, ANALYTIC AND OF COMPLETELY REGULAR GROWTH IN A HALF-PLANE
(Presented by Academician S. N. Bernstein, 8 XII 1964)
In the works of B. Ya. Levin \(({}^{1}),\) Chs. I, III; \(({}^{2})\) and A. Pfluger \(({}^{3})\), the equivalence was established between the existence of an angular density for the set of zeros of an entire function of non-integer order and the complete regularity of its growth (for definitions of these concepts see \(({}^{1})\), pp. 118, 182). In this connection a formula was found expressing the indicator of such a function in terms of the angular density of the set of its zeros, and conversely.
In the present article, instead of entire functions, functions regular in a half-plane are considered. For them, in a special way, the concept of order and analogues of the concept of angular density are introduced, characterizing the complete regularity of growth of the function; moreover, a formula for the indicator of these functions is derived.
Definition 1. Let the function \(f(z)\) be regular in \(\alpha < \arg z < \beta\) and, for some \(\mu > 0\), satisfy the asymptotic estimate
\[ \sup_{|z|\le r,\ \alpha<\arg z<\beta} |f(z)|<\exp(r^\mu); \tag{1} \]
denote by \(\{\mu\}\) the set of all such \(\mu\). Let, furthermore, \(\{\nu\}\) be the set of all such values \(\nu>0\) for which
\[ \overline{\lim_{r\to\infty}}\, \ln |f(re^{i\theta})|/r^\nu \equiv 0 \quad (\alpha<\theta<\beta)^*. \tag{2} \]
Then the number \(\rho=\max(\inf\{\mu\},\inf\{\nu\})\) shall be called the order of the function \(f(z)\) inside the angle \(\alpha<\arg z<\beta\), and the function
\[ h_f(\theta)=\overline{\lim_{r\to\infty}}\, \ln |f(re^{i\theta})|/r^\rho \tag{3} \]
shall be called its indicator.
If (1) is not fulfilled for any \(\mu>0\), then \(f(z)\) is called a function of infinite order.
A definition equivalent to this one is given in \(({}^{4})\), p. 209. For an entire function (with \(0\le \arg z\le 2\pi\)) it coincides with the commonly accepted one:
\[ \rho=\overline{\lim_{r\to\infty}}\, \ln\ln M_f(r)/\ln r, \qquad \text{where } M_f(r)=\max_{|z|=r}|f(z)|. \]
Let \(f(z)\) be regular and of finite order in \(\operatorname{Im} z>0\). Then from (1) it follows \((({}^{5}),\) pp. 66, 83) that it has, almost everywhere for \(-\infty<t<\infty\), finite angular boundary values, and the limiting function \(\ln|f(t)|\) is summable on every finite interval \(-r\le t\le r\).
A function \(f(z)\) of order \(\rho>0\) inside the angle \(\alpha<\arg z<\beta\) shall be called a function of finite type if, asymptotically,
\[ \sup_{|z|\le r,\ \alpha<\arg z<\beta} \ln |f(z)|<K r^\rho, \qquad \text{where } K=\mathrm{const}. \]
Definition 2. A function \(f(z)\), regular and of order \(\rho>0\) in the angle \(\alpha<\arg z<\beta\), continuous in \(\alpha\le \arg z\le \beta\), is called a func-
\[ \text{} \]
* It can be proved that (2) is certainly valid for \(\nu>\max(\mu,\pi/(\beta-\alpha))\).
a function of completely regular growth in the closed angle \(\alpha\leq \arg z\leq \beta\), if the function \(r^{-\rho}\ln |f(re^{i\theta})|\) tends to \(h_f(\theta)\) uniformly in \(\theta\), as \(r\) tends to \(+\infty\), except possibly on a set \(E\), common for all \(\theta\), of zero relative measure ((1), pp. 127, 182).
We shall call \(f(z)\) a function of completely regular growth in the open angle \(\alpha<\arg z<\beta\), if it is of finite type in \(\alpha<\arg z<\beta\) and has completely regular growth in every angle \(\alpha+\varepsilon\leq \arg z\leq \beta-\varepsilon\) \((\varepsilon>0)\).
Denote by \(A_\rho\) (by \(\overline{A}_\rho\)) the class of functions, regular in \(\operatorname{Im} z>0\), having order \(\rho\) and completely regular growth in the open angle \(0<\arg z<\pi\) (in the closed angle \(0\leq \arg z\leq \pi\)). Obviously, \(\overline{A}_\rho\subset A_\rho\).
