In this paper, we obtain new sufficient conditions of boundedness of L-index in joint variables for entire function in Cn functions. They give an estimate of maximum modulus of an entire function by its minimum modulus on a skeleton in a polydisc and describe the behavior of all partial logarithmic derivatives and the distribution of zeros. In some sense, the obtained results are new for entire functions of bounded index and l-index in C too. They generalize known results of Fricke, Sheremeta, and Kuzyk.

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 [1] where some similar restrictions are established. Another approach was used by a slice function F(z0+tb), where z0∈Cn,t∈C,b is a given direction in Cn∖{0},F:Cn→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 Cn function F is of bounded lj-index in every direction 1j=(0,…,0,1︸jthplace,0,…,0), then F is of bounded L-index in joint variables for L=(l1,…,ln),lj:Cn→R+ (Theorem 6, [1]). It helped us to find restrictions by directional logarithmic derivatives and distribution zeros in every direction 1j,j∈{1,…,n}. We assumed that the logarithmic derivative in direction 1j is bounded by a function lj outside some exceptional set, which contains all zeros of entire function F (see definition of Grb(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 Cn. 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 Cn,(n≥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 [9] that for every entire function f with bounded multiplicities of zeros there exists a positive continuity on [0;+∞) function l(r) (r=|z|) such that f is of bounded l-index. This result can be easily generalized for entire functions in Cn. 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 [10], 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 Cn. 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+=[0,+∞). Denote 0=(0,…,0)∈R+n, 1=(1,…,1)∈R+n,2=(2,…,2)∈R+n, 1j=(0,…,0,1︸jthplace,0,…,0)∈R+n.

For R=(r1,…,rn)∈R+n and K=(k1,…,kn)∈Z+n denote R=r1+⋯+rn, K!=k1!·…·kn!. For =(a1,…,an)∈Cn, b=(b1,…,bn)∈Cn, z=(z1,…,zn)∈Cn, we will use formal notations without violating the existence of these expressions:(1)a=a1,a2,…,an,A±B=a1±b1,…,an±bn,AB=a1b1,…,anbn,AB=a1b1,…,anbn,AB=a1b1a2b2·…·anbn,dz=dz1dz2⋯dzn.If a,b∈Rn the notation a<b means that aj<bj(j=1,…,n); similarly, the relation a≤b is defined.

The polydisc {z∈Cn:|zj-zj0|<rj,j=1,…,n} is denoted by Dn(z0,R), its skeleton z∈Cn:zj-zj0=rj,j=1,…,n is denoted by Tn(z0,R), and the closed polydisc z∈Cn:zj-zj0≤rj,j=1,…,n is denoted by Dn[z0,R]. For K=(k1,…,kn)∈Z+n and partial derivatives of entire function F(z)=F(z1,…,zn) we will use the notation (2)FKz=∂KF∂zK=∂k1+⋯+knf∂z1k1⋯∂znkn.

Let L(z)=(l1(z),…,ln(z)), where lj(z) are positive continuous functions of z∈Cn, j∈{1,2,…,n}. An entire function, F(z),z∈Cn, is called a function of bounded L-index in joint variables [1] if there exists a number m∈Z+ such that for all z∈Cn and J=(j1,j2,…,jn)∈Z+n(3)FJzJ!LJz≤maxFKzK!LKz:K∈Z+n,K≤m.

If lj=lj(|zj|) then we obtain a concept of entire functions of bounded L-index in a sense of definition given in [24]. If lj(zj)≡1,j∈{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∈R+n, j∈{1,…,n} and L(z)=(l1(z),…,ln(z)) we define(4)λ1,jR=infz0∈Cninfljzljz0:z∈Dnz0,RLz0,λ2,jR=supz0∈Cnsupljzljz0:z∈Dnz0,RLz0,Λ1R=λ1,jR,…,λ1,nR,Λ2R=λ2,1R,…,λ2,nR.

By Qn we denote a class of functions L(z) which for every R∈R+n and j∈{1,…,n} satisfy the condition(5)0<λ1,jR≤λ2,jR<+∞.If n=1 then Q≡Q1.

Let L~(z)=(l~1(z),…,l~n(z)). A notation L≍L~ means that there exist Θ1=(θ1,j,…,θ1,n)∈R+n, Θ2=(θ2,j,…,θ2,n)∈R+n such that ∀z∈Cnθ1,jl~j(z)≤lj(z)≤θ2,jl~j(z).

3. Auxiliary Propositions

We need the following theorems.

Theorem 1 ([<xref ref-type="bibr" rid="B8">11</xref>, p. 158, Th. 4.2], see also [<xref ref-type="bibr" rid="B4">23</xref>]).

