# Entire Functions of Bounded L-Index: Its Zeros and Behavior of Partial Logarithmic Derivatives.

1. Introduction

In this paper, we find multidimensional sufficient conditions of boundedness of L-index in joint variables, which describe distribution of zeros and behavior of partial logarithmic derivatives. Recently, we published a paper  where some similar restrictions are established. Another approach was used by a slice function F([z.sup.0] + Zb), where [z.sup.0] [member of] [C.sup.n], t [member of] C, b is a given direction in [C.sup.n] \ {0}, F : [C.sup.n] [right arrow] C is an entire function. It is a background for concept of function of bounded L-index in direction (see definition and properties in [2, 3]). We proved that if an entire function in [C.sup.n] function F is of bounded [l.sub.j]-index in every direction [mathematical expression not reproducible], then F is of bounded L-index in joint variables for L = ([l.sub.1], ..., [l.sub.n]), [l.sub.j] : [C.sup.n] [right arrow] [R.sub.+] (Theorem 6, ). It helped us to find restrictions by directional logarithmic derivatives and distribution zeros in every direction [1.sub.j], j [member of] {1, ..., n}. We assumed that the logarithmic derivative in direction [1.sub.j] is bounded by a function [l.sub.j] outside some exceptional set, which contains all zeros of entire function F (see definition of [G.sup.b.sub.r](F) below). Prof. Chyzhykov paid attention in conversation with authors that this exceptional set is too small because it does not contain neighborhoods of some zeros of the function in [C.sup.n]. Thus, it leads to the following question: is there sufficient conditions of boundedness of L-index in joint variables with larger exceptional sets? We give a positive answer to this question (Theorem 10). Moreover, we obtain sufficient conditions of boundedness of L-index in joint variables by estimating the maximum modulus of an entire function on the skeleton in polydisc by minimum modulus (Theorem 7). Theorems 9 and 10 present restrictions by a measure of zero set of an entire function F, under which F has bounded L-index in joint variables. Nevertheless, we do not know whether the obtained conditions in Theorems 7-10 are necessary too in [C.sup.n], (n [greater than or equal to] 2). Note that these propositions are new even for entire functions of bounded index in joint variables, i. e. L = (1, ..., 1) (see definition and properties in [4-8]).

It is known  that for every entire function f with bounded multiplicities of zeros there exists a positive continuity on [0;+[infinity]) function l(r) (r = [absolute value of z]) such that f is of bounded Z-index. This result can be easily generalized for entire functions in [C.sup.n]. Thus, the concept of bounded L-index in joint variables allows the study of growth properties of any entire functions with bounded multiplicities of zero points.

It should be noted that the concepts of bounded L-index in a direction and bounded L-index in joint variables have few advantages in the comparison with traditional approaches to study properties of entire solutions of differential equations. In particular, if an entire solution has bounded index , then it immediately yields its growth estimates, a uniform distribution of its zeros, a certain regular behavior of the solution, and so forth. A full bibliography about application in theory of ordinary and partial differential equations is in [3,11,12].

The paper is devoted to two old problems in theory of entire and meromorphic functions. The first problem is the establishment of sharp estimates for the logarithmic derivatives of the functions in the unit disc outside some exceptional set. Chyzhykov et al. [13-16] considered various formulations of the problem. The obtained estimates were used to study properties of holomorphic solutions of differential equations. Instead, the authors assume that partial logarithmic derivative in every variable satisfies some inequalities (28) or (45).

Another interesting considered problem concerns zero sets of holomorphic function in [C.sup.n]. The different estimates of measure of zero set and its geometrical properties are investigated in [17-22]. We suppose that zero points of entire functions admit uniform distribution in some sense, that is, (29).

Below we use results from Ukrainian papers [23, 24], but they are also included in English monographs [3,11].

2. Main Definitions and Notations

We need some standard notations. Let [R.sub.+] = [0, +[infinity]). Denote 0 = (0, ..., 0) [member of] [R.sup.n.sub.+], 1 = (1, ..., 1) [member of] [R.sup.n.sub.+], 2 = (2, ..., 2) [member of] [R.sup.n.sub.+], [mathematical expression not reproducible].

