Entire Functions of Several Variables: Analogs of Wiman’s Theorem

Abstract

This article considers a class of entire functions of several complex variables that are bounded in the Cartesian product of some half-planes. Each such hyperplane is defined on the condition that the real part of the corresponding variable is less than some r. For this class of functions, there are established analogs of the Wiman theorems. The first result describes the behavior of an entire function from the given class at the neighborhood of the point of the supremum of its modulus. The second result shows asymptotic equality for supremums of the modulus of the function and its real part outside some exceptional set. In addition, the analogs of Wiman’s theorem are obtained for entire multiple Dirichlet series with arbitrary non-negative exponents. These results are obtained as consequences of a new statement describing the behavior of an entire function $F\left(z\right)$ of several complex variables $z=(z_1,\dots ,z_p)$ at the neighborhood of a point w, where the value $F\left(w\right)$ is close to the supremum of its modulus on the boundary of polylinear domains. The paper has two moments of novelty: the results use a more general geometric exhaustion of p-dimensional complex space by polylinear domains than previously known; another aspect of novelty concerns the results obtained for entire multiple Dirichlet series. There is no restriction that every component of exponents is strictly increasing. These statements are valid for any non-negative exponents.

1. Introduction

Wiman’s theorem [1,2,3] states that, for every entire function f of one complex variable $z\in \mathbb{C}$, there exists a set E of finite logarithmic measure ${\int }_{E\cap [1;+\infty )}dlnr<+\infty$, such that the asymptotic relations

$$\begin{array}{c}max\left\{\right|f\left(z\right)|:|z|=r\}=(1+o(1\left)\right)max\left\{\mathrm{Re}f\right(z):|z|=r\}\end{array}$$

(1)

hold as $r\to +\infty$ ($r\notin E$). Remark that, from asymptotic relation (1), it follows that

$$max\left\{\right|f\left(z\right)|:|z|=r\}=-(1+o(1\left)\right)min\left\{\mathrm{Re}f\right(z):|z|=r\}$$

as $r\to +\infty$ ($r\notin E$). Indeed, consider the function $f_1\left(z\right)=-if\left(z\right)$. Then, from (1), we obtain

$$\begin{array}{c}max\left\{\right|f\left(z\right)|:|z|=r\}=max\left\{\right|f_1\left(z\right)|:|z|=r\}=(1+o\left(1\right))max\{\mathrm{Re}{f}_1\left(z\right):\left|z\right|=r\}=\\ =(1+o(1\left)\right)max\{-\mathrm{Re}f(z):|z|=r\}=-(1+o(1\left)\right)min\left\{\mathrm{Re}f\right(z):|z|=r\}\end{array}$$

as $r\to +\infty$ ($r\notin E$). The finiteness of the logarithmic measure of an exceptional set E in Relation (1) is the best possible description of its magnitude in terms of measure [4,5]. The problem of sharpness and finding an unimproved description of the value of an exceptional set in other relations from the Wiman–Valiron theory has been considered in several works (see, for example, Wiman’s inequality in [6,7,8,9] and Borel’s-type relation in [10,11,12]). At the same time, in the papers [13,14], it was established that the description of an exceptional set can be significantly supplemented with information about its so-called asymptotic density at infinity if we consider classes of entire functions of one complex variable with a restriction on the minimum possible growth rate. Various analogs of this theorem for an entire Dirichlet series with a positive monotonically increasing to infinity sequence of exponents can be found in [15,16,17], and for a graph, [18]. Similar results for entire functions of several variables are known for analogs of classical Wiman’s inequality and Borel’s relation for entire functions represented by multiple power series or multiple Dirichlet series. Additionally, some analogs of Wiman’s inequality are known for entire multiple Dirichlet series with arbitrary complex exponents [19], as well as for series in systems of functions [20].

Another interesting property in Wiman–Valiron’s theory concerns the asymptotic behavior of entire functions in certain discs around points of maximum modulus (Wiman’s theorem). There are many papers estimasting the size of these discs from above and below for entire functions [21], as well as for subharmonic functions [22] and meromorphic mappings [23,24]. An extension of the Wiman–Valiron-type reasoning to complex differences gives difference variants of Wiman–Valiron theory [25] and its applications to q-difference equations [26,27] and their systems [28] (see also monograph [29]). Also, this theory was partially developed for fractional derivatives [30], for a polynomial series generated by the Askey–Wilson operator [31,32], and for functions that are analytic in the unit disc [33] and simply connected domain [34]. In the last cases, it was applied to describe the asymptotic behavior of analytic solutions of differential equations [35], to estimate the possible orders of growth of solutions to certain linear differential equations near a finite singular point [36] and to justify that all meromorphic solutions of the third-order non-linear autonomous ordinary differential equations must be rational (or rational in one exponential) [37]. This approach also is well-suited for entire solutions of functional equations [38].

The distance between a maximum modulus point and the zero set of an entire function within Wiman–Valiron’s theory were asymptotically estimated in [39,40]. P. Fenton [41,42,43] developed another approach to deduce the main relations in Wiman–Valiron’s theory in the multidimensional complex case. Also, there is known research concerning other multivariate generalizations. In particular, the development of a generalized Wiman–Valiron theory for Clifford algebra has been initiated [44,45]. Given these results, the theory was generalized for higher-dimensional polynomial Cauchy–Riemann equations [46], which allowed the authors to outline the analogs of the Lindelöf–Pringsheim theorem in this context [47].

The main applications of such multidimensional relations are asymptotic estimations of analytic solutions of partial differential equations (see, for example, the parabolic equations in [48,49,50,51], n-th order linear partial differential equations in [52], and the sixth-order non-linear thin-film equation in [53]).

Note that similar questions for entire functions of several complex functions were either not considered in general formulations or were limited to the consideration of functions represented by multiple power series or multiple Dirichlet series with a vector sequence of exponents, each component of which is a strictly increasing sequence to $+\infty$.

In this article, the class of entire multivariate functions bounded in the Cartesian product of half-planes is considered. For every function from this class, analogs of the Wiman theorems have been established on their behavior in the neighborhood of the point of the supremum of the modulus and on the asymptotic equality between the supremums of its modulus and its real part outside of exceptional set. To achieve this goal, the exhaustion of space by a system of A-like polylinear domains and the characteristics of the growth of functions on these exhaustions are used. In previous research, the geometric exhaustion was defined by a vector $A=(A_1,\cdots ,A_p)>0$, i.e., with the positive components $A_j>0$ $(1\le j\le p).$ The positivity condition ensured that the p-dimensional complex space is the union of A-like polylinear domains above all positive values r, where r is the parameter characterizing the size of the domain. Also, the condition ensured that the A-like polylinear domain with lesser radius is contained in the A-like polylinear domain with larger radius.

In this paper, the positivity condition has been removed and the geometric exhaustion will be defined by a vector $A=(A_1,\cdots ,A_p)$ with arbitrary real components. In particular, such a geometric generality allows one to consider entire multiple Dirichlet series with arbitrary non-negative exponents (see details below).