Theorem 1. Every function, regular and of order \(\rho\geq 0\) in \(\operatorname{Im} z>0\), is representable in the form
\[ \begin{aligned} f(z)=&\exp\left[i(a_0+a_1z+\cdots+a_qz^q)\right]\times\\ &\exp\left\{\frac{1}{\pi i}\left[\int_{-\infty}^{\infty} \frac{(tz+1)^{q+1}\ln |f(t)|}{(t^2+1)^{q+1}(t-z)}\,dt +\int_{-\infty}^{\infty} \frac{(tz+1)^{q+1}}{(t^2+1)^q(t-z)}\,d\varphi(t)\right]\right\} \prod_{|z_n|<1}\frac{z-z_n}{z-\overline{z}_n}\times\\ &\times \prod_{|z_n|\geq 1} \frac{ \left(1-\dfrac{z}{z_n}\right) \exp\left[\dfrac{z}{z_n}+\dfrac{1}{2}\left(\dfrac{z}{z_n}\right)^2+\cdots+\dfrac{1}{q}\left(\dfrac{z}{z_n}\right)^q\right] }{ \left(1-\dfrac{z}{\overline{z}_n}\right) \exp\left[\dfrac{z}{\overline{z}_n}+\dfrac{1}{2}\left(\dfrac{z}{\overline{z}_n}\right)^2+\cdots+\dfrac{1}{q}\left(\dfrac{z}{\overline{z}_n}\right)^q\right] }, \end{aligned} \tag{4} \]
where \(q=[\rho]\); \(a_k\) are real constants; \(z_n\) are the internal, i.e., lying in \(\operatorname{Im} z>0\), zeros of \(f(z)\); \(\varphi(t)\) is a real nondecreasing function for which almost everywhere \(\varphi'(t)=0\), and the integrals \(\displaystyle \int_{-\infty}^{-1}\frac{d\varphi(t)}{t^q}\) and \(\displaystyle \int_1^\infty\frac{d\varphi(t)}{t^q}\) converge.
The theorem follows from previously known results ((7), p. 189, (6)). We now take an arbitrary function \(f(z)\), regular and of order \(\rho>0\) in \(\operatorname{Im} z>0\), and, in the notation of Theorem 1, put
\[ \tau(t)= \begin{cases} \displaystyle \frac{1}{2\pi}\int_1^t \frac{\ln |f(x)|}{x}\,dx+ \frac{1}{2\pi}\int_0^t x\,d\varphi(x), & t>1,\\[2ex] \displaystyle \frac{1}{2\pi}\int_t^{-1}\frac{\ln |f(x)|}{|x|}\,dx+ \frac{1}{2\pi}\int_t^0 |x|\,d\varphi(x), & t<-1; \end{cases} \tag{5} \]
\[ c(r,\eta_1,\eta_2)= \sum_{\substack{\eta_1<\arg z_n\leq \eta_2\\ 1\leq |z_n|\leq r}} \sin\arg z_n,\qquad r>1. \]
Next, compose the function
\[ a(r,\eta_1,\eta_2)= \begin{cases} c(r,\eta_1,\eta_2), & 0<\eta_1<\eta_2<\pi,\\ c(r,0,\eta_2)-\tau(r), & 0=\eta_1<\eta_2<\pi,\\ c(r,\eta_1,\pi)-\tau(-r), & 0<\eta_1<\eta_2=\pi,\\ c(r,0,\pi)-\tau(r)-\tau(-r), & \eta_1=0,\ \eta_2=\pi; \end{cases} \tag{6} \]
\[ a(r,\eta_1,\eta_2)=-a(r,\eta_2,\eta_1)\quad \text{for } \eta_1>\eta_2;\qquad a(r,\eta,\eta)\equiv 0. \]
Definition 3. If, for a function \(f(z)\), regular and of order \(\rho\) in \(\operatorname{Im} z>0\), for all \(\eta_1,\eta_2\in(0\leq \eta\leq \pi)\setminus N\), where \(N\) is at most countable and does not contain the 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 \(\operatorname{Im} z>0\).
The concept of argument-boundary density is closely connected with the concept of angular density introduced by B. Ya. Levin \(({}^{1}, \text{p. }118)\):
\[ \lim_{r\to\infty}\frac{1}{r^\rho} \sum_{\substack{\eta_1<\arg z_n\leq \eta_2\\ 1\leq |z_n|\leq r}} 1 = \Delta(\eta_1,\eta_2). \tag{7} \]
We note that the quantity \(c(r,0,\pi)\) was considered by R. Nevanlinna \(({}^{6})\).
Theorem 2. In order that a function, regular of noninteger order \(\rho\) and finite type in \(\operatorname{Im} z>0\), belong to the class \(A_\rho\), it is necessary and sufficient that the set of its zeros have an argument-boundary density.
Theorem 3. If \(f(z)\in A_\rho\) and \(\rho>0\) is noninteger, then the indicator of the function \(f(z)\) is expressed by the formula
\[ h_f(\theta) = \frac{\pi}{\sin \pi\rho} \int_0^\pi g(\psi,\theta)\,d\lambda(\psi), \qquad 0<\theta<\pi, \]
where \(\lambda(\psi)\equiv \lambda(0,\psi)\) is the argument-boundary density of the zeros of \(f(z)\) and
\[ g(\psi,\theta)= \begin{cases} \dfrac{1}{\sin\psi}\,[\cos\rho(|\theta-\psi|-\pi)-\cos\rho(\theta+\psi-\pi)], & 0<\psi<\pi,\\[6pt] 2\rho\sin\rho(\theta-\pi), & \psi=0,\\ -2\rho\sin\rho\theta, & \psi=\pi. \end{cases} \tag{8} \]
We now introduce concepts characterizing separately the asymptotic distribution of boundary and interior zeros of a function in \(\operatorname{Im} z\geq 0\).