For R = ([r.sub.1], ..., [r.sub.n]) [member of] [R.sup.n.sub.+] and K = ([k.sub.1], ..., [k.sub.n]) [member of] [Z.sup.n.sub.+] denote [parallel]R[parallel] = [r.sub.1] + ... + [r.sub.n], K! = [k.sub.1]! * ... * [k.sub.n]!. For = ([a.sub.1], ..., [a.sub.n]) [member of] [C.sup.n], b = ([b.sub.1], ..., [b.sub.n]) [member of] [C.sup.n], z = ([z.sub.1], ..., [z.sub.n]) [member of] [C.sup.n], we will use formal notations without violating the existence of these expressions:

[mathematical expression not reproducible] (1)

If a, b [member of] [R.sup.n] the notation a < b means that [a.sub.j] < [b.sub.j](j = 1, ..., n); similarly, the relation a [less than or equal to] b is defined.

The polydisc {z [member of] [C.sup.n] : [absolute value of ([z.sub.j] - [z.sup.0.sub.j])] < [r.sub.j], j = 1, ..., n} is denoted by [D.sup.n]([z.sup.0], R), its skeleton {z [member of] [C.sup.n] : [absolute value of ([z.sub.j] - [z.sup.0.sub.j])] = [r.sub.j], j = 1, ..., n} is denoted by [T.sup.n]([z.sup.0], R), and the closed polydisc {z [member of] [C.sup.n] : [absolute value of ([z.sub.j] - [z.sup.0.sub.j])] [less than or equal to] [r.sub.j], j = 1, ..., n} is denoted by [D.sup.n][[z.sup.0], R].

For K = ([k.sub.1], ..., [k.sub.n]) [member of] [Z.sup.n.sub.+] and partial derivatives of entire function F(z) = F([z.sub.1], ..., [z.sub.n]) we will use the notation

[mathematical expression not reproducible]. (2)

Let L(z) = ([l.sub.1](z), ..., [l.sub.n](z)), where [l.sub.j](z) are positive continuous functions of z [member of] [C.sup.n], j [member of] {1, 2, ..., n}. An entire function, F(z), z [member of] [C.sup.n], is called a function of bounded L-index in joint variables  if there exists a number m [member of] [Z.sub.+] such that for all z [member of] [C.sup.n] and J = ([j.sub.1], [j.sub.2], ..., [j.sub.n]) [member of] [Z.sup.n.sub.+]

[[absolute value of ([F.sup.(J)](z))]/J![L.sup.J](z)] [less than or equal to] max {[[absolute value of ([F.sup.(K)](z))]/K![L.sup.K](z)] : K [member of] [Z.sup.n.sub.+], [parallel]K[parallel] [less than or equal to] m}. (3)

If [l.sub.j] = [l.sub.j]([absolute value of ([z.sub.j])]) then we obtain a concept of entire functions of bounded L-index in a sense of definition given in . If [l.sub.j]([z.sub.j]) = 1, j [member of] {1, 2, ..., n}, then the entire function is called a function of bounded index in joint variables [4-8, 25].

The least integer m for which inequality (3) holds is called L-index in joint variables of the function F and is denoted by N(F, L).

For R [member of] [R.sup.n.sub.+], j [member of] {1, ..., n} and L(z) = ([l.sub.1](z), ..., [l.sub.n](z)) we define

[mathematical expression not reproducible] (4)

By [Q.sup.n] we denote a class of functions L(z) which for every R [member of] [R.sup.n.sub.+] and j [member of] {1, ..., n} satisfy the condition

0 < [[lambda].sub.1,j] (R) [less than or equal to] [[lambda].sub.2,j] (R) < +[infinity]). (5)

If n = 1 then Q [equivalent to] [Q.sup.1].

Let [mathematical expression not reproducible]. A notation [mathematical expression not reproducible] means that there exist [[THETA].sub.1] = ([[theta].sub.1,j], ..., [[theta].sub.1,n]) [member of] [R.sup.n.sub.+], [[THETA].sub.2] = ([[theta].sub.2,j], ..., [[theta].sub.2,n]) [member of] [R.sup.n.sub.+] such that [for all]z [member of] [C.sup.n] [[theta].sub.1,j][[??].sub.j](z) [less than or equal to] [l.sub.j](z) [less than or equal to] [[theta].sub.2,j][[??].sub.j](z).

3. Auxiliary Propositions

We need the following theorems.

