UDC 517.55

MATHEMATICS

L. S. MAERGOIZ

ON SCALES OF GROWTH OF ENTIRE FUNCTIONS OF SEVERAL VARIABLES

(Presented by Academician M. A. Lavrent'ev on 4 XI 1969)

Below we use the following notation: \(R^n\) is \(n\)-dimensional Euclidean space; \(r=(r_1,\ldots,r_n)\), \(u=(u_1,\ldots,u_n)\), etc.; \(R_+^n=\{r\in R^n: r_i\geq 0\}\), \(R_0^n=\{r\in R^n: r_i>0\}\); \(\varphi(r)=\varphi(r_1,\ldots,r_n)\); \(W(e^u)=W(e^{u_1},\ldots,e^{u_n})\); \(\varphi(r^\gamma)=\varphi(r_1^{\gamma_1},\ldots,r_n^{\gamma_n})\);

\[ |r|=\left(\sum_1^n r_i^2\right)^{1/2}; \qquad \|r\|=\sum_{i=1}^n r_i; \qquad (k,u)=\sum_{i=1}^n k_i u_i; \]

\((C,V(u))\) is a pair consisting of a function \(V(u)\) and a set \(C\), where \(V(u)\) is defined and finite; \(\{(u,u_{n+1})\in R^{n+1}: u\in C,\ u_{n+1}\geq V(u)\}=[R^n,V(u)]\) is the epigraph of the function \(V(u)\).

\(1^\circ\). The growth of an entire function \(f(z)=f(z_1,\ldots,z_n)\) of \(n\) complex variables \(z_1,\ldots,z_n\) is very often compared with the growth of its majorant

\[ M_f(r)=\max_{|z_i|\leq r_i}|f(z)|. \]

Typical for investigation is the class \(\mathfrak M_n=\{f(z)\}\) of entire functions transcendental in at least one variable:

\[ \mathfrak M_n=\{f(z):0<\gamma_f \stackrel{\mathrm{def}}{=}\varlimsup_{t\to+\infty}(\ln t)^{-1}\ln\ln M(t,\ldots,t)<+\infty\}. \]

For \(n=1\), \(\mathfrak P_1=\{\exp\{r^\gamma\},\ \gamma>0\}\) is the widely known scale of growth of functions of the class \(\mathfrak M_1\). For each function \(f(z)\in\mathfrak M_1\) in the scale \(\mathfrak P_1\) there exists a unique function \(\exp\{r^{\gamma_f}\}\) asymptotically equivalent to \(M_f(r)\) in the known sense, and the quantity \(\gamma_f\) is then called the order of growth of the function \(f(z)\).

The theory of growth of entire functions of several variables, over its more than half-century history, beginning with the work of E. Borel \((^1)\), has accumulated many analogues of the scale \(\mathfrak P_1\) and corresponding analogues of the notion of order of an entire function of one variable. As a scale of growth of functions of the class \(\mathfrak M_n\), functions of the form \(\{\exp\{[\varphi(r)]^\gamma\},\ \gamma>0\}\) are chosen, where \(\varphi(r)\) is a function of relatively simple structure, connected with a definite method of exhausting \(R_+^n\) (for example, for E. Borel \((^1)\), \(\varphi(r)=\max\{r_1,\ldots,r_n\}\); for P. Lelong \((^2)\), \(\varphi(r)=|r|\)), or systems of such functions (for example, for L. I. Ronkin \((^3)\), \(\gamma=1\), \(\varphi(r)=r_1^{\gamma_1}+\cdots+r_n^{\gamma_n}\), \(\gamma_i>0\)). Here results of a very general character are due to A. A. Gol'dberg \((^4)\).

Let \(G\) be a closed bounded domain in \(R_\perp^n\). By the \((G,x)\)-order of an entire function \(f(r)\) with respect to the collection of variables we shall call the quantity

\[ \rho_G(x)=\varlimsup_{t\to+\infty}(\ln t)^{-1}\ln\ln M(t), \quad \text{where } M(t)=\sup_{c\in G} M_f(t^{x_1c_1},\ldots,t^{x_nc_n}), \]

and any element of the set

\[ S_f=\{\gamma\in R^n:\rho_G(\gamma_1^{-1},\ldots,\gamma_n^{-1})=1\} \tag{1} \]

will be called a system of conjugate \(G\)-orders of the function \(f(z)\) (cf. \((^3)\)). Recall that \((R_0^n,\rho_G(x))\) does not depend on \(G\): \(\rho_G(x)\equiv\rho(x)\) \((^4)\). Suppose additionally that \(G\in\mathfrak B\), where \(\mathfrak B\) is the class of closed bounded complete logarithmically convex domains in \(R_+^n\). Then the above definitions lead to scales of growth of the function \(M_f(r)\), consisting of some-

functions in \(R_+^n\), nondecreasing in each of the variables \(r_1,\ldots,r_n\), convex with respect to \(\ln r_1,\ldots,\ln r_n\). It is precisely these properties that the function \(\ln M_f(r)\) possesses \((^5)\).