To describe the size of an exceptional set, for the first time in the multidimensional case, the concept of the asymptotic density of a set at infinity is introduced. This statement describes the asymptotic behavior of an entire function $F\left(z\right)$ of several complex variables $z=(z_1,\dots ,z_p)$ in the neighborhood of a point w, where the value $F\left(w\right)$ is close to the supremum of its modulus on the boundaries of polylinear domains. As a consequence of the proved theorems, we obtain analogs of Wiman’s theorems for entire multiple Dirichlet series with arbitrary non-negative exponents. This is the second moment of novelty in this paper because, for entire multiple Dirichlet series with positive monotonically increasing to infinity exponents, one analog of Wiman’s theorem was obtained in paper [54]. The proof in [54] uses one result from [55].

Note that W. K. Hayman [2,3] used Wiman’s theorem to prove the following theorem for harmonic functions $u(x,y)$ in the entire complex plane, where $z=x+iy.$

Theorem 1 

([2,3]). Let u be a harmonic function in the whole complex plane and $m(r,u):=min\left\{u\right(z):|z|=r\}$, $B(r,u):=max\left\{u\right(z):|z|=r\}$. Then, $B(r,u)=-(1+o(1\left)\right)m(r,u)$ as $r\to +\infty$ outside some set of finite logarithmic measure.

2. Notations and Lemmas

In order to give a clear presentation of the results of this paper, some notations and concepts are introduced.

Let ${\mathbb{R}}^p$ and ${\mathbb{C}}^p$ be real and complex p-dimensional vector spaces, respectively, ${\mathbb{Z}}_{+}=\mathbb{N}\cup \left\{0\right\}$, ${\mathbb{R}}_{+}=(0,+\infty )$, $p\in \mathbb{N}$. For $a=(a_1,\dots ,a_p)\in {\mathbb{R}}^p$, $b=(b_1,\dots ,b_p)\in {\mathbb{R}}^p$, the notation $a<b$ means $a\le b$ 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 $z=(z_1,\dots ,z_p)\in {\mathbb{C}}^p$, $w=(w_1,\dots ,w_p)\in {\mathbb{C}}^p$, $R=(r_1,\dots ,r_p)\in {\mathbb{R}}^p$, the following denotations are used: $\langle z,w\rangle =z_1{w}_1+\dots +z_p{w}_p$, $\parallel z\parallel =z_1+\dots +z_p$, $\mathrm{Re}z=(\mathrm{Re}{z}_1,\dots ,\mathrm{Re}{z}_p)$, ${\Pi }_R=\{z\in {\mathbb{C}}^p:\mathrm{Re}z<R\}$.

Let $A=(A_1,\cdots ,A_p)\in {\mathbb{R}}^p$ be a fixed vector and ${\left\{G_{r,A}\right\}}_{r\ge 0}$ be a system of A-like polylinear domains, which is the exhaustion of ${\mathbb{C}}^p$:

(a)

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

(b)

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

(c)

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

(d)

if $z\in G_{r,A}$, then $(z-rA)\in G_{0,A}$.

This definition is worth comparing with the definition given in ([55], p. 301), in which a similar concept is considered for an exhaustion defined by a vector $A=(A_1,\cdots ,A_p)>0$, i.e., with the positive components $A_j>0$$(1\le j\le p)$. Actually, it was assumed in [55] that the exhaustion ${\left\{G_{r,A}\right\}}_{r\ge 0}$ has some properties. The properties are described in the following steps:

(i)

A domain is called a polylinear domainin ([55], p. 294), if condition (c) from the above definition of exhaustion is satisfied;

(ii)

A polylinear domain G belongs to the class ${\Sigma }_{\pi }$ (for definition, see ([55], p. 299)) if there exist $R_{*},$$R^{*}\in {\mathbb{R}}^p$ such that ${\Pi }_{R_{*}}\subset G\subset {\Pi }_{R^{*}};$

(iii)

For $A=(A_1,\dots ,A_n)>0$$(1\le j\le 0)$), the system of domains $\left\{G_{r,A}^{+}\right\}$ in ([55], p. 301), is called the system of A-like domains if $G_{r,A}^{+}=G+rA$ (i.e., it is the translation of G by the vector $rA$), and G is a polylinear domain.

The class of the exhaustions possessing properties (i)–(iii) is denoted by ${\Sigma }^{+}.$

The condition $A>0$ in the definition just given from [55] ensures the equality ${\mathbb{C}}^p={\bigcup }_{r\ge 0}{G}_{r,A}^{+}$ and the inclusion $G_{r_1,A}^{+}\subset G_{r_2,A}^{+}.$ This means that, in our definition of exhaustion, the conditions (a) and (b) are valid. This allows us to abandon the a priori restriction on the positivity of the components of the vector $A.$

Assume that the family $\left\{G_{r,A}\right\},$ $A\in {\mathbb{R}}^p,$ in addition to conditions (a)–(d), also satisfies condition (ii). The class of such exhaustions is denoted by $\Sigma .$

Let us consider the class ${\mathcal{H}}^p$ of entire functions in ${\mathbb{C}}^p,$ which are bounded in an arbitrary domain ${\Pi }_R,$ $R=(R_1,\dots ,R_p)\in {\mathbb{R}}_{+}^p$. 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 a function $F\in {\mathcal{H}}^p$ and $r>0$ the supremum of its modulus at the polylinear domain $G_{r,A}$ is denoted by

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

It is proven ([55] Theorem 1.4.26) that, if $\left\{G_{r,A}\right\}\in {\Sigma }^{+},$ where the entire function F is such that $M(x,F)<+\infty$ and, in each polylinear domain ${\Pi }_R$, the function F is bounded, then $ln{S}_F(r,a)$ is a convex function of $r>0.$

One should observe that $S_F(r,A)<+\infty$$(\forall r\ge 0);$ in the case when $F\in {\mathcal{H}}^p$ and the exhaustion is such that ${\Pi }_{R_{*}}\subset G_{r,A}\subset {\Pi }_{R^{*}}$ for some $R_{*}<R^{*}$, i.e., condition (ii) is satisfied.

Let us first prove the following statement. Its proof almost verbatim repeats the considerations on pp. 302–303 in [55].

Lemma 1. 

Let $F\in {\mathcal{H}}^p$ and $\left\{G_{r,A}\right\}\in \Sigma .$ Then, $ln{S}_F(r,A)$ is a convex function of $r>0.$

Proof. 

