Entire Functions of Several Variables: Wiman–Valiron-Type Results Without Exceptional Sets

Abstract

This article is devoted to establishing analogs of the main theorems of the Wiman–Valiron theory on asymptotic relations for entire functions of several complex variables that are satisfied without exceptional sets when the complex space is exhausted by a system of multiple-circular domains or by a system of A-like polylinear domains. These results concern: (i) analogs of the asymptotic equality of the logarithms of the maximum modulus of an entire function on the boundary of multiple-circular domains of exhaustion and the corresponding maximal term; (ii) a statement describing the behavior of the entire function $F\left(z\right)$ of several complex variables $z=(z_1,\dots ,z_p)$ in the vicinity of the point w, where the value of $F\left(w\right)$ is close to the supremum of its modulus on the boundary of polylinear domains; (iii) the so-called fundamental relation of the Wiman–Valiron theory, which establishes the asymptotic behavior of directional derivatives for a given fixed direction of the vector A, which specifies the direction of space exhaustion.

1. Introduction

The study of entire functions of several complex variables occupies a central position in modern complex analysis and is deeply connected with functional analysis, probability theory, and differential equations. Foundational contributions by Bitlyan and Gol’dberg established the first multidimensional formulations of the Wiman–Valiron theory for entire functions of several variables [1], while the classical monograph of Ronkin provided a comprehensive analytic and geometric framework for this theory in ${\mathbb{C}}^p$ [2]. Subsequent multidimensional advances [3,4], including the development of diagonal maximal term techniques [5] and convergence classes for analytic functions in the Reinhardt domains [6], extended methods for studying growth phenomena in several complex variables.

The origins of the Wiman–Valiron theory date back to the foundational works of Hadamard and Valiron [7], as well as classical inequalities by Pólya and Szegö [8]. Later investigations by Wittich [9] and Hayman [10] described local growth of entire functions of one complex variable in terms of Wiman–Valiron’s theory, suggested applications to differential equations, and motivated the pursuit of higher-dimensional analogs. Many results in transcendental function theory, including the growth of solutions of differential equations, rely fundamentally on these classical insights [11]. Likewise, the early contributions of Borel [12] on the relationship between maximum modulus and maximal term initiated a research direction on Borel-type relations for Dirichlet series and entire functions.

A substantial body of subsequent works deepened this direction. Skaskiv established the existence of exceptional sets in Borel-type relations for entire Dirichlet series [13], while Sheremeta clarified conditions for full equivalence (i.e., without exceptional sets) between the logarithms of the maximum modulus and maximal term [14]. More recent progress includes multidimensional analogs of Wiman’s theorem [15], generalizations of Picard-type theorem [16], and Wiman–Valiron methods for Dirichlet series [17,18]. Additional results concern absolutely convergent Dirichlet series [19] and the asymptotic properties of analytic solutions of differential equations [20].

Extensions of Wiman-type theorems to multiple Dirichlet series were later obtained in [21]. The behavior of directional derivatives of entire functions outside some exceptional sets, crucial in the multidimensional setting, was further explored in [22]. Classical tools such as those presented in the problem compendium of Pólya and Szegö [23] continue to play an important supporting role. Lévy’s probabilistic insights into the growth of entire functions [24,25], together with the probabilistic perspectives of Erdös and Réǹyi [26] and multidimensional approaches from [27], influenced recent approaches to growth descriptions, including those in [28,29].

The multidimensional Wiman–Valiron theory has undergone further refinement. Recent progress includes new inequalities in multiple-circular domains [30], analyses of Lévy-type phenomena on polydiscs [31], and results for functions with rapidly oscillating coefficients [32]. Important advances have also been made in understanding exceptional sets and their asymptotic densities [33,34]. Additional geometric investigations of Wiman–Valiron discs appear in [35,36], with connections to dynamical systems and Julia sets in [37]. Related developments include results for meromorphic functions in the unit disc [38], applications of complex delay-differential equations [39], variants of logarithmic derivative estimates [40], and analyses of maximal terms of general series in systems of functions [41]. Furthermore, Wiman-type inequalities for multiple Dirichlet series with arbitrary complex exponents were recently established in [42], while detailed studies of maximum modulus points and zero sets continue to appear in [43]. Non-power-series approaches to the Wiman–Valiron theory were proposed in [44], and refined multidimensional formulations were developed in [45].

In various applications (in particular, in the analytical theory of differential equations), the so-called Ostrovsky problem of establishing an implicit description of the size of exceptional sets in a certain asymptotic relation of the theory comes to the fore. Despite significant progress, in the case of many relations, this problem remains open or far from being completely solved. In particular, the following question remains open: under what natural assumptions can the basic Wiman–Valiron asymptotic relations be guaranteed without exceptional sets? The formulation of classical results in the case of entire functions of one variable does not guarantee the absence of exceptional sets. For many variables, this issue becomes even more delicate, where the geometry of multiple-circular and polylinear domains leads to strongly anisotropic growth behavior. We note that one-dimensional results concerning asymptotic behavior without exceptional sets of derivatives of entire functions were obtained in [46,47].

It was a certain surprise for the authors that it is possible to describe the conditions for the fulfillment of the main asymptotic relations without exceptional sets in terms of the directional logarithmic derivative of the maximum modulus of an entire function on the boundary of the exhaustion domain. We develop an improved analytical structure that ensures the validity of the main Wiman–Valiron asymptotic relations everywhere, without exceptional sets, for entire functions of several complex variables bounded in the corresponding polylinear domains. By synthesizing the methods of multidimensional complex analysis, the geometry of polylinear domains, and the asymptotic structure of directional derivatives, we obtain new results that significantly extend the classical Wiman–Valiron theory and clarify the global behavior of entire functions of several complex variables.

Papers [15,22] are devoted to asymptotic estimates in Wiman–Valiron’s theory with some exceptional sets. In the current paper, we select a subclass of entire multivariate functions satisfying similar estimates for all values of radius $r\to +\infty .$

In mathematical analysis, in the broad sense of the concept, as well as in the applications of mathematical analysis to other branches of mathematics, the following problem quite often arises (which we formulate here for the case of real functions of a single real variable): given a known relation (very often, an asymptotic one) between two functions, it is required to obtain a similar relation between their derivatives. The statements obtained in this way are usually called Tauberian-type theorems. This is in contrast to statements that indicate conditions under which relations between the functions themselves can be deduced from relations between their derivatives—these are the so-called Abelian-type statements. A characteristic example of such statements is the well-known elementary theorems referred to in mathematical analysis courses as L’Hôpital’s rules. Statements of the first type are customarily called converse L’Hôpital rules in this context. In the case of complex-valued functions, even for holomorphic functions of a single variable, the situation becomes significantly more complicated. In particular, it is well known that the main asymptotic relations in the classical Wiman–Valiron theory hold outside of exceptional sets. This latter circumstance very often acts as a factor that limits (sometimes very severely) the applicability of the main results of this theory. Since even for the general class of entire functions the presence of an exceptional set is an essential feature, several research directions have emerged. The first direction is to find an unimprovable (sharp) description of the size of the exceptional set (I. Ostrovskii’s problem [24,43]), or at least, whenever possible, to find the most complete description of such sets [22]. Another aspect of this problem—finding and describing the conditions under which exceptional sets are absent—is examined in this article with respect to certain fundamental relations in specific classes of entire functions of several variables. Regarding the main relation for the class of entire functions of a single variable, some approaches to this latter problem were developed by Gorenflo [46], London [47], and Dubei [34]. The conditions in these theorems are of a different nature in a certain sense. However, Gorenflo’s result follows from Dubei’s theorem. We emphasize that the class of entire functions satisfying the conditions of Gorenflo’s theorem is obviously non-empty, and therefore, the class of functions satisfying the conditions of Dubei’s theorem is also non-empty. This article relies on the idea from Dubei’s paper, which consists of considering classes of entire functions for whose characteristics certain relations from the Borel–Nevanlinna lemma hold. We should note that the necessity for Borel–Nevanlinna-type relations is, on the one hand, a determining factor for obtaining asymptotic relations in the Wiman–Valiron theory. On the other hand, for all positive functions of regular growth in a certain sense, the exceptional set is completely absent in the relations from the Borel–Nevanlinna lemma ($u\left(x\right)=e^x,u\left(x\right)=\exp \left\{\exp \left\{x\right\}\right\},$ etc.). Therefore, the question of whether the considered classes are empty, or whether such classes are extremely small, should not even arise.

2. Notations

We use the following standard notation [1,5]. Let ${\mathbb{C}}^p$ be the p-dimensional $(p\ge 1)$ complex vector space, ${\mathbb{Z}}_{+}=\mathbb{N}\cup \left\{0\right\},$ $z^n=z_1^{n_1}\dots z_p^{n_p},\parallel n\parallel =n_1+\cdots +n_p$ for $n=(n_1,\dots ,n_p)\in {\mathbb{Z}}_{+}^p$ and $z=(z_1,\dots ,z_p)\in {\mathbb{C}}^p$, ${\mathbb{R}}_{+}=[0,+\infty )$. Let $\lambda =\left({\lambda }_k\right)$ be a sequence such that $0 = {\lambda }_0 < {\lambda }_k\uparrow +\infty$ $(1\le k\uparrow +\infty ),$ $P_0\left(z\right) \equiv a_0\in \mathbb{C},$ and $P_k\left(z\right)={\sum }_{\parallel n\parallel = {\lambda }_k}{a}_n{z}^n$ be a homogeneous polynomial of degree ${\lambda }_k,$ that is $P_k\left(\tau z\right)={\tau }^{{\lambda }_k}{P}_k\left(z\right)$ for all $\tau \in \mathbb{C},$ $z\in {\mathbb{C}}^p$ $(k\in \mathbb{N})$. By ${\mathcal{E}}^p\left(\lambda \right)$, we denote the class of entire functions $f:{\mathbb{C}}^p\to \mathbb{C},$ (i.e., entire functions of p complex variables), represented by a power series of the form

$$\begin{array}{c}f\left(z\right)={\displaystyle \sum _{k=0}^{+\infty }}{P}_k\left(z\right),z\in {\mathbb{C}}^p.\end{array}$$

(1)

In the case ${\lambda }_k\equiv k(k\ge 0)$, we obtain the class ${\mathcal{E}}^p$ of all entire functions of p complex variables. Denote by ${\mathcal{E}}^1,$ ${\mathcal{E}}^1\left(\lambda \right)$ the classes of all entire functions of one complex variable and entire functions of one complex variable represented by gap power series of the form

$$\begin{array}{c}f\left(z\right)=a_0+{\displaystyle \sum _{k=1}^{+\infty }}{a}_k{z}^{{\lambda }_k},z\in \mathbb{C},\end{array}$$