In the present paper we investigate the growth scales of A. A. Gol'dberg. Starting from this analysis and developing results on the asymptotics of convex functions in \((^6)\), we propose qualitatively new growth scales for functions of the class \(\mathfrak M_n\). These scales completely take into account certain asymptotic properties of the function \(M_f(r)\).

\(2^\circ\). Let \(H=\{\varphi(r)\}\) be the class of functions in \(R^n\) such that: 1) \(\varphi(r)>0\) for \(r\in R_+^n\setminus 0\); 2) \(\varphi(\lambda r)=\lambda\varphi(r)\) for \(\forall\lambda>0,\ r\in R_+^n\); 3) \(\varphi(r)\) is nondecreasing in each variable; 4) \(\varphi(r)\) is continuous in \(R_+^n\); 5) \(\varphi(r)\) is convex with respect to \(\ln r_1,\ldots,\ln r_n\) for \(r\in R_0^n\); \(W_f(r)=\ln\ln M_f^+(r)\), \(M_f^+(r)=\max\{M_f(r),e\}\) *.

Theorem 1. Let \(x\) be an arbitrary element of \(R_0^n\). In order that the number \(a\ge 0\) be the \(G\)-order of an entire function \(f(z)\) for some \(G\in\mathfrak B\), it is necessary and sufficient that there exist a function \(\varphi(r)\in H\) such that

\[ a=\lim_{\|r\|\to+\infty} W_f(r)[\ln\varphi(r^\gamma)]^{-1}, \tag{2} \]

where \(\gamma_i=x_i^{-1},\ i=1,\ldots,n\), and

\[ G=\{r\in R_+^n:\ \varphi(r^\gamma)\le 1\}. \]

The definitions of orders according to A. A. Gol'dberg therefore lead to the following system of growth scales for functions of the class \(\mathfrak M_n\): \(E=\{E(\varphi;\gamma),\ \varphi\in H,\ \gamma\in R_0^n\}\), where \(E(\varphi;\gamma)=\{\exp[(\varphi(r^\gamma))^\tau],\ \tau>0\}\), and for every function \(f(z)\) from \(\mathfrak M_n\) there exists, in each scale \(E(\varphi;\gamma)\), a unique function
\[ a(r;\gamma)=\exp\{[\varphi(r^\gamma)]^{\rho(x)}\}, \]
where \(x_i=\gamma_i^{-1},\ i=1,\ldots,n\); \(\rho(x)\) is the \(x\)-order of the function \(f(z)\), asymptotically equivalent to \(M_f(r)\) in the following sense:

\[ \lim_{\|r\|\to+\infty} W_f(r)\cdot[\ln\ln a(r;\gamma)]^{-1}=1. \]

\(3^\circ\). In studying the asymptotics of the growth of functions of the class \(\mathfrak M_n\), a natural object of investigation is the quasiconvex function \((^7)\)
\[ V(u)=W_f(e^u) \]
(i.e.,
\[ V(\lambda u+\mu v)\le \max\{V(u),V(v)\},\quad \forall\lambda+\mu=1;\ \lambda,\mu>0;\ u,v\in R^n). \]

Definition 1. Let \(\Omega_V=\{K\}\) be the collection of cones with vertex at \(O\in R^{n+1}\), some shifts of which belong to the epigraph \([R^n,V(u)]\) of the quasiconvex function \((R^n,V(u))\). The asymptotic cone \(\Pi(V)\) of the epigraph \([R^n,V(u)]\) is the set
\[ \overline{\bigcup_{K\in\Omega_V} K} \]
**.

Let us introduce the following characteristic of growth \((^6)\), p. 584):

Definition 2. The function of growth orders of an entire function \(f(z)\) is the function \((D,\rho_f(u))\), where

\[ \rho_f(u)=\lim_{t\to+\infty} W_f(e^{u_1t},\ldots,e^{u_nt})\,t^{-1},\qquad D=\{u\in R^n:\ \rho_f(u)<+\infty\}. \]

1) If \(f\in\mathfrak M_n\), then \(D=R^n\), and \(\Pi(W_f)=[R^n,\rho_f(u)]\), where \(\Pi(W_f)\) is the asymptotic cone of the epigraph of the function \(\widetilde W_f(e^u)\).