Let L∈Qn and L≍L~. An entire function F:Cn→C has bounded L~-index in joint variables if and only if F has bounded L-index in joint variables.

Theorem 2 (see [<xref ref-type="bibr" rid="B3">1</xref>]).

Let L∈Qn. 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 p1=p1(R′,R′′)≥1 such that for every z0∈Cn inequality(6)maxFz:z∈Tnz0,R′′Lz0≤p1maxFz:z∈Tnz0,R′Lz0.holds.

Remark 3.

It was also proved that the condition “for any R′, R′′, 0<R′<R′′, there exists a number p1=p1R′,R′′≥1” in Theorem 2 can be replaced by the condition “there exist R′, R′′, 0<R′<1<R′′, and p1=p1R′,R′′≥1”. It is Theorem 5 in [1].

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

Theorem 4.

Let L∈Qn, F:C→Cn be an entire function. If there exist R′, R′′, 0<R′<R′′, and p1=p1R′,R′′≥1 such that for every z0∈Cn inequality (6) holds; then the function F has bounded L-index in joint variables.

Proof.

From (6) with 0<R′<R′′ it follows that(7)maxFz:z∈Tnz0,2R′′R′+R′′R′+R′′2Lz0≤P1maxFz:z∈Tnz0,2R′R′+R′′R′+R′′2Lz0.Denoting L~z=2Lz/R′+R′′, we obtain (8)maxFz:z∈Tnz0,2R′′R′+R′′L~z0≤P1maxFz:z∈Tnz0,2R′R′+R′′L~z0,where 0<2R′/R′+R′′<1<2R′′/R′+R′′. In view of Remark 3, F has bounded L~-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)≡1.

Lemma 5.

If L:Cn→R+ is a continuous function such that (∀R∈R+n)Λ2(R)<∞ then (∀R∈R+n)Λ1(R)≥1/Λ2(RΛ2(R))>0.

Proof.

Let (∀R∈R+n)Λ2(R)<∞ i.e. ∀j∈{1,…,n}λ2,j(R)<+∞. Hence, we have lj(z)≤λ2,j(R)lj(z0) for z∈Dn(z0,R/L(z0)). This means that zj-zj0≤rj/lj(z0)≤rjλ2,j(R)/lj(z). Using definition of λ1,j(R), we deduce(9)infljzljz0:z∈Dnz0,RLz0=1supljz0/ljz:z∈Dnz0,R/Lz0≥1supljz0/ljz:zj0-zj≤rjλ2,jR/ljz,j∈1,…,n≥1λ2,jRΛ2R.Thus, λ1,j(R)≥1/λ2,j(RΛ2(R)).

Remark 6.

By Lemma 5 the left inequality in (5) is excessive because the condition λ2,j(R)<+∞ implies λ1,j(R)>0. But in our considerations we will use so Λ1(R) as Λ2(R). It is convenient.

4. Estimate Maximum Modulus on a Skeleton in Polydisc

Let ZF be a zero set of entire function F. We denote(10)GRF=⋃z0∈ZFz∈Cn:zj-zj0<rjljz0∀j∈1,2,…,n=⋃z0∈ZFDnz0,RLz0.

Theorem 7.

Let L∈Qn,F be an entire in Cn function. If ∃R>0∃p2≥1∃Θ∈R+n, 0<Θ<R, ∃R′>0, (R′=0 for ZF=∅) such that ∀z0∈Cn∃R0=R0(z0)∈R+n, Θ≤R0≤R, for which(11)measTnz0,R0Lz0∩GR′F<2π3n∏j=1nθjλ2,j2R+1ljz0,(12)maxFz:z∈Tnz0,R0Lz0≤p2minFz:z∈Tnz0,R0Lz0∖GR′Fthen 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 ∃p1>0∀z0∈Cn(13)maxFz:z∈Tnz0,R+1Lz0≤p1maxFz:z∈Tnz0,RLz0.Denote lj∗=maxlj(z):z∈Dnz0,2(R+1)/L(z0), ρj,0=rj/lj(z0), ρj,k=ρj,0+k·θj/lj∗, k∈N,j∈{1,…,n}. The following estimate holds (14)θjlj∗<rjlj∗≤rjljz0<2rj+2ljz0-rj+1ljz0.Hence, there exists S∗=(s1∗,…,sn∗)∈N independent of z0 such that(15)ρj,mj-1<rj+1ljz0<ρj,mj≤2rj+2ljz0for some mj=mj(z0)≤sj∗ because L∈Qn. Indeed,(16)2rj+2/ljz0-ρj,0θj/lj∗=2rj+2-rjlj∗θjljz0=rj+2θjmaxljzljz0:z∈Dnz0,2R+1Lz0≤rj+2θjλ2,j2R+1.Thus, sj∗=rj+2/θjλ2,j(2(R+1)), where x is the integer part of x∈R.