Theorem 1 ([11, p. 158, Th. 4.2], see also ). Let L [member of] [Q.sup.n] and [mathematical expression not reproducible]. An entire function F : [C.sup.n] [right arrow] C has bounded [??]-index in joint variables if and only if F has bounded L-index in joint variables.

Theorem 2 (see ). Let L [member of] [Q.sup.n]. An entire function F is of bounded L-index in joint variables if and only if, for any R', R", 0 < R' < R", there exists a number [p.sub.1] = [p.sub.1](R', R") [greater than or equal to] 1 such that for every [z.sup.0] [member of] [C.sup.n] inequality

[mathematical expression not reproducible] (6)

holds.

Remark 3. It was also proved that the condition "for any R', R", 0 < R' < R", there exists a number [p.sub.1] = [p.sub.1](R', R") [greater than or equal to] 1" in Theorem 2 can be replaced by the condition "there exist R', R", 0 < R' < 1 < R", and [p.sub.1] = [p.sub.1](R', R") [greater than or equal to] 1". It is Theorem 5 in .

Now we relax the restriction R' < 1 < R" in sufficient conditions.

Theorem 4. Let L [member of] [Q.sup.n], F : C [right arrow] [C.sup.n] be an entire function. If there exist R', R", 0 < R' < R", and [p.sub.1] = [p.sub.1](R', R") [greater than or equal to] 1 such that for every [z.sup.0] [member of] [C.sup.n] inequality (6) holds; then the function F has bounded L-index in joint variables.

Proof. From (6) with 0 < R' < "n it follows that

[mathematical expression not reproducible] (7)

Denoting [??](z) = 2L(z)/(R' + R"), we obtain

[mathematical expression not reproducible] (8)

where 0 < 2R'/(R' + R") < 1 < 2R"/(R' + R"). In view of Remark 3, F has bounded [??]-index in joint variables. By Theorem 1, the function F is bounded L-index in joint variables.

Note that Theorem 4 is new even if L(z) [equivalent to] 1.

Lemma 5. If L : [C.sup.n] [right arrow] [R.sub.+] is a continuous function such that ([for all]R [member of] [R.sup.n.sub.+]) [[LAMBDA].sub.2](R) < [infinity] then ([for all]R [member of] [R.sup.n.sub.+]) [[LAMBDA].sub.1](R) [greater than or equal to] 1/[[LAMBDA].sub.2](R[[LAMBDA].sub.2](R)) > 0.

Proof. Let ([for all]R [member of] [R.sup.n.sub.+]) [[LAMBDA].sub.2](R) < [infinity] i.e. [for all]j [member of] {1, ..., n} [[lambda].sub.2,j](R) < +[infinity]. Hence, we have [l.sub.j](z) [less than or equal to] [[lambda].sub.2,j](R)[l.sub.j]([z.sup.0]) for z [member of] [D.sup.n]([z.sup.0], R/L([z.sup.0])). This means that [absolute value of ([z.sub.j] - [z.sup.0.sub.j])] [less than or equal to] [r.sub.j]/[l.sub.j]([z.sup.0]) [less than or equal to] [r.sub.j][[lambda].sub.2,j](R)/[l.sub.j](z). Using definition of [[lambda].sub.1,j](R), we deduce

[mathematical expression not reproducible] (9)

Thus, [[lambda].sub.1,j](R) [greater than or equal to] 1/[[lambda].sub.2,j](R[[LAMBDA].sub.2](R)).

Remark 6. By Lemma 5 the left inequality in (5) is excessive because the condition [[lambda].sub.2,j]R) < +[infinity] implies [[lambda].sub.1,j](R) > 0. But in our considerations we will use so [[LAMBDA].sub.1](R) as [[LAMBDA].sub.2](R). It is convenient.

4. Estimate Maximum Modulus on a Skeleton in Polydisc

Let [Z.sub.F] be a zero set of entire function F. We denote

[mathematical expression not reproducible] (10)

Theorem 7. Let L [member of] [Q.sup.n], F be an entire in [C.sup.n] function. If [there exists]R > 0 [there exists][p.sub.2] [greater than or equal to] 1 [there exists][THETA] [member of] [R.sup.n.sub.+], 0 < [THETA] < R, [there exists]R' > 0, (R' = 0 for [Z.sub.F] = 0) such that [for all][z.sup.0] [member of] [C.sup.n] [there exists][R.sup.0] = [R.sup.0]([z.sup.0]) [member of] [R.sup.n.sub.+], [THETA] [less than or equal to] [R.sup.0] [less than or equal to] R, for which

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

then the function F has bounded L-index in joint variables (meas is the Lebesgue measure on the skeleton in the polydisc).

