F. I. GECHE

ON CHARACTERISTICS OF THE GROWTH OF ENTIRE FUNCTIONS OF SEVERAL COMPLEX VARIABLES

(Presented by Academician S. N. Bernstein on 25 V 1965)

For simplicity we shall restrict ourselves to the case of two complex variables; the case of an arbitrary finite number of variables is treated in the same way.

In studying the growth of entire functions of two complex variables, the space \(C^2\) is usually exhausted either by the one-parameter family of surfaces
\(S(R): |z_1|^\alpha + |z_2|^\alpha = R^\alpha,\ \alpha > 0\), or by the two-parameter family
\(S(r_1, r_2): \{|z_1| = r_1\} \times \{|z_2| = r_2\}\). In the first case, the growth of a function \(f(z_1,z_2)\) is measured by means of the function \(\ln M(R)\), where \(M(R)\) denotes the maximum modulus of the function \(f(z_1,z_2)\) on \(S(R)\), and one obtains global characteristics of growth—order and type. In the second case one compares the growth of the function \(\ln M(r_1,r_2)\) with the growth of \(\sigma_1 r_1^{\rho_1} + \sigma_2 r_2^{\rho_2}\), where

\[ M(r_1,r_2)= \max_{(z_1,z_2)\in S(r_1,r_2)} |f(z_1,z_2)|, \]

and arrives at the definition of a system of conjugate orders (s.c.o.) and a system of conjugate types (s.c.t.) \((^1)\).

For a more complete characterization of the growth of an entire function of one complex variable the notion of refined order is used \((^2)\). It carries over automatically to the case of functions of two variables if their growth is measured by means of \(\ln M(R)\). Less trivial is the problem of studying refined orders when the space \(C^2\) is exhausted by the family of surfaces \(S(r_1,r_2)\).

Let an entire function \(f(z_1,z_2)\) of finite order \(\rho\) have s.c.o. \((\rho_1,\rho_2)\). As a system of conjugate refined orders (s.c.r.o.) it is natural to choose systems of functions \((\rho_1(r_1,r_2), \rho_2(r_1,r_2))\) possessing certain properties. We propose the following

Definition 1. A system of nonnegative continuous functions \((\rho_1(r_1), \rho_2(r_2))\), defined for \(0 \le r_i < \infty,\ i=1,2\), will be called an s.c.r.o. if: 1) \(\rho_i(r_i)\) is a differentiable function of the variable \(r_i\), except, perhaps, for isolated points at which one-sided derivatives exist; 2)

\[ \lim_{r_i\to\infty}\rho_i(r_i)=\rho_i<\infty; \]

3)

\[ \lim_{r_i\to\infty} r_i \rho_i'(r_i)\ln r_i = 0 \quad (i=1,2). \]

Definition 2. Let \((\rho_1(r_1), \rho_2(r_2))\) be an s.c.r.o. If for the entire function \(f(z_1,z_2)\) there exists a system of positive numbers \((\sigma_1,\sigma_2)\) that satisfies the condition

\[ \lim_{r_1+r_2\to\infty} \frac{\ln M(r_1,r_2)} {\sigma_1 r_1^{\rho_1(r_1)}+\sigma_2 r_2^{\rho_2(r_2)}}=1, \]

then the system of functions \((\rho_1(r_1), \rho_2(r_2))\) is called an s.c.r.o. of the function \(f(z_1,z_2)\), and the system \((\sigma_1,\sigma_2)\) is called the system of conjugate types (s.c.t.) corresponding to the s.c.r.o. \((\rho_1(r_1), \rho_2(r_2))\).

The existence of an s.c.r.o. for any entire function \(f(z_1,z_2)\) follows from the following theorem, which generalizes Theorem 16 of Ch. I in \((^2)\).