Let M0=(m1,…,mn) and τK∗∗ be such a point in Cn that(17)FτK∗∗=maxFz:z∈Tnz0,RK,where K=(k1,…,kn), RK=(ρ1,k1,…,ρn,kn) and τj,K∗ be the intersection point in C of the segment [zj0,τj,K∗∗] with |zj-zj0|=ρj,kj-1. We construct a sequence of polydisc Dn(z0,RK) with K≤M0, R0=R/L(z0)=(ρ1,0,…,ρn,0) and Θ/L(z0)=(θ1/l1∗,…,θn/ln∗) (see Figures 1 and 2).

Denote αK(j)=(τ1,K∗∗,…,τj-1,K∗∗,τj,K∗,τj+1,K∗∗,…,τn,K∗∗). Hence, for every rj>θj and K≤S∗:τj,K∗-τj,K∗∗=θj/lj∗≤rj/lj(αK(j)). Thus, for some R0=R0(αK(j))∈R+n, Θ≤R0≤R, we deduce (18)FτK∗∗≤maxFz:z∈TnαKj,R0LαKj≤p2minFz:z∈TnαKj,R0LαKj∖GR′F≤p2minFz:z∈TnαKj,R0LαKj∖GR′F,z∈Dnz0,RK-1j≤p2maxFz:z∈Tnz0,RK-1j.To deduce (18), we implicitly used that(19)TnαKj,R0LαKj∖GR′F∩Dnz0,RK-1j≠∅.Condition (11) provides (19). Indeed, we will find a lower estimate of measure of the set Tn(αK(j),R0/L(αK(j)))∩Dn[z0,RK-1j] and will show that the measure is not lesser than a left part of inequality (11).

The set Tn(αK(j),R0/L(αK(j)))∩Dn[z0,RK-1j] is a Cartesian product of the following arcs on circles: for every m∈{1,…,n},m≠j (see Figure 3)(20)zm∈C:zm-τm,K∗∗=rm0lmαKj∩zm∈C:zm-zm0≤ρm,kmand for m=j (see Figure 4)(21)zj∈C:zj-τj,K∗=rj0ljαKj∩zj∈C:zj-zj0≤ρj,kj-1.It is easy to prove that the length of arc equals (22)2rm0lmαKj·arccosrm02lmαKjρm,kmfor m≠j,(23)2rj0ljαKj·arccosrj02ljαKjρj,kj-1for m=j.But for m≠jrm0/lm(αK(j))≤ρm,km and rj0/lj(αK(j))≤ρj,kj-1 the argument in arccosine from (23) and (22) does not exceed 1/2. This means that the length of arc is not lesser than(24)2rm0lmαKjarccos12≥2θmπ3lmz0λ2,m2R+1for every m∈1,2,…,n,because L∈Qn. Accordingly, the measure of the set(25)TnαKj,R0LαKj∩Dnz0,RK-1jon the skeleton of polydisc is always not lesser than ∏m=1n2θmπ/3lm(z0)λ2,m(2(R+1)). Assuming a strict inequality in (11), we deduce that (19) is valid.

Applying (18) mjth times in every variable zj, we obtain(26)maxFz:z∈Tnz0,R+1Lz0≤maxFz:z∈Tnz0,RM0≤p2maxFz:z∈Tnz0,RM0-1n≤p2mnmaxFz:z∈Tnz0,RM0-mn1n≤⋯≤p2mn+1maxFz:z∈Tnz0,RM0-mn1n-1n-1≤p2mn+mn-1maxFz:z∈Tnz0,RM0-mn1n-mn-11n-1≤⋯≤p2∥M0∥maxFz:z∈Tnz0,R0≤p2∥S∗∥maxFz:z∈Tnz0,RLz0.By Theorem 2 the function F has bounded L-index in joint variables.

With r′=rm0/lmαKj.

With r′=rj0/ljαKj.

Let us denote c(z′,r)={z∈C:z-z′=r/l(z′)}. For n=1 Theorem 7 implies the following corollary.

Corollary 8.

Let l∈Q,f be an entire function. If ∃r>0, ∃r′≥0, ∃p2≥1∃θ∈(0,r), such that ∀z0∈C∃r0=r0(z0)∈[θ;r], and measc(z0,r0)∩GR′(F)<2πθ/3l(z0)λ2(2r+2) and (27)maxfz:z∈cz0,r0≤p2minfz:z∈cz0,r0∖Gr′fthen 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(z0,r0) can contain zeros of the function f. Meanwhile, in corresponding theorem from [26, 27] the circle c(z0,r0) is chosen such that f(z)≠0 for all z∈c(z0,r0).