Proof. By Theorem 4, we will show that [there exists][p.sub.1] > 0 [for all][z.sup.0] [member of] [C.sup.n]

max{[absolute value of (F(z))] : z [member of] [T.sup.n] ([z.sup.0], [R + 1/L([z.sup.0])]} [less than or equal to] [p.sub.1] max{[absolute value of (F(z))] : z [member of] [T.sup.n] ([z.sup.0], [R/L([z.sup.0])]}. (13)

Denote [l.sup.*.sub.j] = max{[l.sub.j](z) : z [member of] [D.sup.n][[z.sup.0], 2(R + 1)/L([z.sup.0])]}, [[rho].sub.j,0] = [r.sub.j]/[l.sub.j]([z.sup.0]), [[rho].sub.j,k] = [[rho].sub.j,0] + (k * [[theta].sub.j])/[l.sup.*.sub.j], k [member of] N, j [member of] {1, ..., n}. The following estimate holds

[[theta].sub.j]/[l.sup.*.sub.j] < [r.sub.j]/[l.sup.*.sub.j] [less than or equal to] [r.sub.j]/[l.sub.j]([z.sup.0]) < 2[r.sub.j] + 2/[l.sub.j]([z.sup.0]) - [r.sub.j] + 1/[l.sub.j]([z.sup.0]). (14)

Hence, there exists [S.sup.*] = ([s.sup.*.sub.1], ..., [s.sup.*.sub.n]]) [member of] N independent of [z.sup.0] such that

[mathematical expression not reproducible] (15)

for some [m.sub.j] = [m.sub.j]([z.sup.0]) [less than or equal to] [s.sup.*.sub.j] because L [member of] [Q.sup.n]. Indeed,

[mathematical expression not reproducible] (16)

Thus, [s.sup.*.sub.j] = [(([r.sub.j]+2)/[[theta].sub.j])[[lambda].sub.2,j](2(R+1))], where[x] is the integer part of x [member of] R.

Let [M.sub.0] = ([m.sub.1], .... [m.sub.n]) and [[tau].sup.**.sub.K] be such a point in [C.sup.n] that

[absolute value of (F([[tau].sup.**.sub.K]))] = max {[absolute value of F(z))] : z [member of] [T.sup.n] ([z.sup.0], [R.sub.K])}, (17)

where K = ([k.sub.1], ..., [k.sub.n]), [mathematical expression not reproducible] and [[tau].sup.**.sub.j,K] be the intersection point in C of the segment [[z.sup.0.sub.j], [t.sup.**.sub.j,K]] with [mathematical expression not reproducible]. We construct a sequence of polydisc [D.sup.n]([z.sup.0], [R.sub.K]) with K [less than or equal to] [M.sub.0], [R.sub.0] = R/L([z.sup.0]) = ([[rho].sub.1,0], ..., [[rho].sub.n,0]) and [THETA]/L([z.sup.0]) = ([[theta].sub.1]/[l.sup.*.sub.1], ..., [[theta].sub.n]/[l.sup.*.sub.n]]) (see Figures 1 and 2).

Denote [[alpha].sup.(j)].sub.K] = ([[tau].sup.**.sub.1,K], ... [[tau].sup.**.sub.j-1], [[tau].sup.*.sub.j,K], [[tau].sup.**.sub.j+1,K], ..., [[tau].sup.**.sub.n,K]). Hence, for every [r.sub.j] > [[theta].sub.j] and K [less than or equal to] [S.sup.*] : [absolute value of ([[tau].sup.*.sub.j,K] - [[tau].sup.**.sub.j,K])] = [[theta].sub.j]/[l.sup.*.sub.j] [less than or equal to] [r.sub.j]/[l.sub.j]([[alpha].sup.(j).sub.K]). Thus, for some [R.sup.0] = [R.sup.0]([[alpha].sup.(j)].sub.K]) [member of] [R.sup.n.sub.+], [THETA] [less than or equal to] [R.sup.0] [less than or equal to] R, we deduce

[mathematical expression not reproducible] (18)

To deduce (18), we implicitly used that

[mathematical expression not reproducible] (19)

Condition (11) provides (19). Indeed, we will find a lower estimate of measure of the set [mathematical expression not reproducible] and will show that the measure is not lesser than a left part of inequality (11).