(2)

respectively. According to [1] (see also [5,6]), we consider the one-parameter exhaustion of the space ${\mathbb{C}}^p$ by a system $\mathbb{G}={\left({\mathbf{G}}_r\right)}_{r\ge 0}$ of bounded complete multiple-circular domains ${\mathbf{G}}_r$ centered at the point $\mathbf{0}=(0,\dots ,0)\in {\mathbb{C}}^p$; that is

(1)

${\displaystyle \bigcup _{r\ge 0}}{\mathbf{G}}_r={\mathbb{C}}^p;$

(2)

$(\forall r_1<r_2):{\mathbf{G}}_{r_1}\subset {\mathbf{G}}_{r_2};$

(3)

$(z_1,\dots ,z_p)\in {\mathbf{G}}_1⟺(\forall r>0):(r{z}_1,\dots ,r{z}_p)\in {\mathbf{G}}_r;$

(4)

$(z_1,\dots z_p)\in {\mathbf{G}}_r⟹(\forall \theta =({\theta }_1,\dots ,{\theta }_p)\in {\mathbb{R}}^p):z{e}^{i\theta }=(z_1{e}^{i{\theta }_1},\dots ,z_p{e}^{i{\theta }_p})\in {\mathbf{G}}_r.$

For example, the following systems ${\left({\mathbf{G}}_r\right)}_{r\ge 0}$ of domains are contained in the class $\mathbb{G}$:

(i)

${\mathbf{G}}_r=C_{r,a} = \{(z_1,\dots ,z_p)\in {\mathbb{C}}^p:|z_j|<a_{j}r,1\le j\le p\}$;

(ii)

${\mathbf{G}}_r={\mathbb{B}}_{r,a} = \{(z_1,\dots ,z_p)\in {\mathbb{C}}^p:a_1|z_1{|}^2+\cdots +a_p|z_p{|}^2<r^2\}$;

(iii)

${\mathbf{G}}_r={\mathsf{\Pi }}_{r,a} = \{(z_1,\dots ,z_p)\in {\mathbb{C}}^p:a_1|z_1|+\cdots +a_p|z_p|<r\};$

(iv)

${\mathbf{G}}_r={\mathbf{G}}_{r,a}=\left\{(z_1,\dots ,z_p)\in {\mathbb{C}}^p: |z_1{|}^{a_1}·\dots ·{|z_2|}^{a_p}<r^{a_1+\cdots +a_p}\right\};$

where $a=(a_1,\dots ,a_p)$, $a_j>0$ $(1\le j\le p),$ $r>0.$ For $p=1$, the family of the disks ${\mathbb{D}}_r=\{z\in \mathbb{C}:|z|<r\},r>0,$ belongs to $\mathbb{G}$.

For $n=(n_1,\dots ,n_p)\in {\mathbb{Z}}_{+}^p$ and a domain ${\mathbf{G}}_1\in \mathbb{G}$ denote

$$d_n=d_n\left({\mathbf{G}}_1\right)=\max \left\{\right|z^n| = |z_1^{n_1}\cdots z_p^{n_p}|:z=(z_1,\dots ,z_p)\in {\overline{\mathbf{G}}}_1\}.$$

Remark,

$$\begin{array}{c}{d}_n=a_1^{n_1}·\dots ·a_p^{n_p}\mathrm{for}{\mathbf{G}}_1=C_{1,a},\\ d_n=\exp \left\{-1/2{\displaystyle \sum _{j=1}^p}{n}_j\ln \left(\parallel n\parallel /\left(e{n}_j\right)\right)\right\}\mathrm{for}{\mathbf{G}}_1=\mathbb{B}:={\mathbb{B}}_{1,a},a=(1,\dots ,1)\\ d_n=\exp \left\{-{\displaystyle \sum _{j=1}^p}{n}_j\ln \left(\parallel n\parallel /\left(e{n}_j\right)\right)\right\}\mathrm{for}{\mathbf{G}}_1={\mathsf{\Pi }}_1:={\mathsf{\Pi }}_{1,a},a=(1,\dots ,1).\end{array}$$

We denote by $\left|\mathbf{G}\right|$ the image in ${\mathbb{R}}_{+}^p$ of a multiple-circular domain $\mathbf{G}\subset {\mathbb{C}}^p$ at the mapping $r=(r_1,\dots ,r_p)=\left(\right|z_1|,\dots ,|z_p\left|\right):\mathbf{G}\to {\mathbb{R}}_{+}^p$ ([2], p. 38).

We call a set $E\subset {\mathbb{R}}_{+}^p$ a complete domain in ${\mathbb{R}}_{+}^p$ if it is a domain that has the following properties—$r^0=(r_1^0,\dots ,r_p^0)\in E$⟺ $\forall r=(r_1,\dots ,r_p)\in E,0\le r_j\le r_j^0(1\le j\le p)$—and does not contain the closure points of its complement in ${\mathbb{R}}_{+}^p$. For $r>0$ and an entire function $f\in {\mathcal{E}}^p\left(\lambda \right)$ of the form (1), we denote

$$\begin{array}{c}M(r,f)=\max \left\{\right|f\left(z\right)|:z\in {\overline{\mathbf{G}}}_r\},m_k(r,f)=M(r,P_k)(k\ge 0).\end{array}$$

For $r>0$ and an entire function $f\in {\mathcal{E}}^1\left(\lambda \right)$, we denote

$$M_f\left(r\right)=\max \left\{\right|f\left(z\right)|:|z|=r\},{\mu }_f\left(r\right)=\max \left\{\right|a_k|r^{{\lambda }_k}:k\ge 0\},$$

the maximum modulus and the maximal term of power series (2), respectively.

By the maximum modulus principle, there exists a point $s^{\left(k\right)}=(s_1^{\left(k\right)},\dots ,s_p^{\left(k\right)})\in \partial {\mathbf{G}}_1$ such that

$$|P_k\left(s^{\left(k\right)}\right)|=m_k(1,f).$$

The definition of ${\mathbf{G}}_r$ implies $z^{\left(k\right)}:=r{s}^{\left(k\right)}\in \partial {\mathbf{G}}_r.$ But $P_k\left(z\right)$ is a homogeneous polynomial, hence

$$|P_k\left(z^{\left(k\right)}\right)|=r^{{\lambda }_k}\left|P_k\left(s^{\left(k\right)}\right)\right|=m_k(r,f)(r>0,k\ge 0).$$

According to [1,5,6], we define the diagonal maximal term of the series (1)

$$m(r,f):=\max \{m_k(r,f):k\ge 0\}=\max \{r^{{\lambda }_k}{m}_k(1,f):k\ge 0\}.$$

It is easy to see that $m(r,f)\equiv {\mu }_f\left(r\right)$ for $p=1$.

It is well known that the main asymptotic relations in the Wiman–Valiron theory hold outside some exceptional sets (for example, see [9,10]). The questions arise, first, about the existence of examples of entire functions for which exceptional sets exist in the basic relations, and, second, under what additional conditions exceptional sets in one or another asymptotic relation exceptional sets are nevertheless absent. It is clear that the second question should be understood in the sense that it is necessary to find sufficiently broad classes of functions such that for each representative of this class, the exceptional set is absent.

3. Borel Relation

Let us analyze in this context some asymptotic relations from this theory. If $f\in {\mathcal{A}}^1\left(\lambda \right)$, that is, an entire function of the form (2) of finite order

$${\rho }_f:=\underset{r\to +\infty }{\overline{\lim }}{\displaystyle \frac{\ln \ln M_f\left(r\right)}{\ln r}}<+\infty ,$$

then it is well known that (see [8], Part IV, Ch. 1, § 3, Problem 54) the Borel relation

$$\ln M(r,f)\sim \ln m(r,f).$$

(3)

holds as $r\to +\infty$. This elementary fact is often attributed to E. Borel. Let us point out that E. Borel ([12], pp. 62–70) established (in the absence of the restriction ${\rho }_f<\infty$) that for each entire function, the relation (3) holds along some sequence $r_j\to +\infty$, i.e., as $r=r_j\to +\infty$.

From the results established for entire Dirichlet series in the articles [13], it follows that on the one hand, for any sequence $\lambda =\left({\lambda }_n\right)$, there exists a function $f\in {\mathcal{E}}^1\left(\lambda \right)$ such that ([13]) the Borel relation (3) cannot be satisfied even on some sequence $r=r_n\to +\infty$. On the other hand ([14]), let us recall another result by Sheremeta:

Lemma 1 

([14]). Let $f\in {\mathcal{E}}^1\left(\lambda \right)$ admit representation (2). Suppose that a function $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is such that $\psi \left(x\right)\to +\infty$ $(x\to +\infty )$; if the conditions $|a_k|\le \exp \{-{\lambda }_k\psi \left({\lambda }_k\right)\}$ $(k\ge k_0)$ and $\ln k=O\left(\psi \left({\lambda }_k\right)\right)$ are satisfied, then relation (3) holds as $r\to +\infty$.

Note that the just-formulated statement is correct [14,48] for an entire Dirichlet series of the form

$$F\left(z\right)=a_0+{\displaystyle \sum _{k=1}^{+\infty }}{a}_k{e}^{z{\lambda }_k},z\in \mathbb{C};$$

in exactly the same formulation; here, $\lambda =\left({\lambda }_k\right)$ is a sequence of positive real numbers such that ${\lambda }_k\uparrow +\infty$ $(k\uparrow +\infty ).$ At the same time, in article [48], it is proved that the condition $\ln n=O\left(\psi \left({\lambda }_n\right)\right)$ $(n\to +\infty )$ cannot be replaced by any weaker condition. For such an entire Dirichlet series satisfying $|a_k| \le \exp \{-{\lambda }_k\psi \left({\lambda }_k\right)\}$ $(k\ge k_0)$, a sequence of exponents $\left({\lambda }_k\right)$ was constructed, for which the condition $\ln n=O\left(\psi \left({\lambda }_n\right)\right)$ $(n\to +\infty )$ does not hold. However, it does not follow that a similar statement (counterexample) can be constructed with integer exponents ${\lambda }_n$ for at least one function $\psi$.

Denote

$$\begin{array}{c}{F}_1\left(R\right)={\displaystyle \sum _{k=0}^{\infty }}{A}_k{R}^{{\lambda }_k},A_k:={\displaystyle \sum _{\parallel n\parallel ={\lambda }_k}}\left|a_n\right|d_n\left({\mathbf{G}}_1\right)\\ F_2\left(R\right)={\displaystyle \sum _{k=1}^{\infty }}{B}_k{R}^{{\lambda }_k},F_3\left(R\right)={\displaystyle \sum _{k=1}^{\infty }}{B}_k{({\lambda }_k+1)}^p{R}^{{\lambda }_k},B_k=:\underset{\parallel n\parallel ={\lambda }_k}{\max }\left|a_n\right|d_n\left({\mathbf{G}}_1\right).\end{array}$$

(4)

We now state and prove the following results.

Proposition 1. 