5. Behavior of Partial Logarithmic Derivatives

Denote J={(j1,…,jn):ji∈{0,1},i∈{1,…,n}}∖0.

Theorem 9.

Let L∈Qn. If an entire function F satisfies the following conditions

for every R>0 there exists p1=p1(R)>0 such that for all z∈Cn∖GR(F) and for all J∈J(28)lnFzJ≤p1LJz,

where lnF(z) is the principal value of logarithm.

for every R>0 and R′≥0 exists p2=p2(R,R′)≥1 that for all z0∈Cn such that Tn(z0,R/L(z0))∖GR′(F)=⋃iCi≠∅, where the sets Ci are connected disjoint sets, and either (a) maximinz∈Ci|F(z)|≤p2miniminz∈Ci|F(z)|, or (b) maximaxz∈Ci|F(z)|≤p2minimaxz∈Ci|F(z)|, or (c) |F(z∗)|=maximaxz∈Ci|F(z)|, |F(z∗∗)|=miniminz∈Ci|F(z)|, and z∗,z∗∗ belong to the same set Ci0

for every R>0 there exists n∗(R)>0 such that for all z∈Cn(29)measZF∩Dnz,RLz≤n∗R.

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

Let z0∈Cn be arbitrarily chosen point. In view of Theorem 7 we need to prove that(30)measTnz0,R0Lz0∩GR′F<2π3n∏j=1nθjλ2,j2R+1ljz0for some R0=R0z0.Let R>0 be arbitrary radius. We choose Θ, R′∈R+n such that θj<2rj/2+3λ2,j(2(R+1)), (31)rj′<minθj,2λ1,j2R+RΛ2R/Λ1Rθjrj-θj3n∗R+RΛ2R/Λ1R1/nλ2,j2R+1,j∈1,…,n.Let dS=ds1·…·dsn, S=(s1,…,sn), ωz be a volume measure in R2n. Clearly, (see [28, p. 75-76])(32)∫Dnz0,Ruzdωz=∫0r1⋯∫0rns1⋯sn∫02π⋯∫02πuz0+SeiΘdθ1⋯dθnds1⋯dsn=∫0R∫02π⋯∫02πuz0+SeiΘds1θ1⋯dsnθndS,where u is plurisubharmonic function. Hence, (33)∫0R/Lz0measTnz0,S∩GR′FdS=measDnz0,RLz0∩GR′F.Obviously, there can exist points z′∈ZF∖Dn[z0,R/L(z0)] such that (34)Dnz0,RLz0∩Dnz′,R′Lz′≠∅.Let zj′′ be the intersection point of the segment [zj0,zj′] and the circle zj-zj0=rj/lj(z0),j∈{1,…,n}. Then zj′′-zj′≤rj′/ljz′ and z′′∈Tn(z0,R/L(z0)). Using L∈Qn, we estimate maximum distance between zj0 and zj′:(35)ljz′≥ljz′ljz′′·ljz′′ljz0·ljz0≥λ1,jRλ2,jR′·ljz0≥λ1,jRλ2,jR′ljz0,zj0-zj′≤z0-zj′′+zj′′-zj′≤rjljz0+rj′ljzj′≤rjljz01+λ2,jRλ1,jR′.Denote R′′=R+RΛ2(R)/Λ1(R). Let V2n-2 be a (2n-2)-dimensional volume, χF(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 Dn[z0,R′′/L(z0)]:(36)∫0R/Lz0measTnz0,S∩GR′FdS≤∫ZF∩Dnz0,R′′/Lz0χFzπn∏j=1nrj′2lj2zdV2n-2≤πn∏j=1nrj′2λ1,j2R′′lj2z0∫ZF∩Dnz0,R′′/Lz0χFzdV2n-2≤n∗R′′πn∏j=1nrj′2λ1,j2R′′lj2z0<n∗R′′πn∏j=1n2λ1,j2R+RΛ2R/Λ1Rθjrj-θj3n∗R+RΛ2R/Λ1R1/nλ2,j2R+1λ1,j2R′′lj2z0=2π3n∏j=1nθjrj-θjλ2,j2R+1lj2z0.Besides, we have that(37)∫0Θ/Lz0measTnz0,S∩GR′FdS=measDnz0,ΘLz0∩GR′F≤πn∏j=1nθj2lj2z0.Hence, the following difference is positive(38)2π3n∏j=1nθjrj-θjλ2,j2R+1lj2z0-∫0Θ/Lz0measTnz0,S∩GR′FdS≥2π3n∏j=1nθjrj-θjλ2,j2R+1lj2z0-πn∏j=1nθj2lj2z0=πn∏j=1nθjlj2z02rj-θj2+3λ2,j2R+13λ2,j2R+1>0because θj<2rj/2+3λ2,j(2(R+1)). From (36) it follows that(39)∫Θ/Lz0R/Lz0measTnz0,S∩GR′FdS<2π3n∏j=1nθjrj-θjλ2,j2R+1lj2z0-∫0Θ/Lz0measTnz0,S∩GR′FdS≤2π3n∏j=1nθjrj-θjλ2,j2R+1lj2z0.By mean value theorem there exists R0=R0(z0) with rj∈[θj,rj] such that(40)∫Θ/Lz0R/Lz0measTnz0,S∩GR′FdS=measTnz0,R0Lz0∩GR′F∏j=1nrj-θjljz0.Hence, in view of (39) we obtain a desired inequality(41)measTnz0,R0Lz0∩GR′F<2π3n∏j=1nθjλ2,j2R+1ljz0.