Theorem 1. Let the entire function \(f(z_1,z_2)\) have s.c.o. \((\rho_1,\rho_2)\) \((\rho_i<\infty,\ i=1,2)\). Then there exists an s.c.r.o. \((\rho_1(r_1), \rho_2(r_2))\) of the function \(f(z_1,z_2)\), which satisfies the conditions: 4)
\[ \ln M(r_1,r_2) \le r_1^{\rho_1(r_1)} + r_2^{\rho_2(r_2)} \]
for \(r_1+r_2>R_0\); 5) on some sequence \(\{(r_1^{(k)}, r_2^{(k)})\}\)

\[ (r_1^{(k)}+r_2^{(k)}\to\infty) \]
the equality
\[ \ln M\bigl(r_1^{(k)},r_2^{(k)}\bigr) =(r_1^{(k)})^{\rho_1(r_1^{(k)})}+(r_2^{(k)})^{\rho_2(r_2^{(k)})} \]
holds.

On the other hand, it can be shown that for any s.s.r.o. \((\rho_1(r_1),\rho_2(r_2))\) there exists an entire function \(f(z_1,z_2)\) for which \((\rho_1(r_1),\rho_2(r_2))\) is an s.s.r.o. In this case it is necessary to require that when \(\rho_i(r_i)\to 0\), \(\rho_i(r_i)\ne0\), \(i=1,2\), for at least one of the values \(i=1,2\) condition 6) hold:
\[ \lim_{r_i\to\infty} r_i^{\rho_i(r_i)}(\ln r_i)^{-1}>0. \]

Obviously, the s.s.r.o. \((\rho_1(r_1),\rho_2(r_2))\) of the function \(f(z_1,z_2)\) is not determined uniquely. In some cases it may be more convenient to use another definition of s.s.r.o., which is obtained if in definitions 1 and 2 the system of functions \((\rho_1(r_1),\rho_2(r_2))\) is replaced by the system \((\rho_1(R),\rho_2(R))\), where \(\rho_i(R)=\rho_i(r_1+r_2)\), as functions of the variable \(R\), possess properties 1)—3). For an s.s.r.o. \((\rho_1(R),\rho_2(R))\) Theorem 1 also holds under one additional condition. Namely, it is required that for an entire function \(f(z_1,z_2)\) having s.r.o. \((\rho_1,\rho_2)\) and not depending on the variable \(z_1(z_2)\), the system \((\rho_1,\rho_2')\) with \(\rho_2'<\rho_2\) \(( (\rho_1',\rho_2)\) with \(\rho_1'<\rho_1)\) not be an s.r.o.

In what follows, when considering the s.s.r.o. \((\rho_1(r_1),\rho_2(r_2))\) \(( (\rho_1(R),\rho_2(R)))\), we assume that \(\rho_i(r_i)\to\rho_i>0\) \((\rho_i(R)\to\rho_i>0)\), \(i=1,2\), without mentioning this specially.

As is known (see (2)), the function \(t_i^{\rho_i(r_i)}\) increases monotonically for all sufficiently large values of \(r_i\). Denote the inverse function by \(\chi_i(t_i)\) \((t_i>t_i^0)\) and extend it to the interval \([0,t_i^0]\), setting it equal to a certain positive constant. The following analogue of Theorem 2, Ch. I from (2), establishes a connection between s.s.r.o., s.t., and the coefficients of the expansion
\[ f(z_1,z_2)=\sum_{k,l=0}^{\infty} a_{kl} z_1^k z_2^l . \tag{1} \]

Theorem 2. In order that the positive numbers \(\sigma_1,\sigma_2\) constitute an s.t. for an s.s.r.o. \((\rho_1(r_1),\rho_2(r_2))\) of the function \(f(z_1,z_2)\), it is necessary and sufficient that
\[ \lim_{k+l\to\infty}\sqrt[k+l]{\,|a_{kl}|\,\chi_1(k)^k\chi_2(l)^l (e\sigma_1\rho_1)^{-k/\rho_1}(e\sigma_2\rho_2)^{-l/\rho_2}\,}=1. \tag{2} \]

For \(\rho_i(r_i)\equiv\rho_i\), \(i=1,2\), this theorem implies the known relation for s.s.r.o. and s.t., established by L. I. Ronkin ((1), p. 390).