If $f\in {\mathcal{E}}^p$ is a function of form (1) with a finite order of growth, then asymptotic relation (3) holds as $r\to +\infty$.

Let us denote by $n\left(u\right)={\displaystyle \sum _{{\lambda }_k\le u}}1$ the counting function of the sequence $\lambda =\left({\lambda }_k\right)$.

Proposition 2. 

Let $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a function such that $\ln x=O\left(\psi \right(x\left)\right)$ $(x\to +\infty )$ and $\lambda =\left({\lambda }_k\right)$ be a sequence such that $\ln n\left(u\right)=O\left(\psi \left(u\right)\right)$ $(u\to +\infty )$. If a function $f\in {\mathcal{E}}^p\left(\lambda \right)$ of form (1) satisfies the following condition

$$A_k\le \exp \{-{\lambda }_k\psi \left({\lambda }_k\right)\}(k\ge k_0)$$

then asymptotic relations (3) and

$$\ln F_1\left(r\right)\sim \ln F_2\left(r\right)\sim \ln m(r,f)$$

hold as $r\to +\infty$.

Proposition 3. 

Let $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a function such that $\ln x=O\left(\psi \right(x\left)\right)$ $(x\to +\infty )$ and $\lambda =\left({\lambda }_k\right)$ be a sequence such that $\ln n\left(u\right)=O\left(\psi \left(u\right)\right)$ $(u\to +\infty )$, where $n\left(u\right)$ is the counting function of the sequence $\lambda =\left({\lambda }_k\right)$. If a function $f\in {\mathcal{E}}^p\left(\lambda \right)$ of form (1) satisfies the following condition

$$|a_n|\le \exp \{-\parallel n\parallel \psi (\parallel n\parallel )\left\}\right(\parallel n\parallel \ge k_0)$$

then asymptotic relations (3) and

$$\ln F_1\left(r\right)\sim \ln F_2\left(r\right)\sim \ln m(r,f)$$

hold as $r\to +\infty$.

Proof of Proposition 2. 

We need the following lemma.

Lemma 2. 

Let $f\in {\mathcal{E}}^p\left(\lambda \right)$.

1⁰.

(see [2])

$${\displaystyle m(r,f)=\max \{M(1,P_k)R^{{\lambda }_k}:k\ge 0\}\le M(R,f)\le F_1\left(R\right)(R>0).}$$

(5)

2⁰.

$$B_k\le M(1,P_k)\le A_k\le B_k·{({\lambda }_k+1)}^p(k\ge 0).$$

Proof of 2⁰ in Lemma 2. 

Indeed, we consistently obtain

$$\begin{array}{c}M(1,P_k)\le {\displaystyle \sum _{\parallel n\parallel ={\lambda }_k}}|a_n\left|\max \right\{|z^n|:z\in \overline{{\mathbf{G}}_1}\}={\displaystyle \sum _{\parallel n\parallel ={\lambda }_k}}\left|a_n\right|d_n\left({\mathbf{G}}_1\right)=A_k\\ \le \underset{\parallel n\parallel ={\lambda }_k}{\max }\left|a_n\right|d_n\left({\mathbf{G}}_1\right)·{\displaystyle \sum _{\parallel n\parallel ={\lambda }_k}}1\le B_k·{({\lambda }_k+1)}^p.\end{array}$$

Let $h=(h_1,\dots ,h_p)\in \partial {\mathbf{G}}_1\cup {\mathbb{R}}_{+}^p$ be a given point. To prove the first inequality with $2^0$ by Cauchy’s integral formula for $P_k$, we have

$$a_n={\displaystyle \frac{1}{{\left(2\pi i\right)}^p}}\underset{C_{1,h}^{*}}{\int }{\displaystyle \frac{P_k\left(\tau \right)}{{\tau }^{n+\mathbf{1}}}}d\tau ,$$

where $\mathbf{1}=(1,\dots ,1)\in {\mathbb{R}}_{+}^p$, $C_{1,h}^{*}$ is the skeleton of a polydisk $C_{1,h},h=(h_1,\dots ,h_p).$ In other words, $C_{1,h}=\{(z_1,\dots ,z_p)\in {\mathbb{C}}^p:|z_j|<h_j,1\le j\le p\}$ is the polydisk with polyradius h and $C_{1,h}^{*}$ denotes its distinguished boundary.

By definition of the set system from $\mathbb{G}$, one has $C_{1,h}^{*}\subset \partial C_{1,h}\subset \partial {\mathbf{G}}_1,$ and thus $C_{1,h}\subset {\mathbf{G}}_1.$ Therefore,

$$\begin{array}{c}|a_n| \le {\displaystyle \frac{1}{{\left(2\pi \right)}^p}}\underset{\theta \in {[0,2\pi ]}^p}{\int }{\displaystyle \frac{\max \left\{\right|P_k\left(\tau \right)|:\tau \in \partial C_{1,h}\}}{h^{n+\mathbf{1}}}}{h}^{\mathbf{1}}d\theta \\ \le {\displaystyle \frac{\max \left\{\right|P_k\left(\tau \right)|:\tau \in \partial C_{1,h}\}}{h^n}}\le {\displaystyle \frac{\max \left\{\right|P_k\left(\tau \right)|:\tau \in \partial {\mathbf{G}}_1\}}{h^n}}={\displaystyle \frac{M(1,P_k)}{h^n}},\end{array}$$

hence,

$$|a_n|h^n\le M(1,P_k)=m_k(1,f)(\forall h=(h_1,\dots ,h_p)\in \partial {\mathbf{G}}_1\cup {\mathbb{R}}_{+}^p).$$

It remains to be seen that $\sup \{h^n:h\in \partial {\mathbf{G}}_1\cup {\mathbb{R}}_{+}^p\}=\sup \left\{\right|z^n|:z\in \overline{{\mathbf{G}}_1}\}.$ So, the first inequality with $2^0$ is proved.  □

Now let us continue proving Proposition 2. Since, by Lemma 2,

$$M(1,P_k)\le {\displaystyle \sum _{\parallel n\parallel ={\lambda }_k}}\left|a_n\right|d_n\left({\mathbf{G}}_1\right)=A_k,$$

the function $F_1$ satisfies the conditions of Lemma 1. Therefore,

$$\ln F_1\left(R\right)\sim \ln {\mu }_{F_1}\left(R\right)(R\to +\infty ).$$

(6)

Note,

$$m(r,f)=\max \{M(1,P_k)R^{{\lambda }_k}:k\ge 0\}={\mu }_{F_1}\left(R\right)(R>0).$$

Finally, by inequality (5) and relation (6), we obtain relation (3) and the relation

$$\ln F_1\left(r\right)\sim \ln m(r,f)$$

as $r\to +\infty$.

To prove the relation $\ln F_2\left(r\right)\sim \ln m(r,f)$ as $r\to +\infty$, we will need one auxiliary statement.

Let us consider the class $\mathcal{I}\left(\nu \right)$ of the functions $F:\mathbb{R}\to {\mathbb{R}}_{+}$ defined by the integral of the form

$$F\left(x\right)={\int }_{{\mathbb{R}}_{+}}a\left(u\right)e^{xu}\nu \left(du\right),$$

(7)

where $\nu$ is a countably additive measure on the $\sigma$-algebra $\mathcal{B}\left({\mathbb{R}}_{+}\right)$ of Borel sets on ${\mathbb{R}}_{+}$ (Borel measure) with unbounded support such that $\nu \left(\right\{x:0\le x\le b\left\}\right)<+\infty$ for any $b>0,$ $a:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a positive measurable function. Denote by $\mathrm{supp}\nu$ the support of the measure $\nu ,$ i.e., the closed set $E=:\mathrm{supp}\nu$ such that $\nu (\mathbb{R}\setminus E)=0$ and $\nu \left(\right\{u\in \mathbb{R}:|u-u_0| < r\})>0$ for any $u_0\in E$ and $r>0.$ For $x\in \mathbb{R}$ and $F\in \mathcal{I}\left(\nu \right)$, we set

$${\mu }_{*}(x,F)=\sup \{a\left(u\right)e^{xu}:u\in \mathrm{supp}\nu \}.$$

In article [49], we find the following statement.

Theorem 1 

([49]). Let $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a function such that $\psi \left(x\right)\uparrow +\infty$ $(x\to +\infty )$. If

$$\ln a\left(u\right)\le -u\psi \left(u\right)(u\ge u_0),\ln \nu \left([0,u]\right)=O\left(\psi \left(u\right)\right)(u\to +\infty ),$$

then for each function F of form (7)

$$\ln F\left(x\right)\le (1+o\left(1\right))\ln {\mu }_{*}(x,F)(x\to +\infty ).$$

Let us continue our proof. Define a continuous function $a\left(u\right):{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that

$$a\left({\lambda }_k\right)=B_k(k\ge 0),\ln a\left(u\right)\le -u\psi \left(u\right)(u\ge 0),{\mu }_{F_2}\left(e^x\right)={\mu }_{*}(x,F).$$

In particular, we can choose $a\left({\lambda }_k\right)=B_k$ $(k\ge 0)$ and $a\left(u\right)=0$ $(u\notin \{{\lambda }_k:k\ge 0\}).$ Denote

$$\nu \left(u\right)={\displaystyle \underset{0}{\overset{u}{\int }}}{(t+1)}^{p}dn\left(t\right).$$

We write the function $F_3$ in the form of an integral

$$F_3\left(e^x\right)={\displaystyle \underset{0}{\overset{+\infty }{\int }}}a\left(u\right)e^{xu}d\nu \left(u\right),$$

Then,

$$\begin{array}{c}{\mu }_{F_2}\left(e^x\right)\le F_2\left(e^x\right)\le F_3\left(e^x\right)={\displaystyle \sum _{k=0}^{+\infty }}{B}_k·{({\lambda }_k+1)}^p{e}^{x{\lambda }_k}={\int }_0^{+\infty }a\left(u\right)e^{ux}d\nu \left(u\right):=F\left(x\right).\end{array}$$

Since, by condition, $\ln \nu \left([0,u]\right)\le p\ln (u+1)+\ln n\left(u\right)\le O\left(\psi \left(u\right)\right)$ $(u\to +\infty )$, Theorem 1 implies

$$\begin{array}{c}\ln {\mu }_{F_2}\left(e^x\right)\le \ln F_2\left(e^x\right)\le \ln F_3\left(e^x\right)=\ln F\left(x\right)\le (1+o\left(1\right))\ln {\mu }_{*}\left(x\right)\\ =(1+o\left(1\right))\ln {\mu }_{F_2}\left(e^x\right)\end{array}$$

as $(x\to +\infty )$, hence,

$$\ln {\mu }_{F_2}\left(e^x\right)\sim \ln F_3\left(e^x\right)\sim \ln F_2\left(e^x\right)(x\to +\infty ).$$

(8)