As in [55], for a fixed $z\in {\mathbb{C}}^p$, it is possible to consider the entire function $g\left(\tau \right)=g(\tau ,z):=F(z+A\tau )$ of one complex variable $\tau .$ Since the function F is bounded in the polylinear domain ${\Pi }_R$ with $R=(\mathrm{Re}{z}_1^0+At,\dots ,\mathrm{Re}{z}_p^0+At)$ for each $t>0,$ the function $g\left(\tau \right)=g(\tau ,z)$ is bounded in the half-plane ${\Pi }_t^1:=\{\tau \in \mathbb{C}:\mathrm{Re}\tau <t\}$. Therefore, $g\left(\tau \right)$ satisfies the conditions of Theorem 1.1.17 in ([55], p. 145), according to which

$$ln{S}_g\left(t\right)=ln{S}_{g(·,z)}\left(t\right):=sup\{ln|g\left(\tau \right)|:\mathrm{Re}\tau <t\}=sup\{ln|g\left(\tau \right)|:\mathrm{Re}\tau =t\}$$

is a convex function of $t>0.$ That is, for all $t\in [0;1]$ and $0<r_1<r_2$, one has

$$ln{S}_g(t{r}_1+(1-t)r_2)\le tln{S}_g\left(r_1\right)+(1-t)ln{S}_g\left(r_2\right).$$

(2)

For an arbitrary polylinear domain G by Theorem 2.3.26 ([55] p. 298),

$$sup\left\{\right|F\left(z\right)|:z\in G\}=sup\left\{\right|F\left(z\right)|:z\in \partial G\},$$

i.e., it is attained at the boundary $\partial G$ of G. Thus, it yields

$$\begin{array}{c}{S}_F(r,A)=sup\left\{sup\left\{\right|g(\tau ,z)|:\mathrm{Re}\tau =r\}:z\in G_{0,A}\right\}=\\ =sup\left\{S_{g(·,z)}\left(r\right):z\in G_{0,A}\right\}.\end{array}$$

(3)

Then, from (2) and (3), this implies that the following inequality holds

$$ln{S}_F(t{r}_1+(1-t)r_2)\le tln{S}_F\left(r_1\right)+(1-t)ln{S}_F\left(r_2\right),t\in [0;1],0<r_1<r_2.$$

This yields that $ln{S}_F(r,A)$ is a convex function of $r>0$. □

Since $ln{S}_F(r,A)$ is a convex function, it has a right-hand derivative everywhere:

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

which is a nondecreasing function. It is obvious that, for every $r_0>0$, there exists

$$L_{+}\left(r_0\right):=\underset{r\to r_0+0}{lim}{L}_F(r,A)\ge \underset{r\to r_0-0}{lim}{L}_F(r,A).$$

Therefore, without any loss of the generality, it is possible to assume that the function $L_F(r,A)$ is right semicontinuous. Then, one has $L_F(r_0,A)=L_{+}\left(r_0\right)$ at every point $r_0>0$.

The following statement was essentially established in [55] pp. 304–305. Now, it is formulated for the class ${\mathcal{H}}^p$, although, in the original, it was formulated in the case of exhaustion with $A=(A_1,\dots ,A_p)>0$.

Lemma 2. 

Let $F\in {\mathcal{H}}^p$ and $A\in {\mathbb{R}}^p$ be such that $L_F(r,A)\uparrow +\infty$$(r\to +\infty )$. Then, there exists a set $E\subset {\mathbb{R}}_{+}$ of finite measure such that, for all $r\in {\mathbb{R}}_{+}\setminus E$ and all $\eta \in \mathbb{C}$,

$$\left|\eta \right|\le 1/4L_F^{-\beta \left(r\right)-1/2}(r,A){ln}^{(-1+\alpha )/2}{L}_F(r,A),$$

it holds that

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

where $\left|\omega \left(\eta \right)\right|<\left|\eta \right|{\left(L_F(r,A){ln}^{1+\alpha }{L}_F(r,A)\right)}^{1/2}$, $\alpha >0$, and the point $w\in \partial G_{r,A}$ is such that

$$\left|F\left(w\right)\right|\ge L_F^{-\beta \left(r\right)}(r,A)S_F(r,A),$$

where $\beta \left(r\right)$ is an arbitrary function, $0\le \beta \left(r\right)\le q<1/2$.

The proof in [55], pp. 304–305, uses only the general properties of the function $g(z,\tau )=F(z+A\tau )$ of one complex variable $\tau$ as an entire function, the properties of the monotonicity and right-continuity of the function $L_F(r,A)$, and some variant of the classical Borel and Nevanlinna Lemma for positive, nondecreasing, right continuous functions. By Lemma 1 and the assumption regarding the function $L_F(r,A)$, all described conditions are satisfied. This means that the statement in Lemma 2 has been proven.

To justify the main results, a slightly different and somewhat simpler version of Lemma 2 is needed. Given this, a full proof of this version is given in the auxiliary statement below. Regarding the proof of Lemma 2, the considerations are limited to the above.

Let $\mathcal{L}$ be the class of positive continuous functions increasing to $+\infty$ on $[0;+\infty )$ and ${\mathcal{L}}_1$ be the class of continuous positive nondecreasing on $[0,+\infty )$ functions h, such that

$$h(x+O(1\left)\right)=O\left(h\right(x\left)\right)(x\to +\infty ).$$

Let ${\mathcal{L}}_2$ be the class of continuous, positive, nondecreasing on $[0;+\infty )$ functions h, such that

$$h\left(x+{\displaystyle \frac{1}{h\left(x\right)}}\right)=O\left(h\left(x\right)\right)(x\to +\infty ).$$

Let h be a positive continuous nondecreasing function and $E\subset [0;+\infty )$ be a locally Lebesgue-measurable set of finite measure $\mathrm{meas}E={\int }_{E}dx<+\infty .$ Then, the asymptotic h-density of E is defined as

$$D_h\left(E\right)=\underset{R\to +\infty }{\overline{lim}}h\left(R\right)·\mathrm{meas}(E\cap [R,+\infty )).$$

The following lemma is needed.

Lemma 3. 

Let $F\in {\mathcal{H}}^p$ and $A\in {\mathbb{R}}^p$ be such that $L_F(r,A)\uparrow +\infty$$(r\to +\infty )$. Then, there exists a set $E\subset {\mathbb{R}}_{+}$ of zero asymptotic h-density (i.e., $D_h\left(E\right)=0$) and a function $\delta \left(u\right)\uparrow +\infty$$(u\to +\infty )$ such that, for all $r\in {\mathbb{R}}_{+}\setminus E$ and all $\eta \in \mathbb{C}$,

$$\left|\eta \right|\le \delta \left(r\right)/L_F(r,A),$$

it holds that

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

where

$$\left|\omega \left(\eta \right)\right|<\left|\eta \right|c\left(r\right)L_F(r,A)/\delta \left(r\right),c\left(r\right):=1+e(1+\epsilon \left(r\right))$$

and the point $w\in \partial G_{r,A}$ exists such that

$$\left|F\left(w\right)\right|\ge S_F(r,A)/(1+\epsilon \left(r\right)),$$

$\epsilon \left(r\right)$ is a given arbitrary function such that $\epsilon \left(r\right)\to +0$$(r\to +\infty )$.

Various analogs of Wiman’s theorem need the corresponding version of the Borel–Nevanlinna Lemma. Below, its variant from [14] is stated.

Lemma 4 

([14]). Let $u\left(r\right)$ be a right, semicontinuous, increasing on $[r_0,+\infty )$ function and $\Phi \in \mathcal{L}$, $h\in {\mathcal{L}}_2$ be functions such that

$$u\left(r\right)\ge \Phi \left(r\right)(r\ge r_0),h\left(r\right)=o(\Phi \left(r\right))(r\to +\infty ).$$

Then, there exists a function $\delta \left(u\right)\uparrow +\infty$$(u\to +\infty )$ such that the set

$$E^1=\left\{r\ge r_0:u\left(r+{\displaystyle \frac{\delta \left(u\right(r\left)\right)}{u\left(r\right)}}\right)\ge \left(1+{\displaystyle \frac{1}{\delta \left(u\right(r\left)\right)}}\right)u\left(r\right)\right\}$$

(4)

has zero asymptotic h-density, i.e., $D_h\left(E^1\right)=0$.