Definition 4. 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 (5). Then the limits
\[ \lim_{r\to+\infty}\frac{\tau(r)}{r^\rho}=l_1, \qquad \lim_{r\to+\infty}\frac{\tau(-r)}{r^\rho}=l_2, \tag{9} \]
if they exist and are finite, are called, respectively, the right- and left-sided boundary densities of the set of zeros of the function \(f(z)\). If in (9) upper limits are taken, then we shall call them upper boundary densities \((l_1^*\text{ and }l_2^*)\).
Definition 5. Let in \(\operatorname{Im} z>0\) a set of points \(\{z_n\}\), \(n=1,2,\ldots\), be given, 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 a finite limit
\[ \lim_{r\to\infty} c(r,\eta_1,\eta_2)/r^\rho = \mu(\eta_1,\eta_2), \tag{10} \]
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 (10) an upper limit is taken).
Theorem 4. If \(f(z)\) is a function of the class \(A_\rho\), \(\rho>0\), then: 1) its upper boundary densities are finite, and the upper argument density is bounded; 2) its indicator can be expressed by the formula \((0<\theta<\pi)\)
\[ h_f(\theta) = \frac{\pi}{\sin \pi\rho} \left[ \int_0^\pi g(\psi,\theta)\,d\mu^*(\psi) - 2\rho l_1^* \sin\rho(\theta-\pi) + 2\rho l_2^* \sin\rho\theta \right], \]
where \(\mu^*(\psi)=\mu^*(0,\psi)\), and \(g(\psi,\theta)\) is defined by relation (8).
The argument density of zeros of a function of the class \(A_\rho\) exists if and only if there exists for it a two-sided boundary density. For functions of the class \(\bar A_\rho\), however, there holds
* If \(\rho\geq 1\), then 1 is true for any function of finite type in \(\operatorname{Im} z<0\).
Theorem 5. In order that a function \(f(z)\) of noninteger order \(\rho>0\) and finite type in \(\operatorname{Im} z>0\) belong to the class \(\bar A_\rho\), it is necessary and sufficient that the set of its zeros have an angular density, continuous at \(\psi=0\) and \(\psi=\pi\), and a two-sided limiting density satisfying the equalities
\[ \lim_{r\to+\infty}\frac{\ln |f(r)|}{r^\rho}=\rho l_1,\qquad \lim_{r\to+\infty}\frac{\ln |f(-r)|}{r^\rho}=\rho l_2. \]
The indicator of the function \(f(z)\) is expressed by the formula \((0\leq \theta\leq \pi)\)
\[ h_f(\theta)=\frac{\pi}{\sin \pi\rho}\left[ \int_0^\pi g(\psi,\theta)\,d\mu(\psi) -2\rho l_1\sin\rho(\theta-\pi)+2\rho l_2\sin\rho\theta \right]. \]
Let us dwell further on the properties of the angular density (7).
Theorem 6. If \(f(z)\in A_\rho\), \(\rho>0\), and \(\Delta(\psi)\) is the angular density of the set of zeros of \(f(z)\), then the Stieltjes integral
\[ \int_0^\pi \sin\psi\,d\Delta(\psi) = \lim_{\delta,\varepsilon\to+0} \int_\delta^{\pi-\varepsilon}\sin\psi\,d\Delta(\psi) \]
(generally speaking, improper) converges.
Theorem 7. If \(f(z)\in \bar A_\rho\) and \(\rho\) is noninteger, then the formula holds
\[ h_f(\theta)=\frac{1}{\sin\pi\rho} \left\{ \pi\int_0^\pi [\cos\rho(|\theta-\psi|-\pi)-\cos\rho(\theta+\psi-\pi)]\,d\Delta(\psi) \right. \]
\[ \left. -h_f(0)\sin\rho(\theta-\pi)+h_f(\pi)\sin\rho\theta \right\} \qquad (0\leq \theta\leq \pi). \]
In conclusion I express my deep gratitude to Prof. F. D. Gakhov, who supervised the present work, and to Prof. B. Ya. Levin for valuable comments.
Novocherkassk
Polytechnic Institute
Received
6 XII 1964
CITED LITERATURE
- B. Ya. Levin, Distribution of Zeros of Entire Functions, Moscow, 1956.
- B. Ya. Levin, Matem. sborn., 2 (44), 6, 1097 (1937).
- A. Pfluger, Comm. Math. Helv., 12, 25 (1939).
- E. Titchmarsh, Theory of Functions, Moscow–Leningrad, 1951.
- I. I. Privalov, Boundary Properties of Analytic Functions, Moscow–Leningrad, 1950.
- R. Nevanlinna, Acta Soc. Sci. Fenn., 50, No. 12 (1925).
- K. Hoffman, Banach Spaces of Analytic Functions, Moscow, 1963.