Similarly, we have

$$F_1\left(e^x\right)={\displaystyle \underset{0}{\overset{+\infty }{\int }}}A\left(u\right)e^{xu}dn\left(u\right)$$

for some positive continuous function $A:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that $\ln A\left(u\right)\le -u\psi \left(u\right)$ $(u\ge 0)$, $A\left({\lambda }_k\right)=A_k$ $(k\ge 0)$ and ${\mu }_{*}(e^x,F_1)={\mu }_{F_1}\left(e^x\right)$. In particular, we can choose $A\left({\lambda }_k\right)=A_k$ $(k\ge 0)$ and $A\left(u\right)=0$ $(u\notin \{{\lambda }_k:k\ge 0\}).$ So, by Theorem 1, we get

$$\ln {\mu }_{F_1}\left(e^x\right)\le \ln F_1\left(e^x\right)\le (1+o\left(1\right))\ln {\mu }_{*}(e^x,F_1)=(1+o\left(1\right))\ln {\mu }_{F_1}\left(e^x\right)$$

as $x\to +\infty$. Therefore,

$$\ln F_1\left(e^x\right)\sim \ln \mu (e^x,F_1)(x\to +\infty ).$$

(9)

But, by $2^0$ in Lemma 2,

$$\ln \mu (e^x,F_2)\le \ln \mu (e^x,F_1)\le \ln \mu (e^x,F_3)\le (1+o\left(1\right))\ln \mu (e^x,F_2)$$

as $x\to +\infty$, i.e.,

$$\ln \mu (e^x,F_2)\sim \ln \mu (e^x,F_1)(x\to +\infty ).$$

Hence, using asymptotic relations (8) and (9), we obtain

$$\ln F_1\left(e^x\right)\sim \ln F_2\left(e^x\right)(x\to +\infty ).$$

The proof of Proposition 2 is complete.  □

Proof of Proposition 1. 

If ${\varrho }_f<+\infty$, then by the A.A. Gol’dberg theorem (see [2], Theorem 3.1.1, p.131), we have

$${\varrho }_f=\underset{\parallel n\parallel \to +\infty }{\overline{\lim }}{\displaystyle \frac{\parallel n\parallel \ln \parallel n\parallel }{-\ln |a_n|}},$$

hence,

$$\ln |a_n|\le -c\parallel n\parallel \ln \parallel n\parallel$$

(10)

for all n and some $c\in (0,+\infty ).$

1. 

Remark, in ([2], p. 20), it is noted that the image of a multiple-circular domain $\mathbf{G}\subset {\mathbb{C}}^p$ in ${\mathbb{R}}_{+}^p$ is a complete domain in ${\mathbb{R}}_{+}^p$.

2. 

For each pair $E^{\left(1\right)},$ $E^{\left(2\right)}$ of complete domains in ${\mathbb{R}}_{+}^p$, there exist $l>0,L>0$ such that ([2], p. 106)

$$l·E^{\left(1\right)}\subset E^{\left(2\right)}\subset L·G^{\left(1\right)}.$$

If we take $G^{\left(1\right)}=\left|C_{1,a}\right|$, $E^{\left(2\right)}=\left|{\mathbf{G}}_1\right|$ with $a=(1,\dots ,1)$ then we get that $(\exists l>0,\exists L>0):$

$$|C_{l,a}|=\{r:r_1<l,\dots ,r_p<l\}\subset |{\mathbf{G}}_1|\subset |C_{L,a}|=\{r:r_1<L,\dots ,r_p<L\}.$$

From 2., we obtain

$$\begin{array}{c}{d}_n\left({\mathbf{G}}_1\right)=\sup \left\{\right|z^n|:z\in {\mathbf{G}}_1\}=\sup \{r^n:r\in |{\mathbf{G}}_1\left|\right\}\le \sup \{r^n:r\in |C_{L,a}\left|\right\}\\ \le L^{\parallel n\parallel }\sup \left\{{\left({\displaystyle \frac{r}{L}}\right)}^n:r\in \left|C_{L,a}\right|\right\}=L^{\parallel n\parallel }\sup \left\{{\left({\displaystyle \frac{r}{L}}\right)}^n:{\displaystyle \frac{r}{L}}\in \left|C_{1,a}\right|\right\}\\ =L^{\parallel n\parallel }\sup \{{\tau }^n:\tau \in |C_{1,a}|\}=L^{\parallel n\parallel }.\end{array}$$

Applying now the last inequality to inequality (10), we have

$$\begin{array}{c}{A}_k=\sum _{\parallel n\parallel =k}\left(|a_n|d_n\left({\mathbf{G}}_1\right)\right)\le \sum _{\parallel n\parallel =k}\exp \{-c\parallel n\parallel \ln \parallel n\parallel +\parallel n\parallel \ln L\}\\ \le {(k+1)}^p·\exp \{-ck\ln k+k\ln L\}\le \exp \{-(c/2)k\ln k\}(k\ge k_0).\end{array}$$

(11)

Let us now note that for ${\lambda }_k\equiv k$ $(k\ge 0)$ one has

$$u\le n\left(u\right)=\sum _{{\lambda }_k\le u}1=\left[u\right]+1\le u+1$$

Now choose $\psi \left(x\right)=(c/2)\ln x$. Then, $\ln n\left(u\right)=O\left(\psi \right(u\left)\right)$ $(u\to +\infty )$, $\ln u=O\left(\psi \right(u\left)\right)$ $(u\to +\infty )$, and the inequality (11) means

$$A_k\le \exp \{-k\psi \left(k\right)\},$$

(12)

these together imply that the conditions of Proposition 2 are satisfied. Hence, in particular, we obtain the statement of Proposition 1. In some sense, Proposition 1 can be traited as a corollary of Proposition 2 with $\psi \left(x\right)=(c/2)\ln x$ and ${\lambda }_k=k$.  □

Proof of Proposition 3. 

Similarly to inequality (11), we obtain by conditions

$$\begin{array}{c}{A}_k=\sum _{\parallel n\parallel ={\lambda }_k}\left(|a_n|d_n\left({\mathbf{G}}_1\right)\right)\le \sum _{\parallel n\parallel ={\lambda }_k}\exp \{-c\parallel n\parallel \ln \parallel n\parallel +\parallel n\parallel \ln L\}\\ \le {({\lambda }_k+1)}^p·\exp \{-c{\lambda }_k\ln {\lambda }_k+{\lambda }_k\ln L\}\le \exp \{-(c/2){\lambda }_k\ln {\lambda }_k\}(k\ge k_0).\end{array}$$

(13)

That is, the conditions of Proposition 2 are satisfied. Applying Proposition 2 completes the proof.  □

4. Wiman’s Theorem

By Wiman’s theorem [10], for every non-identically constant entire function f of one complex variable $z\in \mathbb{C}$, there exists a set E of finite logarithmic measures (i.e., ${\int }_{E\cap [1;+\infty )}d\ln r<+\infty$) such that the relations

$$M_f\left(r\right)=(1+o\left(1\right))B_f\left(r\right)=-(1+o\left(1\right))C_f\left(r\right)$$

satisfy $r\to +\infty$ ($r\notin E$), where

$$\begin{array}{c}{B}_f\left(r\right)=\max \{\mathrm{Re}f\left(z\right):|z|=r\},C_f\left(r\right)=\min \{\mathrm{Re}f\left(z\right):|z|=r\}.\end{array}$$

For entire functions from the class ${\mathcal{E}}^p$, an analog of this theorem in the case of an exhaustion of the space ${\mathbb{C}}^p$ by a system $\mathbb{G}={\left({\mathbf{G}}_{r,a}\right)}_{r\ge 0}$ of bounded complete multiple-circular domains centered at the point $\mathbf{0}$ such that ${\mathbf{G}}_{r,a}=\left\{(z_1,\dots ,z_p)\in {\mathbb{C}}^p:|z_1{|}^{a_1}·\dots ·{\left|z_2\right|}^{a_p}<r^{\parallel a\parallel }\right\}$, follows from the results obtained in article [15]. Let for an entire function $f\in {\mathcal{E}}^p$ and $r>0$

$$\begin{array}{c}M(r,f)=\sup \left\{\right|f\left(z\right)|:z\in {\mathbf{G}}_{r,a}\},B(r,f)=\sup \{\mathrm{Re}f\left(z\right):z\in {\mathbf{G}}_{r,a}\},\\ C(r,f)=\inf \{\mathrm{Re}f\left(z\right):z\in {\mathbf{G}}_{r,a}\}.\end{array}$$

Then asymptotic relations

$$M(r,f)=(1+o(1\left)\right)B(r,f)=-(1+o(1\left)\right)C(r,f)$$

hold as $r\to +\infty$ outside some set of finite logarithmic measures.

Let us consider again the problem of establishing conditions under which the latter relations hold without exceptional sets. For convenience, we will move on to a somewhat more general situation. In fact, we will consider the class of entire functions of several complex variables bounded in complete polylinear domains. Such domains are, in particular, the domains ${\mathsf{\Pi }}_R:=\{z\in {\mathbb{C}}^p:\mathrm{Re}z<R\},$ $R\in {\mathbb{R}}^p.$

This section of the article is devoted to proving the one analog of Wiman’s theorem for entire functions of several complex variables bounded in polylinear domains. This theorem will be obtained as a consequence of the theorem describing the asymptotic behavior of the entire function $F\left(z\right)$ of several complex variables $z=(z_1,\dots ,z_p)$ in the neighborhood of the point w, where the value of $F\left(w\right)$ is close to the supremum of its modulus on the boundaries of polylinear domains. For entire multiple Dirichlet series with positive monotonically increasing exponents to infinity, we found one analog of Wiman’s theorem in the article [21]. The proof in [21] uses one result from [20]. Remark, an entire multiple Dirichlet series with positive monotonically increasing exponents to infinity is bounded in the polylinear domains ${\mathsf{\Pi }}_R,$ $R\in {\mathbb{R}}^p.$

First, let us give a few definitions.

For $a=(a_1,\dots ,a_p)\in {\mathbb{R}}^p$, $b=(b_1,\dots ,b_p)\in {\mathbb{R}}^p$ we write $a<b$, respectively $a\le b$, if $(\forall j,1\le j\le p):a_j<b_j$, and $(\forall j,1\le j\le p):a_j\le b_j$, respectively. For $z=(z_1,\dots ,z_p)\in {\mathbb{C}}^p$, $w=(w_1,\dots ,w_p)\in {\mathbb{C}}^p$, we denote $⟨z,w⟩=z_1{w}_1+\cdots +z_p{w}_p$, $\parallel z\parallel =z_1+\cdots +z_p$, $\mathrm{Re}z=(\mathrm{Re}{z}_1,\dots ,\mathrm{Re}{z}_p)$.