In order to prove Lemma 3, the following modified version of the Borel–Nevanlinna Lemma is used.

Lemma 5. 

Let $u\left(r\right)$ be a right, semicontinuous, increasing on $[r_0,+\infty )$ function and $\Phi \in \mathcal{L}$, $h\in {\mathcal{L}}_2$ be the functions such that

$$u\left(r\right)\ge \Phi \left(r\right)(r\ge r_0),h\left(r\right)=o(\Phi \left(r\right))(r\to +\infty ).$$

Then, there exists a function $\delta \left(u\right)\uparrow +\infty$$(u\to +\infty )$ such that the set

$$E=\left\{r\ge r_0:\left|u\left(r\pm {\displaystyle \frac{\delta \left(u\right(r\left)\right)}{u\left(r\right)}}\right)-u\left(r\right)\right|\ge u\left(r\right)/\delta \left(u\left(r\right)\right)\right\}$$

(5)

has zero asymptotic h-density, i.e., $D_h\left(E\right)=0$. In other words, the inequality

$$\left|u\right(r+\tau )-u(r\left)\right|<u\left(r\right)/\delta \left(u\right(r\left)\right)$$

(6)

holds for all $r\in (r_0,+\infty )\setminus E$ and all $\tau \in \mathbb{R},$ $\left|\tau \right|\le \psi \left(r\right):=\delta \left(u\right(r\left)\right)/u\left(r\right)$, where the set E has zero asymptotic h-density.

Proof. 

Given the inequality in (4), it remains to prove that the set

$$E^2=\left\{r\ge r_0:u\left(r-{\displaystyle \frac{\delta \left(u\right(r\left)\right)}{u\left(r\right)}}\right)-u\left(r\right)>-u\left(r\right)/\delta \left(u\left(r\right)\right)\right\}$$

with the same function $\delta \left(u\right)$ has zero asymptotic h-density.

Of course, it is possible to consider only this case in the proof. But, for completeness, we provide the proof of both parts of the inequality

$$\left|u\left(r\pm {\displaystyle \frac{\delta \left(u\right(r\left)\right)}{u\left(r\right)}}\right)-u\left(r\right)\right|<u\left(r\right)/\delta \left(u\left(r\right)\right)$$

simultaneously.

Let $\delta >1.$ Without loss of generality, assume $u\left(r\right)-\delta /u\left(r\right)>r_0$$(r\ge r_0).$ Consider the sets

$$E^1\left(\delta \right)=\left\{r\in (r_0,+\infty ):u\left(r+{\displaystyle \frac{\delta }{u\left(r\right)}}\right)\ge \left(1+{\displaystyle \frac{1}{\delta }}\right)u\left(r\right)\right\},$$

$$E^2\left(\delta \right)=\left\{r\in (r_0,+\infty ):u\left(r\right)\ge \left(1+{\displaystyle \frac{1}{\delta }}\right)u\left(r-{\displaystyle \frac{\delta }{u\left(r\right)}}\right)\right\}.$$

Let us prove that $D_h(E^1\left(\delta \right)\cup E^2\left(\delta \right))=0$. Suppose that $E^1\left(\delta \right)$ and $E^2\left(\delta \right)$ are unbounded. Otherwise, for $r\notin E^1\left(\delta \right),r>r_1,$ for some $r_1\ge r_0$, one has

$$u\left(r+{\displaystyle \frac{\delta }{u\left(r\right)}}\right)<\left(1+{\displaystyle \frac{1}{\delta }}\right)u\left(r\right),$$

hence, $u\left(r+{\displaystyle \frac{\delta }{u\left(r\right)}}\right)-u\left(r\right)<u\left(r\right)/\delta$ and

$$\mathrm{meas}(E^1\left(\delta \right)\cap [r_1,+\infty ))=0.$$

Similarly, for $r\notin E^2\left(\delta \right)$, the following estimate is valid:

$$u\left(r\right)<\left(1+{\displaystyle \frac{1}{\delta }}\right)u\left(r-{\displaystyle \frac{\delta }{u\left(r\right)}}\right)(r\ge r_2)$$

and $\mathrm{meas}(E^2\left(\delta \right)\cap [r_2,+\infty )=0.$

Now, denote $E^j(\delta ,r)=E^j\left(\delta \right)\cap [r,+\infty ),$ $j\in \{1,2\}.$ Let us define the following sequences:

$$r_1^{\left(j\right)}=\mathrm{essinf}\{r:r\in E^j(\delta ,r_0)\},R_1^{\left(j\right)}=r_1^{\left(j\right)}+{\displaystyle \frac{\delta }{u(r_1^{\left(j\right)}+0)}},j\in \{1,2\}.$$

Assume that $r_k^{\left(j\right)},R_k^{\left(j\right)}$ are already defined for $k\le n$. Now define

$$r_{n+1}^{\left(j\right)}=\mathrm{essinf}\{r:r\in E^j(\delta ,R_n^{\left(j\right)})\},R_{n+1}^{\left(j\right)}=r_{n+1}^{\left(j\right)}+{\displaystyle \frac{\delta }{u(r_{n+1}^{\left(j\right)}+0)}}.$$

It is clear that (see also [14,56])

$$E^j\left(\delta \right)\setminus E^{*}\subset \bigcup _{n=1}^{+\infty }[r_n^{\left(j\right)},R_n^{\left(j\right)}],$$

where $E^{*}$ is at most a countable set.

Let us prove that, if

$${\displaystyle r^{\left(j\right)}=\mathrm{essinf}\{r:r\in E^j(\delta ,r)\},a^{\left(j\right)}=u(r^{\left(j\right)}+0)=\underset{r\to r^{\left(j\right)}+0}{lim}u\left(r\right),}$$

then

$${\displaystyle u\left(r^{\left(1\right)}+\frac{\delta }{a^{\left(1\right)}}+0\right)\ge \left(1+\frac{1}{\delta }\right)a^{\left(1\right)}}$$

and

$${\displaystyle a^{\left(2\right)}\ge \left(1+\frac{1}{\delta }\right)u\left(r^{\left(2\right)}-\frac{\delta }{a^{\left(2\right)}}+0\right).}$$

At first, suppose that, for a certain sequence $r_k\downarrow r^{\left(1\right)}$ from the set $E^1(\delta ,r)$, one has

$${\displaystyle v_k=r_k+\frac{\delta }{u\left(r_k\right)}\to v_0+0,v_0=r^{\left(1\right)}+\frac{\delta }{a^{\left(1\right)}}.}$$

Then, progressing to the limit as $k\to +\infty$ in the inequality

$${\displaystyle u\left(v_k\right)\ge \left(1+\frac{1}{\delta }\right)u\left(r_k\right)}$$

one has

$${\displaystyle u(v_0+0)\ge \left(1+\frac{1}{\delta }\right)u(r^{\left(1\right)}+0),}$$

i.e., the required inequality. If, for $r_k\downarrow r^{\left(1\right)}$ from the set $E^1(\delta ,r)$, the sequence $v_k$ tends to $v_0-0$, then such an upper estimate will be valid after passing to the limit as $k\to +\infty$:

$${\displaystyle u(v_0-0)\ge \left(1+\frac{1}{\delta }\right)u(r^{\left(1\right)}+0).}$$