An analogous theorem holds when considering the s.s.r.o. \((\rho_1(R),\rho_2(R))\). The role of the functions \(\chi_i(t_i)\) in this case is played by the functions \(r_i=\chi_i(t_1,t_2)\), \(i=1,2\), which are the unique solution of the system of equations \((r_1+r_2\ge R_0)\)
\[ t_1=r_1^{\rho_1(r_1+r_2)},\qquad t_2=r_2^{\rho_2(r_1+r_2)}. \]

In this case equality (2) is replaced by the following:
\[ \lim_{k+l\to\infty}\sqrt[k+l]{\,|a_{kl}|\,\chi_1(k,l)^k\chi_2(k,l)^l (e\sigma_1\rho_1)^{-k/\rho_1}(e\sigma_2\rho_2)^{-l/\rho_2}\,}=1. \]

Hence the following assertion follows: whatever the s.s.r.o. \((\rho_1(R),\rho_2(R))\) may be, there exists an entire function \(f(z_1,z_2)\) for which this system is an s.s.r.o. It can be shown that this assertion is also true in the case when only one of the functions \(\rho_1(R),\rho_2(R)\) tends to zero as \(R\to\infty\), or both tend to zero, but condition 6) is then satisfied for them.

Using Theorem 2, one can prove the following theorem, analogous to Theorem 26.4 from (1).

Theorem 3. In order that a system of positive numbers \((\sigma_1,\sigma_2)\) be an s.c.t. for an s.c.r.o. \((\rho_1(r_1),\rho_2(r_2))\) of some entire function \(f(z_1,z_2)\), it is necessary and sufficient that the curve with coordinates \((\ln \sigma_1,\ln \sigma_2)\) constitute the boundary of some convex quadrant-like* domain \(D\).

An analogous theorem is valid for an s.c.r.o. \((\rho_1(R),\rho_2(R))\).

Following V. K. Ivanov \((^{3,4})\), we give the following

Definition 3. Let \(f(z_1,z_2)\) be an entire function with s.c.r.o. \((\rho_1(r_1),\rho_2(r_2))\). By \(T(\varphi)=T(\varphi_1,\varphi_2)\), \(0\leq \varphi_1,\varphi_2\leq 2\pi\), we shall denote the set of points \((\nu_1,\nu_2)\) of the real plane for which there exists a constant \(A=A(\varphi_1,\varphi_2,\nu_1,\nu_2)\) such that for all nonnegative \(r_1,r_2\) the inequality

\[ \left|f\left(r_1 e^{i\varphi_1},\, r_2 e^{i\varphi_2}\right)\right| \leq A\exp\left[\nu_1 r_1^{\rho_1(r_1)}+\nu_2 r_2^{\rho_2(r_2)}\right]. \tag{3} \]

is satisfied.

The boundary of the set \(\overline{T}(\varphi)\) is a generalization of the concept of the indicator of an entire function of one variable to the case of two variables. From Definition 3 it follows that for any \(\varphi_i=\mathrm{const}\) \((i=1,2)\) \(\overline{T}(\varphi)\) is convex and quadrant-like. For a function of one variable the basic property of the indicator is trigonometric convexity. A certain analogous assertion also holds in the case of two variables.

Denote by \(K(\varphi)=K(\varphi_1,\varphi_2)\) the set of points of the real plane \((\nu_1,\nu_2)\) satisfying the condition \(\nu_1\geq k_1(\varphi_1)\), \(\nu_2\geq k_2(\varphi_2)\), where \(k_i(\varphi_i)\) is defined by the equality

\[ k_i(\varphi_i)= \frac{ h_i^{(1)}\sin \rho_i\left(\varphi_i^{(2)}-\varphi_i\right) + h_i^{(2)}\sin \rho_i\left(\varphi_i-\varphi_i^{(1)}\right) }{ \sin \rho_i\left(\varphi_i^{(2)}-\varphi_i^{(1)}\right) }, \]

\(h_i^{(1)}, h_i^{(2)}\) are certain constants \((i=1,2)\).