Let $A=(A_1,\dots ,A_p)\in {\mathbb{R}}^p$ be a fixed vector, and ${\left\{\mathbf{G}(r,A)\right\}}_{r\ge 0}$ be a system of A-like polylinear domains, which is the monotone exhaustion of the space ${\mathbb{C}}^p$, i.e.,

(a)

${\displaystyle \bigcup _{r\ge 0}}\mathbf{G}(r,A)={\mathbb{C}}^p$;

(b)

$\mathbf{G}(r_1,A)\subset \mathbf{G}(r_2,A)$ $(0\le r_1<r_2<+\infty )$;

(c)

if $z=(z_1,\dots ,z_p)\in \mathbf{G}(r,A)$, then for every $y=(y_1,\dots ,y_p)\in {\mathbb{R}}^p$, we get $z+iy=(z_1+i{y}_1,\dots ,z_p+i{y}_p)\in \mathbf{G}(r,A)$;

(d)

$z\in \mathbf{G}(r,A)⟺$ $(z-rA)\in \mathbf{G}(0,A)$;

(e)

there exist $R_{*},$ $R^{*}\in {\mathbb{R}}^p$ such that ${\mathsf{\Pi }}_{R_{*}}\subset \mathbf{G}(0,A)\subset {\mathsf{\Pi }}_{R^{*}}.$

We will denote the class of such exhaustions by $\mathsf{\Sigma }.$

Remark, if $\mathbb{G}={\left(\mathbf{G}(r,A)\right)}_{r\ge 0}$ is a system of A-like complete polylinear domains in ${\mathbb{C}}_z^p$ and ${\mathbf{D}}_1$ is the image of $\mathbf{G}(0,A)$ by the mapping $\tau =({\tau }_1,\dots ,{\tau }_p)$, ${\tau }_j=e^{z_j}$ $(1\le j\le p)$, $z=(z_1,\dots ,z_p)$, then ${\left({\mathbf{D}}_r\right)}_{r\ge 0}$ is a system of bounded complete multiple-circular domains.

Let us consider the class ${\mathcal{H}}^p$ of entire functions in ${\mathbb{C}}^p,$ which are bounded in an arbitrary domain ${\mathsf{\Pi }}_R,$ $R=(R_1,\dots ,R_p)\in {\mathbb{R}}_{+}^p$. For a function $F\in {\mathcal{H}}^p$ and $x\in {\mathbb{R}}^p$ it is obvious that

$$M(x,F):=\sup \left\{\right|F(x+iy)|:y\in {\mathbb{R}}^p\}<+\infty .$$

Moreover, if an analytic function F is bounded in a polylinear domain $G,$ then for every x such that $\{z\in {\mathbb{C}}^p:\mathrm{Re}{z}_1=x_1,\dots ,\mathrm{Re}{z}_p=x_p\}\subset G$, one has $M(x,F)<+\infty .$

For $r>0$ and a function $F\in {\mathcal{H}}^p$, we denote

$$S_F(r,A):=\sup \left\{\right|F\left(z\right)|:z\in \mathbf{G}(r,A)\}.$$

We note that $S_F(r,A)<+\infty$ $(\forall r\ge 0),$ in the case where $F\in {\mathcal{H}}^p,$ and the exhaustion is such that ${\mathsf{\Pi }}_{R_{*}}\subset \mathbf{G}(r,A)\subset {\mathsf{\Pi }}_{R^{*}}$ for some $R_{*}<R^{*}$; i.e., condition (e) is satisfied.

The following Lemma 3 is proved in [15]. It is a slightly more general statement than the same statement in ([20], pp. 302–303) since this article, like article [15], considers a more general class of space exhaustions.

Lemma 3 

([15]). Let $\left\{\mathbf{G}\right(r,A\left)\right\}\in \mathsf{\Sigma }.$ If $F\in {\mathcal{H}}^p$, then $\ln S_F(r,A)$ is a convex function of $r>0.$

By Lemma 3, the function $\ln S_F(r,A)$ is a convex function, so it has a nondecreasing right-hand derivative everywhere

$$L_F(r,A):={\left(\ln S_F(r,A)\right)}_{+}^{\prime }.$$

Let $\mathcal{R}$ be an unbounded set on ${\mathbb{R}}_{+}$.

Definition 1. 

We say that a function $F\in {\mathcal{H}}^p$ is a function from the class ${\mathcal{S}}_0^p\left(\mathcal{R}\right)$ if there exists a function $\delta \left(r\right):{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that $0<\delta \left(r\right)\uparrow +\infty$, $1/r\le L_F(r,A)/\delta \left(r\right)\uparrow +\infty$ $(0\le r_0\le r\to +\infty )$ and the inequality

$$\left|L_F\left(r\pm {\displaystyle \frac{\delta \left(r\right)}{L_F(r,A)}},A\right)-L_F(r,A)\right|\le L_F(r,A)/\delta \left(r\right)$$

(14)

holds for all $r\ge r_0,r\in \mathcal{R},$ and some $r_0\ge 0.$ That is, the inequality

$$|L_F(r+\tau ,A)-L_F(r,A)|<L_F(r,A)/\delta \left(r\right)$$

(15)

is satisfied for all $r\in (r_0,+\infty )\cap \mathcal{R}$ and all $\tau \in \mathbb{R},$ $\left|\tau \right|\le \psi \left(r\right):=\delta \left(r\right)/L_F(r,A)$.

In case $\mathcal{R}=(r_0,+\infty )$, we denote ${\mathcal{S}}_0^p:={\mathcal{S}}_0^p\left(\mathcal{R}\right)$.

In this case, the class ${\mathcal{S}}_0^p$ contains all functions whose directional logarithmic derivative $L_F(r,A)$ is one of elementary functions or its compositions in the variable r. Particularly, in the one-dimensional case, the class contains functions of the form $\exp \left\{f\right(r\left)\right\},$ where f is a function obtained from elementary functions by a finite number of main arithmetic operations and its compositions. Moreover, the iterated exponent also belongs to the class. Denote $l\left(r\right)=L_F(r,A)$. Then, in the one-dimensional case, the class contains F for which one has $l(r+O(1/l\left(r\right)\left)\right)=O\left(l\right(r\left)\right)$ $r\to +\infty .$ It is not difficult to show that ${\mathcal{S}}_0^p$ also contains all functions F for which $L_F(r,A)\in Q_A^p$, where the class $Q_A^p$ (mostly known under notation $Q_{\mathbf{b}}^n$) is a class of some positive continuous functions from the theory of holomorphic functions with a bounded index in a direction in many papers of A. Bandura, O. Skaskiv, and their co-authors.

Let us first prove the statement about the asymptotic behavior of a function F from our class ${\mathcal{S}}_0^p\left(\mathcal{R}\right)$ in the neighborhood of maximum modulus points (with absence of exceptional sets in case $F\in {\mathcal{S}}_0^p$).

Theorem 2. 

Let $F\in {\mathcal{S}}_0^p$. Then for each $A\in {\mathbb{R}}^p$ such that $L_F(r,A)\uparrow +\infty$ $(r\to +\infty )$, arbitrary function $\epsilon \left(r\right)$ such that $\epsilon \left(r\right)\to +0$ $(r\to +\infty )$, and for every point $w\in \partial \mathbf{G}(r,A)$ such that

$$\left|F\left(w\right)\right|\ge S_F(r,A){(1+\epsilon \left(r\right))}^{-1},$$

(16)

the inequality

$$|{\displaystyle \frac{F(w+A\eta )}{F\left(w\right)}}{\mathrm{e}}^{-\eta L_F(r,A)}-1|<\left|\eta \right|\varkappa \left(r\right)L_F(r,A)/\delta \left(r\right)$$

(17)

holds as $r\to +\infty$ for all $\eta \in \mathbb{C}$, $\left|\eta \right|\le \delta \left(r\right)/L_F(r,A)$, where $\varkappa \left(r\right):=1+e(1+\epsilon (r\left)\right).$

Theorem 2 directly follows from the following Proposition 4 if we choose $\mathcal{R}=(r_0,+\infty )$ in it.

Proposition 4. 

Let $\mathcal{R}$ be an unbounded set on ${\mathbb{R}}_{+}$. If $F\in {\mathcal{S}}_0^p\left(\mathcal{R}\right)$, then for each $A\in {\mathbb{R}}^p$ such that $L_F(r,A)\uparrow +\infty$ $(r\to +\infty )$, of arbitrary function $\epsilon \left(r\right)$ such that $\epsilon \left(r\right)\to +0$ $(r\to +\infty )$ and for every point $w\in \partial \mathbf{G}(r,A)$ that satisfies inequality (16), the inequality (17) holds as $r\to +\infty$ $(r\in \mathcal{R})$ for all $\eta \in \mathbb{C}$, $\left|\eta \right|\le \delta \left(r\right)/L_F(r,A),$ where $\varkappa \left(r\right):=1+e(1+\epsilon (r\left)\right).$

Proof of Proposition 4. 

We denote $\psi \left(r\right):=\delta \left(r\right)/L_F(r,A)$. Since, by definition of the class ${\mathcal{S}}_0^p\left(\mathcal{R}\right),$ the condition $\psi \left(r\right)\downarrow$ holds for all $r\ge r_0$, inequality $r-\psi \left(r\right)\ge r_0-\psi \left(r_0\right)>0$ holds for all $r\ge R_0$. Similar to what was done in article [15], let us consider an auxiliary entire function

$$g\left(\tau \right)=g(z,\tau )=F(z+A\tau ),\tau \in \mathbb{C},$$

an arbitrary point $w\in \partial \mathbf{G}(r,A)$ that satisfies the following inequality (16), and denote

$$\omega \left(\eta \right)={\displaystyle \frac{F(w+\eta )}{F\left(w\right)}}{e}^{-\eta L(x,F)}-1.$$

From the convexity of $\ln S_F(r,A)$, it follows that

$$\ln S_F(r_2,A)-\ln S_F(r_1,A)={\displaystyle \underset{r_1}{\overset{r_2}{\int }}}{L}_F(t,A)dt(\forall r_1,r_2,0\le r_1<r_2).$$

where $L_F(t,A)$ is a nondecreasing and left continuous function. Hence, at $r_2=r+h$, $r_1=r$ due to the nondecreasing of $L_F(t,A)$, and we obtain for all $r>0,h>0$

$$\ln S_F(r+h,A)-\ln S_F(r,A)\le h{L}_F(r+h,A),$$

(18)

and for all $r_1=r>0,r_2=r+h>0,h<0,$ the inequality

$$\ln S_F(r+h,A)-\ln S_F(r,A)\le h{L}_F(r,A).$$

Using (18), for $r>0,h>0$, we have

$$\begin{array}{c}\ln S_F(r+h,A)-\ln S_F(r,A)-h{L}_F(r,A)\le h{L}_F(r+h,A)-h{L}_F(r,A)\\ =\left|h\right||L_F(r+h,F)-L_F(r,A)|,\end{array}$$

because $\left|h\right|=h,$ and $|L_F(r+h,F)-L_F(r,A)|=L_F(r+h,A)-L_F(r,A)$ due to the nondecreasing of $L_F(t,A).$ These modules have been introduced in the estimate to ensure that the resulting estimate has the same form as in the case discussed below $h<0.$ Similarly, for $r>0,h<0$, one has

$$\ln S_F(r+h,A)-\ln S_F(r,A)-h{L}_F(r,A)\le 0\le \left|h\right||L_F(r+h,F)-L_F(r,A)|.$$