Since $u(v_0+0)\ge u(v_0-0),$ the required relation is obtained. Therefore, the statement concerning the first inequality is proven because

$${\displaystyle \underset{r\to r^{\left(1\right)}+0}{lim}\left(r+\frac{\delta }{u\left(r\right)}\right)}$$

exists. Similarly, if $r_k\downarrow r^{\left(2\right)}$ for a certain sequence from $E^2(\delta ,r)$, then

$${\displaystyle w_k=r_k-\frac{\delta }{u\left(r_k\right)}\ge r_{k+1}-\frac{\delta }{u\left(r_{k+1}\right)}=w_{k+1}.}$$

Thus,

$${\displaystyle w_k\downarrow w_0:=r^{\left(2\right)}-\frac{\delta }{a^{\left(2\right)}},}$$

and, by the inequality $u\left(r_k\right)\ge (1+{\displaystyle \frac{1}{\delta }})u\left(w_k\right)$, one has

$${\displaystyle u(r^{\left(2\right)}+0)\ge (1+\frac{1}{\delta })u(w_0+0),}$$

i.e., the desired conclusion.

In view of the above proven facts, one has

$$u(r_{n+1}^{\left(1\right)}+0)\ge u(R_n^{\left(1\right)}+0)=u\left(r_n^{\left(1\right)}+{\displaystyle \frac{\delta }{u(r_n^{\left(1\right)}+0)}}+0\right)\ge \left(1+{\displaystyle \frac{1}{\delta }}\right)u(r_n^{\left(1\right)}+0),$$

(7)

and, since $r_{n+1}^{\left(2\right)}\ge R_n^{\left(2\right)}=r_n^{\left(2\right)}+{\displaystyle \frac{\delta }{u(r_n^{\left(2\right)}+0)}}\ge r_n^{\left(2\right)}+{\displaystyle \frac{\delta }{u\left(r_{n+1}^{\left(2\right)}\right)}},$ the folowing lower estimate is true:

$$u(r_{n+1}^{\left(2\right)}+0)\ge \left(1+{\displaystyle \frac{1}{\delta }}\right)u\left(r_{n+1}^{\left(2\right)}-{\displaystyle \frac{\delta }{u(r_{n+1}^{\left(2\right)}+0)}}+0\right)\ge \left(1+{\displaystyle \frac{1}{\delta }}\right)u(r_n^{\left(2\right)}+0).$$

(8)

It follows from (7) and (8) that $r_n^{\left(j\right)}\uparrow +\infty$$(n\to +\infty )$ and

$$\delta {\displaystyle \frac{u(r_{n+1}^{\left(j\right)}+0)-u(r_n^{\left(j\right)}+0)}{u(r_n^{\left(j\right)}+0)}}\ge 1.$$

Arguing similarly as in [14,56], for $r\in [R_n^{\left(j\right)},r_{n+1}^{\left(j\right)}]$, one has

$$\begin{array}{c}\mathrm{meas}{E}^j(\delta ,r)\le \sum _{k=n+1}^{+\infty }(R_k^{\left(j\right)}-r_k^{\left(j\right)})\le {\displaystyle \frac{\delta }{u(r_{n+1}^{\left(j\right)}+0)}}+\\ +{\delta }^2\sum _{k=n+2}^{+\infty }{\displaystyle \frac{u(r_k^{\left(j\right)}+0)-u(r_{k-1}^{\left(j\right)}+0)}{u(r_k^{\left(j\right)}+0)u(r_{k-1}^{\left(j\right)}+0)}}={\displaystyle \frac{\delta (1+\delta )}{u(r_{n+1}^{\left(j\right)}+0)}}\le {\displaystyle \frac{\delta (1+\delta )}{\Phi \left(r_{n+1}^{\left(j\right)}\right)}}.\end{array}$$

(9)

In the same manner, it is easy to see that, for $r\in [r_n^{\left(j\right)},R_n^{\left(j\right)}]$

$$\begin{array}{c}\mathrm{meas}{E}^j(\delta ,r)\le \sum _{k=n}^{+\infty }(R_k^{\left(j\right)}-r_k^{\left(j\right)})\le {\displaystyle \frac{\delta }{u(r_n^{\left(j\right)}+0)}}+\\ +{\delta }^2\sum _{k=n+1}^{+\infty }{\displaystyle \frac{u(r_k^{\left(j\right)}+0)-u(r_{k-1}^{\left(j\right)}+0)}{u(r_k^{\left(j\right)}+0)u(r_{k-1}^{\left(j\right)}+0)}}={\displaystyle \frac{\delta (1+\delta )}{u(r_n^{\left(j\right)}+0)}}\le {\displaystyle \frac{\delta (1+\delta )}{\Phi \left(r_n^{\left(j\right)}\right)}}.\end{array}$$

(10)

It is clear that $h\left(r\right)\le h\left(r_{n+1}^{\left(j\right)}\right)$ for $r\le r_{n+1}^{\left(j\right)}$ and

$$\begin{array}{c}h\left(r\right)\le h(r_n^{\left(j\right)}+R_n^{\left(j\right)}-r_n^{\left(j\right)})\le h\left(r_n^{\left(j\right)}+{\displaystyle \frac{\delta }{u\left(r_n^{\left(j\right)}\right)}}\right)=\\ =h\left(r_n^{\left(j\right)}+o\left({\displaystyle \frac{1}{h\left(r_n^{\left(j\right)}\right)}}\right)\right)=O\left(h\left(r_n^{\left(j\right)}\right)\right)(n\to +\infty )\end{array}$$

for $r\le R_n^{\left(j\right)}.$ Therefore, from (9) and (10), for $r\in [r_n^{\left(j\right)},r_{n+1}^{\left(j\right)}]$, one has

$$h\left(r\right)·\mathrm{meas}{E}^j(\delta ,r)=O\left(max\left\{{\displaystyle \frac{h\left(r_n^{\left(j\right)}\right)}{\Phi \left(r_n^{\left(j\right)}\right)}},{\displaystyle \frac{h\left(r_{n+1}^{\left(j\right)}\right)}{\Phi \left(r_{n+1}^{\left(j\right)}\right)}}\right\}\right)=o\left(1\right)(n\to +\infty ).$$

Thus, $D_h{E}^j\left(\delta \right)=0,j\in \{1,2\}$, and this yields $D_h(E^1\left(\delta \right)\cup E^2\left(\delta \right))=0.$ Now, let $t_n\uparrow +\infty$$(n\to +\infty )$ be a sequence such that, for all $r\in [t_n,+\infty )$,

$$h\left(r\right)·\mathrm{meas}E(n+1,r)\le {\displaystyle \frac{1}{n^2}},$$

where $E\left(\delta \right)=E^1\left(\delta \right)\cup E^2\left(\delta \right).$