The set [mathematical expression not reproducible] is a Cartesian product of the following arcs on circles: for every m [member of] {1, ..., n}, m [not equal to] j (see Figure 3)

[mathematical expression not reproducible] (20)

and for m = j (see Figure 4)

[mathematical expression not reproducible] (21)

It is easy to prove that the length of arc equals

[mathematical expression not reproducible] (22)

[mathematical expression not reproducible] (23)

But for [mathematical expression not reproducible] and [mathematical expression not reproducible] the argument in arccosine from (23) and (22) does not exceed 1/2. This means that the length of arc is not lesser than

[2[r.sup.0.sub.m]/[l.sub.m]([[alpha].sup.(j)].sub.K])] arccos [1/2] [greater than or equal to] [2[[theta].sub.m][pi]/3[l.sub.m]([z.sup.0])[[lambda].sub.2,m](2 (R + 1)) for every m [member of] {1, 2, ..., n}, (24)

because L [member of] [Q.sup.n]. Accordingly, the measure of the set

[mathematical expression not reproducible] (25)

on the skeleton of polydisc is always not lesser than [[PI].sup.n.sub.m=1](2[[theta].sub.m][pi]/3[l.sub.m]([z.sup.0])[[lambda].sub.2,m](2(R + 1))). Assuming a strict inequality in (11), we deduce that (19) is valid.

Applying (18) [m.sub.j]th times in every variable [z.sub.j], we obtain

[mathematical expression not reproducible] (26)

By Theorem 2 the function F has bounded L-index in joint variables.

Let us denote c(z', r) = {z [member of] C : [absolute value of (z - z')] = r/l(z')}. For n = 1 Theorem 7 implies the following corollary.

Corollary 8. Let l [member of] Q, f be an entire function. If [there exists]r > 0, [there exists]r' [greater than or equal to] 0, [there exists][p.sub.2] [greater than or equal to] 1 [there exists][theta] [member of] (0, r), such that [for all][z.sup.0] [member of] C [there exists][r.sup.0] = [r.sup.0]([z.sup.0]) [member of] [[theta]; r], and meas{c([z.sup.0], [r.sup.0]) [intersection] [G.sub.R'](F)} < 2[pi][theta]/3l([z.sup.0])[[lambda].sub.2](2r + 2) and

max{[absolute value of (f(z))] : z [member of] c ([z.sup.0], [r.sup.0])} [less than or equal to] [p.sub.2] min {[absolute value of (f(z))] : z [member of] c ([z.sup.0], [r.sup.0])\[G.sub.r'](f)} (27)

then the function f has bounded l-index (here meas means the Lebesgue measure on the circle).

In a some sense, this corollary is new even for an entire function of one variable because the circle c([z.sup.0], [r.sup.0]) can contain zeros of the function f. Meanwhile, in corresponding theorem from [26, 27] the circle c([z.sup.0], [r.sup.0]) is chosen such that f(z) [not equal to] 0 for all z [member of] c([z.sup.0], [r.sup.0]).

5. Behavior of Partial Logarithmic Derivatives

Denote J = {([j.sub.1], ..., [j.sub.n]) : [j.sub.i] [member of] {0,1}, i [member of] {1, ..., n}} \ 0.

Theorem 9. Let L [member of] [Q.sup.n]. If an entire function F satisfies the following conditions

(1) for every R > 0 there exists [p.sub.1] = [p.sub.1](R) > 0 such that for all z [member of] [C.sup.n] \ [G.sub.R](F) and for all J [member of] J

[absolute value of ([ln F(z)).sup.(J)])] [less than or equal to] [p.sub.1][L.sup.J](z), (28)

where ln F(z) is the principal value of logarithm.

(2) for every R > 0 and R' [greater than or equal to] 0 exists [p.sub.2] = [p.sub.2](R, R') [greater than or equal to] 1 that for all [z.sup.0] [member of] [C.sup.n] such that [T.sup.n]([z.sup.0], R/L([z.sup.0])) \ [G.sup.R'](F) = [[universal].sub.i] [C.sub.i] [not equal to] 0, where the sets [C.sub.i] are connected disjoint sets, and either (a) [mathematical expression not reproducible], or (b) [mathematical expression not reproducible], or (c) [mathematical expression not reproducible], and [z.sup.*], [z.sup.**] belong to the same set [mathematical expression not reproducible]

(3) for every R > 0 there exists [n.sup.*] (R) > 0 such that for all z [member of] [C.sup.n]

meas {[Z.sub.F] [intersection] [D.sup.n] [z, [R/L(z)]} [less than or equal to] [n.sup.*](R). (29)

then F has bounded L-index in joint variables (here meas is (2n - 2)-dimensional of the Lebesgue measure).