Therefore, for all $\left|\tau \right|\le \psi \left(r\right)$ and $r\ge r_0$ $(r\in \mathcal{R})$, using inequality (15), we obtain

$$\ln S_F(r+\tau ,A)-\ln S_F(r,A)-\tau L_F(r,A)\le 1.$$

Applying the last inequality at $\tau =\mathrm{Re}\eta$ and inequality (16), we get

$$\begin{array}{c}|1+\omega \left(\eta \right)|=\left|{\displaystyle \frac{F(w+A\eta )}{F\left(w\right)}}{e}^{-\eta L_A(x,F)}\right|\\ \le (1+\epsilon \left(r\right))\exp \left\{\ln S_F(r+\tau ,A)-\ln S_F(r,A)-\tau L_F(r,A)\right\}\le (1+\epsilon \left(r\right))e\end{array}$$

for all $r\ge r_0$ $(r\in \mathcal{R})$ and $\eta \in \mathbb{C}$, $\left|\mathrm{Re}\eta \right|\le \psi \left(r\right)$. It is clear that $\left|\mathrm{Re}\eta \right|\le \left|\eta \right|,$ $\eta \in \mathbb{C},$ thus $\{\eta \in \mathbb{C}:|\eta |\le a\}\subset \{\eta \in \mathbb{C}:|\mathrm{Re}\eta |\le a\}$, and the inequality

$$|1+\omega (\eta \left)\right|\le (1+\epsilon (r\left)\right)e$$

(19)

holds for all $\eta \in \mathbb{C},$ $\left|\eta \right|\le \psi \left(r\right),$ and $r\ge r_0$ $(r\in \mathcal{R})$.

We now apply the Schwarz lemma for a fixed $w\in {\mathbb{C}}^p$ to the function $\omega \left(\eta \right)$ in the disk ${\mathbb{D}}_{\psi \left(r\right)}$ for fixed $r\ge r_0$ $(r\in \mathcal{R})$. From inequality (19), the inequality $\left|\omega \right(\eta \left)\right|-1\le e(1+\epsilon (r\left)\right)$ follows, and, thus, the inequality $\left|\omega \right(\eta \left)\right|\le 1+e(1+\epsilon (r\left)\right)=\varkappa \left(r\right)$ holds for all $r\ge r_0$ $(r\in \mathcal{R})$ and $\eta \in \mathbb{C}$, $\left|\eta \right|\le \psi \left(r\right)$. Therefore, by the Schwarz lemma, for all $\eta ,\left|\eta \right|\le \psi \left(r\right)$; finally, we get

$$\left|\omega \left(\eta \right)\right|\le \varkappa \left(r\right)\left|\eta \right|/\psi \left(r\right)=\left|\eta \right|\varkappa \left(r\right)L_F(r,A)/\delta \left(r\right)$$

(20)

for all $r\ge r_0$ $(r\in \mathcal{R})$ and $\eta \in \mathbb{C}$, $\left|\eta \right|\le \psi \left(r\right)$, where $\varkappa \left(r\right)=(1+e(1+\epsilon \left(r\right)\left)\right).$  □

The following theorem is an analog of Wiman’s theorem.

For $F\in {\mathcal{H}}^p$ and $r>0$, let us denote

$$\begin{array}{c}{B}_F(r,A)=\sup \{\mathrm{Re}F\left(z\right):z\in \partial \mathbf{G}(r,A)\},C_F(r,A)=\inf \{\mathrm{Re}F\left(z\right):z\in \partial \mathbf{G}(r,A)\}.\end{array}$$

Theorem 3. 

If $F\in {\mathcal{S}}_0^p$ and a vector $A\in {\mathbb{R}}^p$ is such that $L_F(r,A)\uparrow +\infty$ $(r\to +\infty )$, then asymptotic relations

$$S_F(r,A)=(1+o\left(1\right))B_F(r,A)=-(1+o\left(1\right))C_F(r,A)$$

(21)

hold as $r\to +\infty .$

Theorem 3 directly follows from the following Proposition 5 if we choose $\mathcal{R}=(r_0,+\infty )$ in it.

Proposition 5. 

Let $\mathcal{R}$ be an unbounded set on ${\mathbb{R}}_{+}$. If $F\in {\mathcal{S}}_0^p\left(\mathcal{R}\right)$ and $A\in {\mathbb{R}}^p$ is such that $L_F(r,A)\uparrow +\infty$ $(r\to +\infty )$, then asymptotic relations (21) hold as $r\to +\infty$ $(r\in \mathcal{R})$.

Proof of Proposition 5. 

To prove Proposition 5, we will use the scheme from [15,16] (see also [10,17]). Let us choose $\eta =i(\pi -\arg F\left(w\right))/L_F(r,A)$, $w\in \partial \mathbf{G}(r,A)$ such that $\left|F\left(w\right)\right|=(1+o\left(1\right))S_F(r,A)$ $(r\to +\infty )$, and make sure that the application of Proposition 4 will ultimately lead us to the formulated statement. Indeed, it is clear that $\left|\eta \right|\le \delta \left(r\right)/L_F(r,A)$ and

$$\left|\omega \right(\eta \left)\right|\le \varkappa \left(r\right)|\pi -\arg F(w\left)\right|/\delta \left(r\right)\to 0$$

as $r\to +\infty$ $(r\in \mathcal{R})$. Thus, by Proposition 4,

$$F(w+A\eta )\sim \left|F\left(w\right)\right|e^{i\pi }=-\left|F\left(w\right)\right|,$$

hence, $\mathrm{Re}F(w+A\eta )\sim -S_F(r,A)$ as $r\to +\infty$ $(r\in \mathcal{R})$. So,

$$C_F(r,A)\le \mathrm{Re}F(w+A\eta )=-(1+o\left(1\right))S_F(r,A)$$

as $r\to +\infty$ $(r\in \mathcal{R})$. But, $|C_F(r,A)|\le S_F(r,A)$; therefore

$$C_F(r,A)\sim -S_F(r,A)(r\to +\infty ,r\in \mathcal{R}).$$

(22)

Let us choose now, $F^{*}\left(w\right):=-iF\left(w\right)$. Then, on the one hand, $\mathrm{Re}F=-\mathrm{Im}{F}^{*}$; hence $B_F(r,A)=-C_{F^{*}}(r,A)$. On the other hand,

$$S_F(r,A)=S_{F^{*}}(r,A),\left|F\left(w\right)\right|=|F^{*}\left(w\right)|,F^{*}\in {\mathcal{S}}_0^p.$$

Therefore, applying relation (22) to the function $F^{*}$, finally, we obtain

$$B_F(r,A)=-C_{F^{*}}(r,A)\sim S_{F^{*}}(r,A)=S_F(r,A)(r\to +\infty ,r\in \mathcal{R}).$$

 □

5. Main Relation of the Wiman–Valiron Theory

In this section, we consider the statement for function F in the class ${\mathcal{S}}_0^p$ about the relations of the form

$$F_A^{\left(k\right)}\left(w\right)=(1+o\left(1\right))L_F^k(r,A)F\left(w\right)$$

(23)

as $r\to +\infty$, i.e., without exceptional sets. Here, our proof again makes substantial use of Theorem 2, which we established in the previous section.

For $A\in {\mathbb{R}}^p$ and a function $F\in {\mathcal{S}}_0^p$ by $F_A^{\prime }\left(w\right)$, we denote the derivative of F in the direction A at the point $w\in {\mathbb{C}}^p$; $F_A^{\left(k\right)}\left(w\right)={(F_A^{(k-1)}\left(w\right))}_A^{\prime }$ denotes the k-th derivative in the direction A at the point $w\in {\mathbb{C}}^p$; $F_A^{\prime }(w+A\tau )$ is the derivative of the function F in the direction A at the point $w+A\tau$.

Theorem 4. 

If $F\in {\mathcal{S}}_0^p$ and $A\in {\mathbb{R}}^p$ are such that $L_F(r,A)\uparrow +\infty$ $(r\to +\infty )$, then for each $k\in \mathbb{N}$, asymptotic relation (23) holds as $r\to +\infty$ for every point $w\in \partial \mathbf{G}(r,A)$ that satisfies inequality (16), where $\epsilon \left(r\right)$ is a given positive function such that $\epsilon \left(r\right)\searrow 0$ $(r\to +\infty )$.

Theorem 4 directly follows from the following Proposition 6 if we choose $\mathcal{R}=(r_0,+\infty )$ in it.

Proposition 6. 

Let $\mathcal{R}$ be an unbounded set on ${\mathbb{R}}_{+}$. If $F\in {\mathcal{S}}_0^p\left(\mathcal{R}\right)$ and $A\in {\mathbb{R}}^p$ are such that $L_F(r,A)\uparrow +\infty$ $(r\to +\infty )$, then for each $k\in \mathbb{N}$ asymptotic relation (23) holds as $r\to +\infty$ $(r\in \mathcal{R})$ for every point $w\in \partial \mathbf{G}(r,A)$ that satisfies inequality (16), where $\epsilon \left(r\right)$ is a given positive function such that $\epsilon \left(r\right)\searrow 0$ $(r\to +\infty )$.

Proof of Proposition 6. 

The proof of Proposition 6 largely follows the scheme of the proof of a similar theorem in [22]. We adapt some places in our proof with additions and some necessary clarifications and rewrite other places from the proof almost verbatim, so that in general, we can obtain a completely correct proof. Let a given point $w\in \partial \mathbf{G}(r,A)$ be such that condition (16) is satisfied, and consider as in Section 2 the function

$$\omega \left(\eta \right)={\displaystyle \frac{F(w+A\eta )}{F\left(w\right)}}{e}^{-\eta L_F(r,A)}-1$$

of the variable $\eta \in \mathbb{C},$ $\left|\eta \right|\le \psi \left(r\right)=\delta \left(r\right)/L_F(r,A)$, for fixed $r\ge r_0$.