Define $\delta \left(u\right(r\left)\right)=\delta \left(r\right):=n+1$ for $r\in [t_n,t_{n+1})$ and $E_1={\displaystyle \bigcup _{n=1}^{+\infty }}(E(n+1,t_n)\cap [t_n,t_{n+1}))$. For $r\in [t_n,t_{n+1})$, one has

$$\begin{array}{c}h\left(r\right)·\mathrm{meas}(E_1\cap [r,+\infty ))\le h\left(r\right)·\mathrm{meas}E(n+1,r)+\\ +\sum _{k=n+1}^{+\infty }{\displaystyle \frac{h\left(r\right)}{h\left(t_k\right)}}·h\left(t_k\right)·\mathrm{meas}E(k+1,t_k)\le \sum _{k=n}^{+\infty }{\displaystyle \frac{1}{k^2}}=o\left(1\right)(n\to +\infty ),\end{array}$$

that is, $D_h{E}_1=0.$ For $r\in [t_n,t_{n+1})\setminus E_1$,

$$\begin{array}{c}u\left(r+{\displaystyle \frac{\delta \left(r\right)}{u\left(r\right)}}\right)=u\left(r+{\displaystyle \frac{n+1}{u\left(r\right)}}\right)<\left(1+{\displaystyle \frac{1}{n+1}}\right)u\left(r\right)=\left(1+{\displaystyle \frac{1}{\delta \left(r\right)}}\right)u\left(r\right)⟺\\ u\left(r+{\displaystyle \frac{\delta \left(r\right)}{u\left(r\right)}}\right)-u\left(r\right)\le u\left(r\right)/\delta \left(r\right).\end{array}$$

(11)

Since the inequality

$${\displaystyle u\left(r\right)<\left(1+\frac{1}{\delta }\right)u\left(r-\frac{\delta }{u\left(r\right)}\right),\delta >1}$$

yields

$$u\left(r-{\displaystyle \frac{\delta }{u\left(r\right)}}\right)>u\left(r\right)-{\displaystyle \frac{1}{\delta }}u\left(r-{\displaystyle \frac{\delta }{u\left(r\right)}}\right)\ge \left(1-{\displaystyle \frac{\delta }{u\left(r\right)}}\right)u\left(r\right),$$

for $r\in [t_n,t_{n+1})\setminus E_1$, such an estimate holds:

$$\begin{array}{c}u\left(r-{\displaystyle \frac{\delta \left(r\right)}{u\left(r\right)}}\right)=u\left(r-{\displaystyle \frac{n+1}{u\left(r\right)}}\right)>\left(1-{\displaystyle \frac{1}{n+1}}\right)u\left(r\right)=\left(1-{\displaystyle \frac{1}{\delta \left(r\right)}}\right)u\left(r\right).\end{array}$$

(12)

From (11) and (12), for all $r\in (r_0,+\infty )\setminus E_1$$(D_h{E}_1=0)$ and $\tau \in \mathbb{R},$$\left|\tau \right|\le \delta \left(r\right)/u\left(r\right)$, it follows that

$$|u(r+\tau )-u\left(r\right)|\le \left\{u\left(r+{\displaystyle \frac{\delta \left(r\right)}{u\left(r\right)}}\right)-u\left(r\right),u\left(r\right)-u\left(r-{\displaystyle \frac{\delta \left(r\right)}{u\left(r\right)}}\right)\right\}<{\displaystyle \frac{u\left(r\right)}{\delta \left(r\right)}}.$$

This is precisely the assertion in Lemma 5. □

From Lemma 5, the following consequence is deduced.

Lemma 6. 

Let $u\left(r\right)$ be a right, semicontinuous, increasing on $[r_0,+\infty )$ function such that $u\left(r\right)\to +\infty$$(r\to +\infty ).$ Then, there exists a function $\delta \left(u\right)\uparrow +\infty$$(u\to +\infty )$ such that the set E, defined by (5), has finite Lebesgue measure, i.e., $\mathrm{meas}(E\cap [0,+\infty \left)\right)<+\infty$.

Indeed, let us choose $h\left(r\right)\equiv 1$ and a function $\Phi \in \mathcal{L}$ such that $u\left(r\right)\ge \Phi \left(r\right)$$(r\ge r_0)$. Then, the statement in Lemma 6 directly follows from Lemma 5.

Proof of Lemma 3. 

As above, the entire function $g\left(\tau \right)=g(z,\tau )=F(z+A\tau )$, $\tau \in \mathbb{C}$ will be considered.

Lemma 4 can be applied to the function $u\left(r\right)$ such that $u\left(r\right)=L_F(r,A)$ at the continuity points r of the function $L_F(r,A)$ and $u\left(r\right)={lim}_{r\to x_1+0}{L}_F(r,A)$ at the discontinuity points $x_1$ of the function $L_F(r,A).$

Let us join all discontinuity points $x_1$ of the function $L_F(r,A)$ to the set $E.$ Since there are at most a countable number of the discontinuity points, by Lemma 5, one has $D_h\left(E\right)=0$ and, for each $r\in [0;+\infty )\setminus E$ and for all $\tau$ with $\left|\tau \right|\le \psi \left(r\right)=\delta \left(r\right)/L_F(r,A)$, Inequality (6) is obtained. Hence, it is deduced that

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

(13)

for all $\tau ,$$\left|\tau \right|\le \psi \left(r\right)$ and $r\in [0;+\infty )\setminus E.$

Let a point $w\in \partial G_A\left(r\right)$ be such that

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

From the convexity of $ln{L}_F(r,A)$, it follows, on the one hand, that, for all $r>0,h>0$,

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

and, on the other hand, that, for all $r>0,h<0$,

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

Therefore, for $r>0,h>0$, one has

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

Similarly, for $r>0,h<0$, the difference of logarithms is estimated as follows:

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

So, for all $\left|\tau \right|\le \psi \left(r\right)$ and $r\in [0;+\infty )\setminus E$, using Inequality (13), one has

$$ln{S}_F(r+\tau ,A)-ln{S}_F(r,A)-\tau L_F(r,A)\le 1.$$

Hence, for all $r\in [0;+\infty )\setminus E$ and $\eta \in \mathbb{C}$, $\left|\mathrm{Re}\eta \right|\le \psi \left(r\right)$,

$$\begin{array}{c}\left|{\displaystyle \frac{F(w+A\eta )}{F\left(w\right)}}{e}^{-\eta L_A(x,F)}\right|\le \\ \le (1+\epsilon \left(r\right))exp\left\{ln{S}_F(r+\mathrm{Re}\eta ,A)-ln{S}_F(r,A)-\mathrm{Re}\eta L_F(r,A)\right\}\le (1+\epsilon \left(r\right))e.\end{array}$$

(14)

Since

$$\{\eta \in \mathbb{C}:|\eta |\le a\}\subset \{\eta \in \mathbb{C}:|\mathrm{Re}\eta |\le a\},$$

Inequality (14) holds for all $\eta \in \mathbb{C},$$\left|\eta \right|\le \psi \left(r\right),$ and $r\in [0;+\infty )\setminus E$.