2) For every \(\varepsilon>0\) there exists a number \(C_\varepsilon>0\) such that

\[ \ln M_f(e^u)<C_\varepsilon\exp\{\rho_f(u)+\varepsilon |u|\},\qquad \forall u\in R^n. \]

Thus the function \((R^n,\rho_f(u))\) completely determines the cone \(\Pi(W_f)\), takes into account the growth of \(M_f(r)\) in all directions (for each fixed \(u\in R^n\), \(\rho_f(u)\) is the order of growth of the function \(\Phi_u(r)=M_f(r^{u_1},\ldots,r^{u_n})\)); with the aid of \(\rho_f(u)\) a simple global upper estimate of the function \(M_f(r)\) is possible,

* In studying the asymptotics of the growth of \(M_f(r)\), it is enough to restrict oneself to its truncation from below.

** Simple examples show that it is not always the case that \(\Pi(V)\in\Omega_V\). For convex functions this definition passes over into the previously known one \((^8)\).

i.e., \(\rho_f(u)\) is the growth characteristic of the function \(f(z)\). If \(n=1\), then \(\rho_f(u)=\max\{0,\rho u\}\), where \(\rho\) is the order of growth of \(f(z)\), i.e., the cone \(\Pi(W_f)\) is determined by one number—an element of \(R_0^1\). For \(n>1\) this is not true in the general case. The cone \(\Pi(W_f)\) is determined by the directrix \(T_f\) of its surface, situated in the horizontal hyperplane \(u_{n+1}=1\): \(T_f=\{u\in R^n:\rho_f(u)=1\}\). The hypersurface \(S_f\) of conjugate orders (formula (1)) determines only part of the cone \(\Pi(W_f)\cap R^{n+1}_+\), since \(S_f=T_f^- \cap R_0^n\), where \(T_f^-\) is the hypersurface obtained from \(T_f\) by the transformation \(y_i=u_i^{-1}\), \(i=1,\ldots,n\).

\(4^\circ\). There is a simple connection between the asymptotic cone \(\Pi(W_f)\) of the epigraph of the function \(W_f(e^u)\), \(f\in \mathfrak M_n\), and the asymptotic cone \(\Pi(\Theta_\gamma^+)\) for \(\forall\,\gamma\in R_0^n\), where \(\Theta_\gamma^+(u)=\max\{\ln\ln\alpha(r;\gamma),0\}\) (see \(2^\circ\)).

Proposition 1. The cone \(\Pi(\Theta_\gamma^+)\) is the intersection of a finite number of half-spaces, and \(\Pi(\Theta_\gamma^+)\subset \Pi(W_f)\) for \(\forall\,\gamma\in R_0^n\), and the cone \(\Pi(\Theta_\gamma^+)\) touches the surface of the cone \(\Pi(W_f)\) along the ray
\[ \{(t\gamma_1^{-1},\ldots,t\gamma_n^{-1},t\rho_f(\gamma_1^{-1},\ldots,\gamma_n^{-1})),\ t\ge 0\}. \]

If \(n>1\), then from the preceding and from the results of L. I. Ronkin \((^3)\), the author \((^9)\) concludes that the directrix of the cones \(\{\Pi(W_f)\cap R_+^{n+1},\ f\in \mathfrak M_n\}\), taken in the hyperplane \(u_{n+1}=1\), may be any closed convex complete domain in \(R_+^n\). Therefore the structure of the asymptotic cones of the epigraphs \(\{[R^n,W_f(e^u)],\ f\in \mathfrak M_n\}\) is much more complicated than that of the epigraphs of functions of systems \(\{\ln\ln\psi(e^u),\ \psi\in E(\varphi;\gamma)\},\ \varphi\in H,\ \gamma\in R_0^n\).

\(5^\circ\). Let \(Y\) be the class of functions in \(R^n\) that are nonnegative, convex, nondecreasing in each variable, and positively homogeneous (of degree 1). In addition, assume that the function \(\varphi_0(u)\equiv 0\notin Y\). Consider the following scale of growth \(Q_n\) of functions of the class \(\mathfrak M_n\): \(Q_n=\{\exp(\exp(\tilde\varphi(r))),\ \varphi\in Y\}\), where \(\tilde\varphi(r)\) is the continuous extension to \(R_+^n\) of the function \((R_0^n,\varphi(\ln r_1,\ldots,\ln r_n))\). For \(n=1\), \(Q_1=\{\exp\{\max\{r^\gamma,1\}\},\ \gamma>0\}\) (cf. with \(\mathfrak B_1,1^\circ\)).