Proof. Let [z.sup.0] [member of] [C.sup.n] be arbitrarily chosen point. In view of Theorem 7 we need to prove that

[mathematical expression not reproducible] (30)

for some [R.sup.0] = [R.sup.0]([z.sup.0]).

Let R > 0 be arbitrary radius. We choose 0, R' [member of] [R.sup.n.sub.+] such that [[theta].sub.j] < 2[r.sub.j]/(2 + 3[[lambda].sub.2,j](2(R + 1))),

[mathematical expression not reproducible] (31)

Let dS = d[s.sub.1] * ... * d[s.sub.n], S = ([s.sub.1], ..., [s.sub.n]), [[omega].sub.z] be a volume measure in [R.sup.2n]. Clearly, (see [28, p. 75-76])

[mathematical expression not reproducible] (32)

where u is plurisubharmonic function. Hence,

[mathematical expression not reproducible] (33)

Obviously, there can exist points z' [member of] [Z.sub.F] [D.sup.n][[z.sup.0], R/L([z.sup.0])] such that

[D.sup.n][[z.sup.0], [R/L([z.sup.0])] [intersection] [D.sup.n] [z', [R'/L(z')]] [not equal to] 0. (34)

Let [z".sub.j] be the intersection point of the segment [[z.sup.0.sub.j], [z'.sub.j]] and the circle [absolute value of ([z.sub.j] - [z.sup.0.sub.j])] = [r.sub.j]/[l.sub.j]([z.sup.0]), j [member of] {1, ..., n}. Then [absolute value of ([z".sup.j] - [z'.sub.j])] [less than or equal to] [r'.sub.j]/[l.sub.j](z') and z" [member of] [T.sup.n]([z.sup.0], R/L([z.sup.0])). Using L [member of] [Q.sup.n], we estimate maximum distance between [z.sup.0.sub.j] and [z'.sub.j]:

[mathematical expression not reproducible] (35)

Denote R" = R + R[[LAMBDA].sub.2](R)/[[LAMBDA].sub.1](R). Let [V.sub.2n-2] be a (2n - 2)-dimensional volume, [[chi].sub.F](z) a characteristic function of zero set of the function F. Now we replace the measure in (33) by integrating on zero set in polydisc [D.sup.n][[z.sup.0], R"/L([z.sup.0])]:

[mathematical expression not reproducible]. (36)

Besides, we have that

[mathematical expression not reproducible] (37)

Hence, the following difference is positive

[mathematical expression not reproducible] (38)

because [[theta].sub.j] < 2[r.sub.j]/(2 + 3[[lambda].sub.2,j](2(R + 1))). From (36) it follows that

[mathematical expression not reproducible]. (39)

By mean value theorem there exists [R.sup.0] = [R.sup.0]([z.sup.0]) with [r.sub.j] [member of] [[[theta].sub.j], [r.sub.j]] such that

[mathematical expression not reproducible]. (40)

Hence, in view of (39) we obtain a desired inequality

[mathematical expression not reproducible]. (41)

Clearly, for every point [z.sup.0] [member of] [C.sup.n] we have [T.sup.n]([z.sup.0], [R.sup.0]/L([z.sup.0]))\[Z.sub.F] = [[universal].sub.i] [C'.sub.i], where [C'.sub.i] are connected disjoint sets, [C'.sub.i] [contains] [C.sub.i] and [C.sub.i] is defined in condition (2). Let [z.sup.*] [member of] [T.sup.n]([z.sup.0], R/L([z.sup.0])) be such that [absolute value of (F([z.sup.*]) = max||F(z))] : z [member of] [T.sup.n]([z.sup.0], [R.sup.0]/f([z.sup.0]))}. Then there exists [i.sub.0] such that [mathematical expression not reproducible]. Let [mathematical expression not reproducible] be such that [mathematical expression not reproducible]. We choose J = ([j.sub.1], ..., [j.sub.n]) [member of] J, where

[mathematical expression not reproducible], (42)

and deduce

[mathematical expression not reproducible]. (43)

Hence,

[mathematical expression not reproducible]. (44)

By Theorem 7 the function F has bounded L-index in joint variables.