Let us consider now, for fixed $w\in {\mathbb{C}}^p$, the function

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

of the variable $\eta \in \mathbb{C},$$\left|\eta \right|\le \psi \left(r\right)$ for fixed $r\in [0;+\infty )\setminus E$. Since Inequality (14) implies

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

for all $r\in [0;+\infty )\setminus E$ and $\eta \in \mathbb{C}$, $\left|\eta \right|\le \psi \left(r\right)$, by Schwarz’s Lemma, for all $\eta ,\left|\eta \right|\le \psi \left(r\right)$, such an estimate is true:

$$\left|\omega \left(\eta \right)\right|\le c\left(r\right){\displaystyle \frac{\left|\eta \right|}{\psi \left(r\right)}},$$

where $c\left(r\right)=(1+e(1+\epsilon \left(r\right)\left)\right).$ □

3. Main Result: An Analog of Wiman’s Theorem

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

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

Theorem 2. 

Let $F\in {\mathcal{H}}^p$, $A\in {\mathbb{R}}^p$ be such that $L_F(r,A)\uparrow +\infty$$(r\to +\infty )$. If $\Phi \in \mathcal{L}$, $h\in {\mathcal{L}}_2$ are the functions such that

$$L_F(r,A)\ge \Phi \left(r\right)(r\ge r_0),h\left(r\right)=o(\Phi \left(r\right))(r\to +\infty ),$$

then there exists a set $E\subset {\mathbb{R}}_{+}$ of zero asymptotic h-density (i.e., $D_h\left(E\right)=0$) such that

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

(15)

as $r\to +\infty$$(r\in {\mathbb{R}}_{+}\setminus E).$

Proof. 

To prove the theorem, the scheme from [15] is used (see also [2,3,16]). At first, take $\eta =i(\pi -argF\left(w\right))/L_F(r,A).$ The application of Lemma 3 gives the following estimate:

$$\left|\omega \right(\eta \left)\right|\le c\left(r\right)\left|\right(\pi -argF\left(w\right)\left)\right|/\delta \left(r\right)\to +\infty$$

as $r\to +\infty$$(r\notin E)$,

$$F(w+A\eta )=(1+o\left(1\right))\left|F\left(w\right)\right|e^{i\pi }=-(1+o\left(1\right))\left|F\left(w\right)\right|$$

and, therefore, $m_F(r,A)\le \mathrm{Re}F(w+A\eta )\le -(1+o\left(1\right))S_F(r,A).$ The obvious inequality $|m_F(r,A)|\le S_F(r,A)$ yields

$$m_F(r,A)=-(1+o\left(1\right))S_F(r,A)(r\to +\infty ,r\notin E).$$

Take $\eta =-iargF\left(w\right)/L_F(r,A)$. Then, such an asymptotic equality holds:

$$F(w+A\eta )=(1+o(1\left)\right)\left|F\right(w\left)\right|(r\to +\infty ,r\notin E).$$

Therefore,

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

as $r\to +\infty$$(r\notin E)$ since $B_F(r,A)\le S_F(r,A)$ and

$$|m_F(r,A)|\le S_F(r,A)\le (1+o\left(1\right))B_F(r,A)(r\to +\infty ,r\notin E).$$

This completes the proof of Theorem 2. □

4. Corollaries: Entire Dirichlet Series

Let ${\Lambda }^p=\left({\lambda }_n\right)$, ${\lambda }_n=({\lambda }_{n_1}^{\left(1\right)},\dots ,{\lambda }_{n_p}^{\left(p\right)})$, $n=(n_1,\dots ,n_p)$, and $0\le {\lambda }_k^{\left(j\right)}$$(0\le k\uparrow +\infty )$, $1\le j\le p$. Denote by ${\mathcal{H}}^p\left({\Lambda }^p\right)$ the class of an entire (absolutely convergent in the whole space ${\mathbb{C}}^p$) Dirichlet series of the form

$$F\left(z\right)=\sum _{\parallel n\parallel =0}^{+\infty }{a}_n{e}^{\langle z,{\lambda }_n\rangle },z\in {\mathbb{C}}^p,$$

such that $(\forall j):$$\#\{n_j:a_{(n_1,\dots ,n_j,\dots ,n_p)}\ne 0\}=+\infty$.

Let ${\gamma }_F$ be the real cone of the growth of the maximal term

$$\mu (\sigma ,F)=max\left\{\right|a_n|e^{\langle \sigma ,{\lambda }_n\rangle }:n\in {\mathbb{Z}}_{+}^p\}$$

of a Dirichlet series $F\in {\mathcal{H}}^p\left({\Lambda }^p\right)$, that is,

$${\gamma }_F:=\left\{\sigma \in {\mathbb{R}}^p:\underset{t\to +\infty }{lim}{\displaystyle \frac{ln\mu (t\sigma ,F)}{t}}=+\infty \right\}.$$

Remark 1. 

Let $F\in {\mathcal{H}}^p\left({\Lambda }^p\right)$.

$$A\in {\gamma }_F⟺sup\{\langle A,{\lambda }_n\rangle :n\in {\mathbb{Z}}_{+}^p\}.$$

Lemma 7. 

Let $F\in {\mathcal{H}}^p\left({\Lambda }^p\right)$. In order that $L_F(r,A)\uparrow +\infty$$(r\to +\infty )$, it is necessary and sufficient that $A\in {\gamma }_F$.

Proof. 1.

$A\in {\gamma }_F⟹$$L_F(r,A)\uparrow +\infty$$(r\to +\infty )$.

Since $L_F(r,A)$ is a nondecreasing function of r, the assumption on the falsity of the condition $L_F(r,A)\uparrow +\infty$$(r\to +\infty )$ consistently gives $L_F(r,A)=O\left(1\right)$$(r\to +\infty )$ and $ln{S}_F(r,A)=O\left(r\right)$$(r\to +\infty )$.

Denote by $G_{r,A}^1$ the image of the set $G_{r,A}$ under the mapping $w=\mathrm{Re}z$ in ${\mathbb{R}}^p$, and

$$d_m\left(G\right):=exp\left(sup\left\{\mathrm{Re}(z,{\lambda }_m):z\in G_{0,A}\right\}\right).$$

Note that ([55], p. 353)

$$sup\left\{|a_m|\exp \left(\left(\sigma ,{\lambda }_m\right)\right):\sigma \in \partial G_{r,A}^1\right\}=d_m\left(G\right)\left|a_m\right|\exp \left\{r\left(A,{\lambda }_m\right)\right\}.$$

Therefore, the Cauchy inequality $\mu (\sigma ,F)\le M(\sigma ,F)$ for given $m\in {\mathbb{Z}}_{+}^p$ yields

$$\begin{array}{c}{d}_m\left(G\right)\left|a_m\right|\exp \left\{r\left(A,{\lambda }_m\right)\right\}\le sup\left\{\mu (\sigma ,F):\sigma \in \partial G_{r,A}^1\right\}\le \\ \le sup\left\{M(\sigma ,F):\sigma \in \partial G_{r,A}^1\right\}=\\ =sup\left\{sup\left\{|F(\sigma +iy):y\in {\mathbb{R}}^p\right\}:\sigma \in G_{r,A}^1\right\}=\\ =sup\left\{\left|F\left(z\right)\right|:z\in G_{r,A}\right\}=S_F(r,A).\end{array}$$

So, if $a_m\ne 0$, then the following estimate holds:

$$\left(A,{\lambda }_m\right)\le ln{S}_F(r,A)/r+o\left(1\right)=O\left(1\right)(r\to +\infty ).$$

But, from the condition $A\in {\gamma }_F$, it follows that

$$sup\left\{\left(A,{\lambda }_n\right):n\in {\mathbb{Z}}_{+}^p\right\}=+\infty .$$

Therefore, $L_F(r,A)\uparrow +\infty$$(r\to +\infty )$ for every $A\in {\gamma }_F$.