Theorem 4 (cf. \((^{2})\), p. 96). Let \(f(z_1,z_2)\) be an entire function with s.c.r.o. \((\rho_1(r_1),\rho_2(r_2))\), and let \(T(\varphi)\) and \(K(\varphi)\) be defined as above. If the point with coordinates \((h_1^{(i)},h_2^{(i)})\) belongs to \(\overline{T}(\varphi_1^{(i)},\varphi_2^{(i)})\) for fixed \((\varphi_1^{(i)},\varphi_2^{(i)})\), \(0\leq \varphi_i^{(2)}-\varphi_i^{(1)}\leq \pi/\rho_i\), \(i=1,2\), then for all \((\varphi_1,\varphi_2)\), \(\varphi_i^{(1)}\leq \varphi_i\leq \varphi_i^{(2)}\), \(i=1,2\), the inclusion \(\overline{K}(\varphi)\subset \overline{T}(\varphi)\) holds.

The well-known Borel–Pólya theorem \((^{2})\), p. 114; \((^{5})\), p. 171) asserts that the conjugate diagram of an arbitrary entire function of finite degree is the mirror image in the real axis of its indicator diagram. This theorem was generalized by M. F. Subbotin \((^{6})\) and V. Bernstein \((^{7,8})\) to entire functions of normal type of order \(\rho\), \(0<\rho<\infty\), where instead of the indicator \(h(\varphi)\) the function \(\max(0,h(\varphi))\) appears. A. A. Avetisyan \((^{9})\) and L. F. Lokhin \((^{10})\) independently found this same result. A generalization of the Borel–Pólya theorem to functions of many variables of finite degree was obtained by V. K. Ivanov \((^{3,4})\) and M. Sh. Stavskii \((^{11})\), and to functions with s.c.t. \((\sigma_1,\sigma_2)\) for s.o. \((\rho_1,\rho_2)\), where \(\rho_i\geq 1/2\), \(0<\sigma_i<\infty\), \(i=1,2\), by L. I. Ronkin \((^{12})\) (see also \((^{31})\)). V. Bernstein \((^{7,8})\) also considered a generalization of this theorem for an entire function of one variable with a refined order \(\rho(r)\). We shall give here a generalization of his result to the case of an entire function \(f(z_1,z_2)\) having s.c.r.o. \((\rho_1(r_1),\rho_2(r_2))\).

With an entire function \(f(z_1,z_2)\) represented in the form of the series (1), we associate two functions

\[ F_1(z_1,z_2)= \sum_{k,l=0}^{\infty} \frac{a_{kl}\gamma_1(k)\gamma_2(l)} {z_1^{k+1}z_2^{l+1}}, \qquad F_2(z_1,z_2)= \sum_{k,l=0}^{\infty} \frac{a_{kl}\beta_1(k)\beta_2(l)} {z_1^{k+1}z_2^{l+1}}, \]

* That is, if \((x,y)\in D\), then also \((x',y')\in D\), \(x'\geq x\), \(y'\geq y\).

where

\[ \beta_i(t)=\int_0^\infty x_i^t \exp[-W_i(x_i)]\,dx_i,\qquad \gamma_i(t)=\Gamma[(t+\alpha_i)\omega_i(t+\alpha_i)]; \]

\(W_i(z_i)\) is an analytic function for \(|\arg z_i|<\pi/\rho_i\) such that
\(W_i(r_i e^{i\varphi_i})\sim e^{i\varphi_i} r_i^{\rho_i}\rho_i(r_i)\) as \(r_i\to\infty\),
\(\omega_i(x)=\ln\chi_i(x)/\ln x\), where \(\chi_i(x)\) is the function inverse to \(W_i(x_i)\) for \(x>C=\mathrm{const}\) and equal to some positive constant for \(x\le C\); \(\alpha_i(\ge \alpha_i^0)\) is an arbitrary constant; \(\alpha_i^0\) is a constant depending only on the proximate order \((\rho_1(r_1),\rho_2(r_2))\) (cf. (8)).