Let us to denote [DELTA] as Laplace operator. We will consider [DELTA] ln [absolute value of F] as generalized function. Using some known results from potential theory, we can rewrite Theorem 9 as follows.

Theorem 10. Let L [member of] [Q.sup.n]. If an entire function F satisfies the following conditions

(1) for every R > 0 there exists [p.sub.1] = [p.sub.1](R) > 0 such that for all z [member of] [C.sup.n] \ [G.sub.R](F) and for every j [member of] {1, ..., n}

[absolute value of ([partial derivative] ln F(z)/[partial derivative][z.sub.j])] [less than or equal to] [p.sub.1][l.sub.j](z), (45)

where ln F(z) is the principal value of logarithm.

(2) for every R > 0 and R' [greater than or equal to] 0 exists [p.sub.2] = [p.sub.2](R, R') [greater than or equal to] 1 that for all [z.sup.0] [member of] [C.sup.n] such that [T.sup.n]([z.sup.0], R/L([z.sup.0])) \ [G.sub.R'](F) = [[universal].sub.i][C.sub.i] [not equal to] 0, where the sets [C.sub.i] are connected disjoint sets, and either (a) [mathematical expression not reproducible], or (b) [mathematical expression not reproducible], or (c) [mathematical expression not reproducible], and [z.sup.*], [z.sup.**] belong to the same set [mathematical expression not reproducible]

(3) for every F > 0 there exists [n.sup.*](R) > 0 such that for all z [member of] [C.sup.n]

[mathematical expression not reproducible] (46)

then F has bounded L-index in joint variables.

Proof. Ronkin [28, p. 230] deduced the following formula for entire function:

[mathematical expression not reproducible] (47)

where [[gamma].sub.F](z) is a multiplicity of zero point of the function F at point z, [R.sup.*] [member of] [R.sup.n.sub.+] is arbitrary radius. Let [[chi].sub.F](z) be a characteristic function of zero set of F. Then [[chi].sub.F](z) [less than or equal to] [[chi].sub.F](z). Hence,

[mathematical expression not reproducible] (48)

that is, inequality (29) holds.

Now we want to prove that (45) implies (28). For every J [member of] J\[[universal].sup.n.sub.k=1] [1.sub.k] and [z.sup.0] [member of] [C.sup.n]\[G.sub.R](F), Cauchy's integral formula can be written in the following form

[mathematical expression not reproducible] (49)

where m is such that [j.sub.m] = 1.

If [z.sup.0] [member of] [C.sup.n] \ [G.sub.R](F) and z' [member of] [Z.sub.F] [subset] [G.sub.R](F), then for every j [member of] {1, ..., n}

[absolute value of ([z.sup.0.sub.j] - [z'.sub.j])] [greater than or equal to] [[r.sub.j]/[l.sub.j](z') [greater than or equal to] [[r.sub.j][[lambda].sub.1,j](R)/[l.sub.j]([z.sup.0])] > [[r.sub.j][[lambda].sub.1,j](R)/2[l.sub.j]([z.sup.0])]. (50)

Let us consider the set [mathematical expression not reproducible]. We want to find the greatest radius [R.sup.*] [member of] [R.sup.n.sub.+] such that [mathematical expression not reproducible]:

[R/L(z')] - [R[[LAMBDA].sub.1](R)/2L([z.sup.0])] [greater than or equal to] [R/L(z')] - [R[[LAMBDA].sub.1](R)/2[[LAMBDA].sub.1]L(z')] = [R/2L(z')]. (51)

Thus, for [mathematical expression not reproducible]. Using (45), we obtain that for every [z.sup.0] [member of] [C.sup.n] \ [G.sub.R](F)

[mathematical expression not reproducible] (52)

where C(R) = 0.5[p.sub.1]((1/3)R)[max.sub.J[member of]J]{[[lambda].sub.2,m](0.5R[[LAMBDA].sub.1](R))[r.sub.m] [[lambda].sub.1,m](R)[(2/R[[LAMBDA].sub.1](R)).sup.J]}. Thus, we proved that inequality (28) is valid.