$L_F(r,A)\uparrow +\infty$$(r\to +\infty )⟹$$A\in {\gamma }_F$.

From the condition $L_F(r,A)\uparrow +\infty$$(r\to +\infty )$, it follows that

$${\displaystyle \frac{1}{r}}ln{S}_F(r,A)\to +\infty (r\to +\infty ).$$

Assume

$$sup\left\{\left(A,{\lambda }_n\right):n\in {\mathbb{Z}}_{+}^p\right\}=K<+\infty .$$

Then,

$$\begin{array}{c}{S}_F(r,A)\le \sum _{\parallel n\parallel =0}^{\infty }\left|a_n\right|\exp \left\{sup\left\{\left(\mathrm{Re}z,{\lambda }_n\right):z\in G_{r,A}\right\}\right\}\le \\ \le \sum _{\parallel n\parallel =0}^{\infty }\left|a_n\right|d_n\left(G\right)\exp \left\{r\left(A,{\lambda }_n\right)\right\}\le \\ \le \sum _{\parallel n\parallel =0}^{\infty }|a_n|d_n\left(G\right)\exp \left(rsup\left\{\left(A,{\lambda }_n\right):n\in {\mathbb{Z}}_{+}^p\right\}\right)\le \sum _{\parallel n\parallel =0}^{\infty }\left|a_n\right|d_n\left(G\right)e^{Kr}.\end{array}$$

Hence,

$${\displaystyle \frac{1}{r}}ln{S}_F(r,A)\le K+o\left(1\right)(r\to +\infty ).$$

This is a contradiction.

Thus, $sup\left\{\left(A,{\lambda }_n\right):n\in {\mathbb{Z}}_{+}^p\right\}=+\infty ,$ and, by Remark 1, $A\in {\gamma }_F$. □

Theorem 3. 

Let $F\in {\mathcal{H}}^p\left({\Lambda }^p\right)$ and $A\in {\gamma }_F$. If $\Phi \in \mathcal{L}$, $h\in {\mathcal{L}}_2$ are the functions such that

$$L_F(r,A)\ge \Phi \left(r\right)(r\ge r_0),h\left(r\right)=o(\Phi \left(r\right))(r\to +\infty ),$$

then there exists a set $E\subset {\mathbb{R}}_{+}$ of zero asymptotic h-density (i.e., $D_h\left(E\right)=0$) such that

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

as $r\to +\infty$$(r\in {\mathbb{R}}_{+}\setminus E).$

Proof. 

Since Lemma 7 implies that the conditions of Theorem 2 are satisfied, the application of Theorem 2 completes the proof. □

Corollary 1. 

Let $F\in {\mathcal{H}}^p\left({\Lambda }^p\right)$ and $A\in {\gamma }_F$. Then, there exists a set $E\subset {\mathbb{R}}_{+}$ of finite Lebesgue measure such that

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

as $r\to +\infty$$(r\in {\mathbb{R}}_{+}\setminus E).$

Concrete examples of Dirichlet series can be obtained from the Riemann zeta-function $\zeta \left(z\right)={\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{n^z}}={\sum }_{n=1}^{\infty }{e}^{-zlnn},$ the Hurwitz zeta-function $\zeta (z,q)={\sum }_{n=0}^{\infty }{\displaystyle \frac{1}{{(q+n)}^z}}={\sum }_{n=0}^{\infty }{e}^{-zln(n+q)},$ the Lerch zeta function $L(\lambda ,z,\alpha )={\sum }_{n=0}^{\infty }{\displaystyle \frac{e^{2\pi i\lambda n}}{{(n+\alpha )}^z}}={\sum }_{n=0}^{\infty }{e}^{2\pi i\lambda n}{e}^{-zln(n+\alpha )}$, and so on. They have their own multidimensional generalizations. For example, the mutiple zeta-function

$$\zeta (z_1,\dots ,z_p)=\sum _{n_1>n_2>\dots >n_p>0}{\displaystyle \frac{1}{n_1^{z_1}{n}_2^{z_2}\dots n_p^{z_p}}}=\sum _{n_1>n_2>\dots >n_p>0}{e}^{-z_{1}ln{n}_1-z_{2}ln{n}_2-\dots -z_{n}ln{n}_p}$$

is the natural multivariate counterpart of the Riemann zeta-function, which is also represented by the Dirichlet series. More information about zeta-functions, their connection with the Dirichlet series, and their applications are in recent papers [57,58,59] and in the references within them.

5. Discussion

In the case $p=1$, the finiteness of the Lebesgue measure of the exceptional set in Relation (15) is the best possible description. This follows from the examples in papers [4,5].

Problem 1.

1. 

Is the finiteness of the Lebesgue measure the best possible description of the exceptional set E in the case $p>1$ in Corollary 1?

2. 

Is the description $D_h\left(E\right)=0$ the best possible description of the exceptional set E in the case $p>1$ in Theorem 3?

3. 

Is the description $D_h\left(E\right)=0$ the best possible description of the exceptional set E in the case $p>1$ in Theorem 2?

It would be interesting to obtain the possible extensions of such results to holomorphic functions in Banach spaces. Now, there are similar results for very special cases of Hilbert space with many additional assumptions [48,49,50,51]. Recently, the Wiman–Valiron’s theory was constructed for the Wilson operator [31,32] and for difference operator [29]. But there are still open problems to constructing such a theory for infinite-dimensional complex Banach spaces (for example, for entire functions defined in ${\mathsf{\ell }}_2$).

6. Conclusions

The main applications of the theory are a description of the local and asymptotic behavior of analytic solutions of ordinary and partial differential equations, difference equations, functional equations, and their systems. This allows one to deduce growth estimates and asymptotic relations for maximum modulus and the maximum of real parts for such solutions. Most numerical methods are applicable locally. In the best cases, they can give global estimates of boundary value problem solutions for real lines or n-dimensional real space. But these methods cannot overlap all analytic solutions independently of initial or boundary conditions. The developed approach in this paper is one of the powerful methods for the analytic theory of differential equations. This can be applied to study properties of the analytic solutions of these equations. In particular, one recent paper [53] shows, by the Wiman–Valiron theory, that all meromorphic solutions $w\left(z\right)=u(x-ct)$ under traveling wave transformation $z=x-ct$ of the sixth-order thin-film equation ${\displaystyle \frac{\partial u}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(u^n{u}_{xxxxx}\right)$ are elliptic functions, a rational function of $e^{\alpha z}$, or a rational function of z. And this result is independent of boundary and initial conditions.

The presented results are new in two directions. They use more a general approach to geometrically exhaust n-dimensional complex space. Mostly, various authors used balls for this goal. Strelitz only used the polylinear domains. It has been suggested to use A-like polylinear domains with arbitrary real n-dimensional vector A instead of positive real components. This exhaustion allows one to deduce new estimates for multiple Dirichlet series with arbitrary real exponents without assumptions that their components are strictly increasing.