For all $\eta \in {\mathbb{D}}_R,$ $R=\psi \left(r\right)/\varkappa \left(r\right)$, $r\ge r_0$ $(r\in \mathcal{R})$, by inequality (20), we have

$$\left|\omega \right(\eta \left)\right|\le \varkappa \left(r\right)\left|\eta \right|/\psi \left(r\right)<1.$$

(24)

Thus,

$$\left|{\displaystyle \frac{F(w+A\eta )}{F\left(w\right)}}{e}^{-\eta L_F(r,A)}\right|=|1+\omega \left(\eta \right)|\ge 1-\left|\omega \left(\eta \right)\right|>0$$

for all $\eta \in {\mathbb{D}}_R,$ $R=\psi \left(r\right)/\varkappa \left(r\right)$, $r\ge r_0$ $(r\in \mathcal{R})$. So, for the fixed A and a given point $w\in \partial \mathbf{G}(r,A)$ such that condition (16) is satisfied, we get $F(w+A\eta )\ne 0$ for all $\eta \in {\mathbb{D}}_R,$ $R=\psi \left(r\right)/\varkappa \left(r\right)$, $r\ge r_0$ $(r\in \mathcal{R})$. Therefore, the function $F_A^{\prime }(w+A\tau )/F(w+A\tau )$ is an analytic function of $\tau \in {\mathbb{D}}_R$, $R=\psi \left(r\right)/\varkappa \left(r\right)$, $r\ge r_0$ $(r\in \mathcal{R})$. Hence, the function

$$f_{A,w}\left(\eta \right)={\int }_{[0,\eta ]}{\displaystyle \frac{F_A^{\prime }(w+A\tau )}{F(w+A\tau )}}d\tau -\eta L_F(r,A),f_{A,w}\left(0\right)=0,$$

(25)

is also an analytic function in the disc ${\mathbb{D}}_R=\{\eta \in \mathbb{C}:|\eta |<R\}$, $R=\psi \left(r\right)/\varkappa \left(r\right)$ as a difference in the integral of analytic function $F_A^{\prime }(w+A\tau )/F(w+A\tau )$ in the variable $\tau$ and the linear function $\eta L_F(r,A)$ in the variable $\eta$. So, $f_{A,w}\left(0\right)=0$ and an analytic function $f_{A,w}$ have in the disk ${\mathbb{D}}_R$ a Taylor series expansion of the form $f_{A,w}\left(z\right)={\displaystyle \sum _{k=1}^{+\infty }}{f}_{A,w,k}{z}^k$. It is clear that

$$f_{A,w,1}=f_{A,w}^{\prime }\left(0\right)=F_A^{\prime }\left(w\right)/F\left(w\right)-L_F(r,A),$$

and, by inequality (24), one has

$$\begin{array}{c}\mathrm{Re}{f}_{A,w}\left(\eta \right)=\ln \left|{\displaystyle \frac{F(w+A\eta )}{F\left(w\right)}}{e}^{-\eta L_F(r,A)}\right|=\ln |1+\omega \left(\eta \right)|\le \ln (1+|\omega \left(\eta \right)\left|\right)\\ \le \ln \left(1+\varkappa \left(r\right)\left|\eta \right|/\psi \left(r\right)\right)\le \ln \left(1+q\varkappa \left(r\right)/\psi \left(r\right)\right)\end{array}$$

(26)

for all $\eta \in {\overline{\mathbb{D}}}_q,$ $q<\psi \left(r\right)/\varkappa \left(r\right)$, $r\ge r_0$ $(r\in \mathcal{R})$.

In the solution of Problem 236 (see [23], Part III, Ch. 5, § 2, 236, p.355) and also [15]), we find the formulation of the statement of the following lemma.

Lemma 4. 

If f is an analytic function in a disk ${\mathbb{D}}_R$, $R>0$, of the form $f\left(z\right)={\displaystyle \sum _{n=0}^{+\infty }}{f}_n{z}^n$ and $\mathrm{Re}f\left(z\right)<M$ for all $z\in {\mathbb{D}}_R$, then $|f_n|R^n\le 2(M-\mathrm{Re}{f}_0)$ for all $n\ge 1$.

Let us apply Lemma 4 to the function $f_{A,w}$ in the disc ${\mathbb{D}}_q$ with $q<\psi \left(r\right)/\varkappa \left(r\right)$. Using inequality (26), we have

$$\begin{array}{c}\left|{\displaystyle \frac{F_A^{\prime }\left(w\right)}{F\left(w\right)}}-L_F(r,A)\right|=|f_{A,w}^{\prime }\left(0\right)|=|f_{A,w,1}|\le 2B_{f_{A,w}}\left(q\right)\\ \le 2\ln \left(1+{\displaystyle \frac{q\varkappa \left(r\right)}{\psi \left(r\right)}}\right)\le {\displaystyle \frac{2\varkappa \left(r\right)}{\psi \left(r\right)}};\end{array}$$

(27)

here, $B_f\left(q\right)=\max \{\mathrm{Re}f\left(z\right):|z|=q\}$. Hence, for all $w\in \partial \mathbf{G}(r,A)$ such that condition (16) is satisfied, one has

$$\begin{array}{c}\left|{\displaystyle \frac{F_A^{\prime }\left(w\right)}{F\left(w\right)}}{\displaystyle \frac{1}{L_F(r,A)}}-1\right|\le {\displaystyle \frac{2\varkappa \left(r\right)}{L_F(r,A)\psi \left(r\right)}}={\displaystyle \frac{\varkappa \left(r\right)}{\delta \left(r\right)}}=o\left(1\right)\end{array}$$

(28)

as $r\to +\infty$ $(r\in \mathcal{R})$.

Let us apply now again, as above, Lemma 4 in the disc ${\mathbb{D}}_q,$ $q<\psi \left(r\right)/\varkappa \left(r\right)$, to the function $f_{A,w}$, which is defined in (25). For fixed $k\in \mathbb{N}$, we have

$$\begin{array}{c}{\displaystyle \frac{1}{k!}}\left|f_{A,w}^{\left(k\right)}\left(0\right)\right|=\left|f_{A,w,k}\right|\le {\displaystyle \frac{2}{q^k}}{B}_{f_{A,w}}\left(q\right)\\ \le {\displaystyle \frac{2}{q^k}}\ln (1+q\varkappa \left(r\right)/\psi \left(r\right))\le 2q^{-k+1}\varkappa \left(r\right)/\psi \left(r\right)\end{array}$$

(29)

for $\left|\eta \right|\le q<\psi \left(r\right)/\varkappa \left(r\right)$, $r\ge r_0$ $(r\in \mathcal{R})$. Let us denote $F_0\left(\eta \right):=F(w+\eta A)/F\left(w\right)$. The function $F_0\left(\eta \right)$ is an analytic function in the variable $\eta$, $\left|\eta \right|\le q$, $F_0\left(0\right)=1.$ So, for $\left|\eta \right|\le q$, we get

$$\begin{array}{c}F(w+\eta A)=F\left(w\right)\exp \{f_{A,w}\left(\eta \right)+\eta L_F(r,A)\}\\ =F\left(w\right)\exp \left\{{\displaystyle \frac{F_A^{\prime }\left(w\right)}{F\left(w\right)}}\eta +{\displaystyle \sum _{k=2}^{+\infty }}{\displaystyle \frac{f_{A,w}^{\left(k\right)}\left(0\right)}{k!}}{\eta }^k\right\}=F\left(w\right)F_0\left(\eta \right).\end{array}$$

(30)

Let the function $F_0$ have in the disk ${\mathbb{D}}_q$ a Taylor series expansion of the form

$$F_0\left(\eta \right)=1+{\displaystyle \sum _{k=1}^{+\infty }}{F}_{0,k}{\eta }^k,\left|\eta \right|\le q.$$

(31)

Let us now denote $f_1\left(\eta \right):=f_{A,w}\left(\eta \right)+\eta L_F(r,A)$. It is clear that $f_1^{\left(k\right)}\left(\eta \right)=f_{A,w}^{\left(k\right)}\left(\eta \right)$ $(k\ge 2)$, so

$$F_0\left(\eta \right)=1+{\displaystyle \sum _{s=1}^{+\infty }}{\displaystyle \frac{1}{s!}}{\left({\displaystyle \frac{F_A^{\prime }\left(z\right)}{F\left(z\right)}}\eta +{\displaystyle \sum _{k=2}^{+\infty }}{\displaystyle \frac{f_{A,w}^{\left(k\right)}\left(0\right)}{k!}}{\eta }^k\right)}^s=1+{\displaystyle \sum _{s=1}^{+\infty }}{\displaystyle \frac{1}{s!}}{\left({\displaystyle \sum _{k=1}^{+\infty }}{\displaystyle \frac{f_1^{\left(k\right)}\left(0\right)}{k!}}{\eta }^k\right)}^s.$$

Therefore, the Taylor coefficients in (31) are formed by sums of the following form

$$F_k=\sum _{{\parallel \alpha \parallel }_0=k}{B}_{\alpha }{\displaystyle \prod _{s=1}^k}{\left({\displaystyle \frac{f_1^{\left(s\right)}\left(0\right)}{s!}}\right)}^{{\alpha }_s},$$

(32)

where $\#\{\alpha :\parallel \alpha {\parallel }_0=k\}<+\infty$, ${\parallel \alpha \parallel }_0:={\sum }_{j=1}^{k}j{\alpha }_j$ for a the multi-index $\alpha =({\alpha }_1,\dots ,{\alpha }_k)\in {\mathbb{Z}}_{+}^k$, $B_{\alpha }\in \mathbb{R}.$ Thus, there is a finite number of summands in (32).

From Equalities (30) and (31) and the Taylor expansion $F(w+A\eta )={\sum }_{k=0}^{+\infty }{\displaystyle \frac{F_A^{\left(k\right)}\left(w\right)}{k!}}{\eta }^k$, we obtain $F_A^{\left(k\right)}\left(w\right)=F\left(w\right)k!F_k$ $(k\ge 1).$ Thus,

$$\begin{array}{c}{\displaystyle \frac{F_A^{\left(k\right)}\left(w\right)}{F\left(w\right)}}{\displaystyle \frac{1}{L_F^k(r,A)}}-1={\left({\displaystyle \frac{F_A^{\prime }\left(w\right)}{F\left(w\right)}}{\displaystyle \frac{1}{L_F(r,A)}}\right)}^k-1+\sum _{{\displaystyle \genfrac{}{}{0pt}{}{{\parallel \alpha \parallel }_0=k}{{\alpha }_1<k}}}{B}_{\alpha }{\displaystyle \prod _{j=1}^k}{\left({\displaystyle \frac{f_1^{\left(j\right)}\left(0\right)}{j!}}\right)}^{{\alpha }_j}.\end{array}$$

(33)