The asymptotic properties of the order functions \(\rho_f(u)\), \(f\in \mathfrak M_n\), are clarified by

Theorem 2. If \(f(z)\) is an arbitrary function from \(\mathfrak M_n\), then
\[ \overline{\lim}_{|u|\to+\infty,\ u\in K} W_f(e^u)\cdot [\rho_f(u)]^{-1}=1 \tag{3} \]
for any cone \(K\) with vertex at \(O\) and such that \(\overline K\setminus\{0\}\subset \{u\in R^n:\rho_f(u)>0\}\), and \(\exp(\exp(\mathfrak F_f(r)))\) is the unique function of the scale \(Q_n\) asymptotically equivalent to \(M_f(r)\) in the sense of condition (3).

Theorem 3. For any function \(\varphi(u)\) of the class \(Y\) there exists an entire function \(f(z)\in\mathfrak M_n\) such that \(\rho_f(u)\equiv \varphi(u)\).

The desired function is, for example,
\[ f(z)=\sum_{s=0}^{+\infty}\ \sum_{\|k\|=s}\exp\{-p_s(k)\ln s\}\,z_1^{k_1}\ldots z_n^{k_n}. \]

Here \(p_s(u)\) is the Minkowski functional \((^{10})\) of the set \(O_{s^{-1/2}}(K_\varphi)\); \(O_\varepsilon(M)\) is the \(\varepsilon\)-neighborhood of the set \(M\); \(K_\varphi\) is the convex compact set whose support function is \(\varphi(u)\). If \(\dim K_\varphi=n\), then there also exists a simpler desired function:
\[ f(z)=\sum_{k\in C}\exp\{p(k)\ln\|k\|\}\,z_1^{k_1}\ldots z_n^{k_n}. \]

Here \(p(u)\) is the Minkowski functional of \(K_\varphi\); \(C\) is the smallest convex cone with vertex at \(O\) containing \(K_\varphi\). (Note that \(O\in K_\varphi\subset R_+^n\).) The idea of the example just given was suggested by the results of G. Valiron \((^5)\) and

* The example of the function \((z_1+z_2)e^{z_1z_2}\) shows that this requirement is essential for the fulfillment of condition (3).

L. I. Ronkina (³). In the proof of Theorem 3, the following result, which is of independent interest, is used.

Theorem 4. For all \(u \in R^n\), \(\rho_f(u) \equiv \tau_f(u)\), \(\forall f \in \mathfrak M_n\), where

\[ \tau_f(u)= \begin{cases} \displaystyle \overline{\lim}_{(k,u)\to+\infty}\frac{(k,u)\ln(k,u)}{-\ln|a_k|}, & u\in A,\\[6pt] 0, & u\in R^n\setminus A, \end{cases} \]

\(\{a_k=a_{k_1\ldots k_n};\ k_1,\ldots,k_n=0,1,\ldots\}\) are the Taylor coefficients of the function \(f(z)\),

\[ A=\bigcap_{m>0} A_m,\qquad A_m=\{u\in R^n:\{k:a_k\ne0\}\cap\{k:(k,u)>m\}\ne\varnothing\}. \]

Theorem 4 is a generalization of a well-known result of A. A. Goldberg (⁴).

In conclusion, the author expresses gratitude to L. A. Aizenberg and S. G. Gindikin for valuable comments.

Institute of Physics
Siberian Branch of the Academy of Sciences of the USSR
Krasnoyarsk

Received
23 IX 1969

REFERENCES

¹ E. Borel, Leçons sur les séries à termes positifs. Paris, 1902.
² P. Lelong, Ann. Sci. École Norm. Sup., 58 (1941).
³ L. I. Ronkina, Ukr. Mat. Zh., 14, No. 3 (1964).
⁴ A. A. Goldberg, Dokl. i Soobshch. Uzhgorod. Univ. Ser. Fiz.-Mat. Nauk, 4 (1961).
⁵ G. Valiron, Bull. Sci. Math., 47, No. 1 (1923).
⁶ L. S. Maergoiz, Siberian Math. Zh., 9, No. 3 (1968).
⁷ B. A. Vertgeim, G. Sh. Rubinshtein, in: Mathematical Programming, Moscow, 1966.
⁸ R. T. Rockafellar, Trans. Am. Math. Soc., 123, No. 1 (1966).
⁹ L. S. Maergoiz, Siberian Math. Zh., 7, No. 6 (1966).
¹⁰ A. D. Ioffe, V. M. Tikhomirov, Uspekhi Mat. Nauk, 23, 6 (144).