Denote by \(C_i(\varphi)=C_i(\varphi_1,\varphi_2)\), \(i=1,2\), the set of points \((\nu_1,\nu_2)\) \((\nu_1,\nu_2>0)\) of the real plane for which the function \(F_i(z_1,z_2)=F_i(r_1e^{i\theta_1},r_2e^{i\theta_2})\) is analytic inside the set
\(\{r_j^{\rho_j}\cos\rho_j(\theta_j-\varphi_j)>\nu_j,\ |\theta_j-\varphi_j|\le \min(\pi/2\rho_j,\pi),\ j=1,2\}\).

By \(T^+(\varphi)\) we denote the intersection of the set \(T(\varphi)\) with the first quadrant of the plane.

Theorem 5. For every entire function \(f(z_1,z_2)\) having proximate order \((\rho_1(r_1),\rho_1(r_2))\), the inclusion

\[ \overline{C}_1(-\varphi)\subset \overline{T}^{+}(\varphi)\subset \overline{C}_2(-\varphi) \tag{4} \]

holds.

If \(\rho_i(r_i)\equiv \rho_i,\ i=1,2\), then for \(\alpha_i=\rho_i\)

\[ F(z_1,z_2)=F_1(z_1,z_2)=F_2(z_1,z_2)= \sum_{k,l=0}^{\infty} \frac{a_{kl}\Gamma(1+k/\rho_1)\Gamma(1+l/\rho_2)} {z_1^{k+1}z_2^{l+1}}, \]

and consequently \(C_1(\varphi)=C_2(\varphi)=C(\varphi)\). In this case the following holds.

Theorem 6. For every entire function \(f(z_1,z_2)\) having type \((\sigma_1,\sigma_2)\) with order \((\rho_1,\rho_2)\) \((0<\rho_i<\infty,\ 0<\sigma_i<\infty,\ i=1,2)\), the sets \(\overline{C}(-\varphi)\) and \(\overline{T}^{+}(\varphi)\) coincide.

We note that for \(\rho_1=\rho_2=1\), in the definition of the set \(C(\varphi)\) the requirement of positivity of \(\nu_1,\nu_2\) can be removed and the equality \(\overline{C}(-\varphi)=T(\varphi)\) obtained (see (4)).

The question of the possibility of associating with an entire function a single function so that, instead of the inclusion (4), equality would hold, remains open in the general case. Under certain assumptions concerning the proximate order \((\rho_1(r_1),\rho_2(r_2))\), analogous to conditions considered by V. Bernstein, the question is answered affirmatively.

In conclusion I express my deep gratitude to A. A. Goldberg for valuable remarks.

Lviv State University
named after Ivan Franko

Received
25 V 1965

CITED LITERATURE

1. B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables, 1962.
2. B. Ya. Levin, Distribution of Zeros of Entire Functions, 1956.
3. V. K. Ivanov, Mat. sborn., 47, No. 1, 131 (1959).
4. V. K. Ivanov, Mat. sborn., 43, No. 3, 367 (1957).
5. E. Borel, Séries divergentes, Paris, 1928.
6. M. F. Subbotin, Math. Ann., 104, 377 (1931).
7. V. Bernstein, C. R., 194, 1887 (1932).
8. V. Bernstein, Mem. Reale Accad. Ital., cl. sc. fis., mat. e. natur., 4, 338 (1933).
9. A. A. Avetisyan, DAN, 105, No. 5 (1955).
10. I. F. Lokhin, Uch. zap. Gorky Univ., ser. phys.-mat., 28, 20 (1955).
11. M. Sh. Stavskii, Izv. vyssh. uchebn. zaved., Matem., No. 2, 227 (1959).
12. I. I. Ronkin, DAN, 153, No. 2 (1963).
13. N. I. Kobeleva, Izv. vyssh. uchebn. zaved., Matem., No. 3, 59 (1962).