For n = 1 Theorem 9 implies the following corollary.

Corollary 11. Let l [member of] Q, f be an entire in C function, n(r, [z.sup.0], f) a number of zeros of the f in the disc [absolute value of (z - [z.sub.0])] [less than or equal to] r/l([z.sup.0]). If the function f satisfies the following conditions:

(1) for every r > 0 there exists [p.sub.1] = [p.sub.1](r) > 0 such that for all z [member of] C \ [G.sub.r](f)

[absolute value of (f'(z)/f(z))] [less than or equal to] [p.sub.1]l(z), (53)

(2) for every r > 0 and r' [greater than or equal to] 0 exists [p.sub.2] = [p.sub.2](r, r') [greater than or equal to] 1 that for all [z.sup.0] [member of] C such that [z [member of] C : [absolute value of (z - [z.sup.0])] = r/l([z.sup.0])} [G.sub.r'](f) = [[universal].sub.i] [C.sub.i] [not equal to] 0, where the sets [C.sub.i] are connected disjoint sets, and either (a) [mathematical expression not reproducible], or (b) [mathematical expression not reproducible], or (c) [mathematical expression not reproducible], and [z.sup.*], [z.sup.**] belong to the same set [mathematical expression not reproducible]

(3) for every r > 0 there exists [n.sup.*](r) > 0 such that for all [z.sup.0] [member of] C n(r, [z.sup.0], f) [less than or equal to] [n.sup.*](r),

then f has bounded l-index.

It is known (see [12, 27, 29]) that in one-dimensional case conditions (1) and (3) of Corollary 11 are necessary and sufficient for boundedness of l-index or index. Thus, condition (2) is excessive in the case. But for [C.sup.n] (n [greater than or equal to] 2), it is required because [D.sup.n][[z.sup.0], R/L([z.sup.0])] \ [G.sub.R'](F) is a multiply connected domain, when [D.sup.n][[z.sup.0], R/L([z.sup.0])] contains zeros of the function F.

We need some notations from . Let b [member of] [C.sup.n] \ {0} be a given direction. For a given [z.sup.0] [member of] [C.sup.n] we denote [mathematical expression not reproducible]. If one has [mathematical expression not reproducible] for all t [member of] C, then [G.sup.b.sub.r](F, [z.sup.0]) := 0; if [mathematical expression not reproducible], then [G.sup.b.sub.r](F, [z.sup.0]) := [[z.sup.0] + tb : t [member of] C}. And if [mathematical expression not reproducible] and [a.sup.0.sub.k] are zeros of the function [mathematical expression not reproducible], then [G.sup.b.sub.r](F, [z.sup.0]) := [[universal].sub.k][[z.sup.0] + tb : [absolute value of (t - [a.sup.0.sub.k])] [less than or equal to] r/L([z.sup.0] + [a.sup.0.sub.k]b)}, r > 0. Let

[mathematical expression not reproducible] (54)

Remark 12. In [1, Theorem 8], sufficient conditions of boundedness of L-index in joint variables were obtained, which are similar to Theorem 10. Particularly, we assumed the validity of inequality (45) for all [mathematical expression not reproducible]. However, [mathematical expression not reproducible], where R = ([r.sub.1], ..., [r.sub.n]). Thus, condition (1) in Theorem 10 is weaker than the corresponding assumption in Theorem 8 from .

https://doi.org/10.1155/2017/3253095

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Andriy Bandura (1) and Oleh Skaskiv (2)

(1) Department of Advanced Mathematics, Ivano-Frankivsk National Technical University of Oil and Gas, 15 Karpatska St., Ivano-Frankivsk 76019, Ukraine

(2) Department of Function Theory and Theory of Probability, Ivan Franko National University of Lviv, 1 Universytetska St, Lviv 79000, Ukraine

Correspondence should be addressed to Andriy Bandura; andriykopanytsia@gmail.com

Received 5 August 2017; Accepted 15 October 2017; Published 6 November 2017

Academic Editor: Stanislawa Kanas Caption: Figure 1:

Caption: Figure 2:

Caption: Figure 3: With r' = [r.sup.0.sub.m]/[l.sub.m]([[alpha].sup.(j)].sub.K])).

Caption: Figure 4: With r' = [r.sup.0.sub.j]/[l.sub.j]([[alpha].sup.(j)].sub.K]).
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