Since $\varkappa \left(r\right)=O\left(1\right)$ and $\delta \left(r\right)\nearrow +\infty$ $(r\to +\infty ),$ $\varkappa \left(r\right)/\delta \left(r\right)=o\left(1\right)$ $(r\to +\infty ).$ Let us recall that $t^k-1=(t-1)·{\displaystyle \sum _{j=0}^{k-1}}{t}^j.$ Therefore, from the inequality (28), we consistently have

$$\begin{array}{c}\left|{\left({\displaystyle \frac{F^{\prime }\left(w\right)}{F\left(w\right)}}{\displaystyle \frac{1}{L_F(r,A)}}\right)}^k-1\right|\le \left|{\displaystyle \frac{F^{\prime }\left(w\right)}{F\left(w\right)}}{\displaystyle \frac{1}{L_F(r,A)}}-1\right|{\displaystyle \sum _{j=0}^{k-1}}{\left|{\displaystyle \frac{F^{\prime }\left(w\right)}{F\left(w\right)}}{\displaystyle \frac{1}{L_F(r,A)}}\right|}^j\\ \le {\displaystyle \frac{\varkappa \left(r\right)}{\delta \left(r\right)}}{\displaystyle \sum _{j=0}^{k-1}}{\left(1+{\displaystyle \frac{\varkappa \left(r\right)}{\delta \left(r\right)}}\right)}^j=\left(\left(1+{\displaystyle \frac{\varkappa \left(r\right)}{\delta \left(r\right)}}\right)-1\right){\displaystyle \sum _{j=0}^{k-1}}{\left(1+{\displaystyle \frac{\varkappa \left(r\right)}{\delta \left(r\right)}}\right)}^j\\ ={\left(1+{\displaystyle \frac{\varkappa \left(r\right)}{\delta \left(r\right)}}\right)}^k-1=o\left(1\right)\end{array}$$

(34)

as $r\to +\infty$ $(r\in \mathcal{R}).$ Let us now take

$$q={\displaystyle \frac{1}{2}}{\displaystyle \frac{\psi \left(r\right)}{\varkappa \left(r\right)}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta \left(r\right)}{\varkappa \left(r\right)L_F(r,A)}}.$$

Using inequality (29), we get

$${\displaystyle \prod _{j=1}^k}{\left|{\displaystyle \frac{f_1^{\left(j\right)}\left(0\right)}{j!}}\right|}^{{\alpha }_j}\le {\displaystyle \prod _{j=1}^k}{\left(2q^{-j+1}{\displaystyle \frac{\varkappa \left(r\right)}{\psi \left(r\right)}}\right)}^{{\alpha }_j}=q^{-{\parallel \alpha \parallel }_0}{\left({\displaystyle \frac{2q\varkappa \left(r\right)}{\psi \left(r\right)}}\right)}^{\parallel \alpha \parallel }=q^{-{\parallel \alpha \parallel }_0},$$

hence,

$$\begin{array}{c}\left|\sum _{{\displaystyle \genfrac{}{}{0pt}{}{{\parallel \alpha \parallel }_0=k}{{\alpha }_1<k}}}{B}_{\alpha }{\displaystyle \prod _{j=1}^k}{\left({\displaystyle \frac{f_1^{\left(j\right)}\left(0\right)}{j!}}\right)}^{{\alpha }_j}\right|\le \sum _{{\displaystyle \genfrac{}{}{0pt}{}{{\parallel \alpha \parallel }_0=k}{{\alpha }_1<k}}}|B_{\alpha }|{\displaystyle \prod _{j=1}^k}{\left|{\displaystyle \frac{f_1^{\left(j\right)}\left(0\right)}{j!}}\right|}^{{\alpha }_j}\le \sum _{{\displaystyle \genfrac{}{}{0pt}{}{{\parallel \alpha \parallel }_0=k}{{\alpha }_1<k}}}\left|B_{\alpha }\right|q^{-k}.\end{array}$$

(35)

But $\delta \left(r\right)/L_F(r,A)\to 0$ as $r\to +\infty$ and $\varkappa \left(r\right)>1+e$. Therefore, applying relation (34) and inequality (35), from equality (33), we obtain

$$\left|{\displaystyle \frac{F_A^{\left(k\right)}\left(w\right)}{F\left(w\right)}}{\displaystyle \frac{1}{L_F^k(r,A)}}-1\right|=o\left(1\right)$$

as $r\to +\infty$ $(r\in \mathcal{R})$. Thus, the statement of Proposition 6 is proved.  □

6. Discussion

In the cited article [15], the authors established multidimensional analogs of Wiman’s theorem for entire functions of several complex variables. However, a fundamental limitation of the classical Wiman–Valiron theory—which carried over into [15]—is that the main asymptotic relations do not hold universally. Specifically, ref. [15] proved that asymptotic relations like $M(r,f)=(1+o(1\left)\right)B(r,f)=-(1+o(1\left)\right)C(r,f)$ hold as $r\to +\infty$ only outside some set of a finite logarithmic measure. This means there are unpredictable intervals (the “exceptional sets”) where the function’s growth behaves erratically and the relation fails.

In the current paper, Theorems 2 and 3 (derived from Propositions 4 and 5) upgrade these findings by guaranteeing that the asymptotic relations hold for all values of the radius $r\to +\infty$, completely removing the need for exceptional sets. Theorem 2 describes the asymptotic behavior of the entire function in the neighborhood of the point where its value is close to the supremum of its modulus on the boundary of polylinear domains. Theorem 3 provides the direct analog to Wiman’s theorem, proving that $S_F(r,A)=(1+o\left(1\right))B_F(r,A)=-(1+o\left(1\right))C_F(r,A)$ globally as $r\to +\infty$.

The key new condition is the class ${\mathcal{S}}_0^p.$ To achieve this exceptional-set-free reality, we introduced a specific new constraint. Instead of looking at the broad class of all entire functions bounded in polylinear domains (${\mathcal{H}}^p$), we restricted our focus to a newly defined subclass denoted as ${\mathcal{S}}_0^p$. It was a certain surprise that the condition for removing exceptional sets could be described completely through the directional logarithmic derivative of the maximum modulus of the function, denoted as $L_F(r,A)$. For a function to belong to the class ${\mathcal{S}}_0^p$ (and thus qualify for Theorems 2 and 3), it must satisfy the following strict regularity conditions on its growth:

• Existence of a controlling function: There must exist a positive, monotonically increasing function $\delta \left(r\right)\uparrow +\infty$.
• Specific growth rate of the derivative: The ratio of the directional logarithmic derivative to this controlling function must grow monotonically: $1/r\le L_F(r,A)/\delta \left(r\right)\uparrow +\infty$ as $r\to +\infty$.
• Bounded variation in small neighborhoods: The most crucial condition is that the logarithmic derivative cannot fluctuate wildly. It must satisfy the inequality $|L_F(r\pm {\displaystyle \frac{\delta \left(r\right)}{L_F(r,A)}},A)-L_F(r,A)| \le L_F(r,A)/\delta \left(r\right)$.

Summarizing this analysis, the previous results in [15] applied to broader classes of entire functions but had to carve out exceptional sets to account for unpredictable growth spikes. The new conditions in Theorems 2 and 3 eliminate these exceptional sets by enforcing a highly predictable, regular growth pattern for the function’s directional logarithmic derivative via the ${\mathcal{S}}_0^p$ classification.

We also discuss relation (23) and how it improves upon the results of Gorenflo [46] and London [47]. While the form of the asymptotic relation for the k-th derivative is classical, the mathematical conditions required to guarantee that this relation holds globally as $r\to +\infty$ (without exceptional sets) are fundamentally new and distinct from the approaches of both Gorenflo and London.

Gorenflo established that for a function of perfectly regular growth defined as $\log M\left(r\right)\sim B{r}^{\rho }$, asymptotic relations for derivatives hold without exceptional sets. However, this result contained a major limitation: the non-negative coefficient constraint. Gorenflo’s proof fundamentally required that all Taylor series coefficients of the function be non-negative ($a_n\ge 0$). Gorenflo himself conjectured that his theorem should hold without this restriction, but was unable to prove it.

London directly addressed Gorenflo’s conjecture, successfully proving that the asymptotic relations hold without exceptional sets even when the $a_n\ge 0$ restriction is dropped. London expanded the applicable functions into a new class, G, which accommodated functions of zero and infinite order. To achieve this, London required the function’s maximum modulus to grow strictly in relation to a bounding function $\varphi$, such that $\log M(r,f)\sim \varphi \left(\log r\right)$. Furthermore, $\varphi$ had to satisfy highly specific, rigid differential inequalities, most notably $\alpha {\displaystyle \frac{{\varphi }^{\prime }\left(x\right)}{\varphi \left(x\right)}}<{\displaystyle \frac{{\varphi }^{″}\left(x\right)}{{\varphi }^{\prime }\left(x\right)}}<\beta {\displaystyle \frac{{\varphi }^{\prime }\left(x\right)}{\varphi \left(x\right)}}$. While London’s result was far more general than Gorenflo’s, it still relied on bounding the overall growth of the function against externally defined, structurally strict smooth functions.

In the current paper, we consider our conditions as being “of a different nature in a certain sense” because they discard both Taylor coefficient constraints and explicit $\varphi$-function growth boundaries. Building on the work of Dubei [34], they rely instead on the Borel–Nevanlinna lemma relations. The improvement is achieved through the definition of the class ${\mathcal{S}}_0^p$ (which becomes ${\mathcal{S}}_0^1$ for the classical $p=1$ case): Instead of bounding the maximum modulus itself, the condition for removing exceptional sets is shifted entirely to the behavior of the directional logarithmic derivative, $L_F(r,A)$. For a one-dimensional function to satisfy relation (23) everywhere, its derivative $l\left(r\right)=L_F(r,A)$ simply needs to exhibit bounded variation in small neighborhoods, expressed as $l(r+O(1/l\left(r\right)\left)\right)=O\left(l\right(r\left)\right)$ as $r\to +\infty$. This condition is highly robust. It naturally includes all functions, where $L_F(r,A)$ is a function obtained from elementary functions by a finite number of main arithmetic operations and its compositions. In particular, it can be an iterated exponential that can grow arbitrarily fast.

In summary, relation (23) improves upon Gorenflo and London by replacing cumbersome coefficient restrictions and rigid global growth boundaries with a singular, localized regularity condition based on the stability of the function’s directional logarithmic derivative.

Below, we indicate unsolved problems and potential directions for future research.

Problem 1. 

If it is possible to implement a construction from [48] similar to the above-mentioned construction in Section 4 for integer exponents, then according to the reasoning scheme from [5], it will be easy to obtain similar a Proposition 4 and and Theorems 2 and 3 from Section 4 for series in homogeneous polynomials as in (1). Now they are obtained for entire functions, which are bounded in the polylinear domains. However, we were unable to adapt and improve the construction from [48] for the case of integer exponents. Let us make a cautious assumption that the desired construction of a Dirichlet series with integer exponents (and, therefore, an entire lacunary power series) can be carried out.

While this study focuses on entire functions represented by power series of homogeneous polynomials, it is of natural interest to extend these results to other structures. For instance, recent studies have explored Szász-Beta operators linking general-Appell polynomials [50]. Synthesizing the methods of multidimensional complex analysis to study the main Wiman–Valiron asymptotic relations for series formed by these polynomial sequences remains an open question. We do not know whether the results can be extended to a wider class of exhaustion domains. For example, exhaustions with two fixed directions $A_1$ and $A_2$. The second part of this paper focuses on functions bounded in polylinear domains. Would similar results hold for more general classes of entire functions?