Clearly, for every point z0∈Cn we have Tn(z0,R0/Lz0)∖ZF=⋃iCi′, where Ci′ are connected disjoint sets, Ci′⊃Ci and Ci is defined in condition (2). Let z∗∈Tn(z0,R/L(z0)) be such that |F(z∗)=max|F(z)|:z∈Tn(z0,R0/L(z0)). Then there exists i0 such that z∗∈Ci0′. Let z∗∗∈Ci0⊂Ci0′ be such that |F(z∗∗)|=minz∈Ci0|F(z)|. We choose J=(j1,…,jn)∈J, where(42)ji=1,zi∗≠zi∗∗,0,zi∗=zi∗∗,i∈1,…,n,and deduce(43)lnFz∗Fz∗∗≤∫z∗∗z∗lnFzJdzJ≤∫z∗∗z∗p1LJzdzJ≤p1LJz0Λ2JR·πRJLJz0≤πnp1RJΛ2JR.Hence, (44)maxFz:z∈Tnz0,R0Lz0=Fz∗≤expπnp1RJΛ2JRFz∗∗=expπnp1RJΛ2JRminz∈Ci0Fz≤expπnp1RJΛ2JRp2miniminz∈CiFz=expπnp1RJΛ2JRp2minFz:z∈Tnz0,R0Lz0∖GR′F.By Theorem 7 the function F has bounded L-index in joint variables.

Let us to denote Δ as Laplace operator. We will consider ΔlnF as generalized function. Using some known results from potential theory, we can rewrite Theorem 9 as follows.

Theorem 10.

Let L∈Qn. If an entire function F satisfies the following conditions

for every R>0 there exists p1=p1(R)>0 such that for all z∈Cn∖GR(F) and for every j∈{1,…,n}(45)∂lnFz∂zj≤p1ljz,

where lnF(z) is the principal value of logarithm.

for every R>0 and R′≥0 exists p2=p2(R,R′)≥1 that for all z0∈Cn such that Tn(z0,R/L(z0))∖GR′(F)=⋃iCi≠∅, where the sets Ci are connected disjoint sets, and either (a) maximinz∈Ci|F(z)|≤p2miniminz∈Ci|F(z)|, or (b) maximaxz∈CiFz≤p2minimaxz∈CiFz, or (c) |F(z∗)|=maximaxz∈Ci|F(z)|, |F(z∗∗)|=miniminz∈Ci|F(z)|, and z∗, z∗∗ belong to the same set Ci0

for every R>0 there exists n∗(R)>0 such that for all z∈Cn(46)∫Dnz0,R/Lz0ΔlnFdV2n≤n∗R

then F has bounded L-index in joint variables.
Proof.

Ronkin [28, p. 230] deduced the following formula for entire function:(47)∫Dn0,R∗ΔlnFdV2n=2π∫ZF∩Dn0,R∗γFzdV2n-2,where γF(z) is a multiplicity of zero point of the function F at point z, R∗∈R+n is arbitrary radius. Let χF(z) be a characteristic function of zero set of F. Then χF(z)≤γF(z). Hence, (48)measZF∩Dnz0,RLz0=∫ZF∩Dnz0,R/Lz0χFzdV2n-2≤∫ZF∩Dnz0,R/Lz0γFzdV2n-2=12π∫Dnz0,R/Lz0ΔlnFdV2n≤n∗R2π;that is, inequality (29) holds.

Now we want to prove that (45) implies (28). For every J∈J∖⋃k=1n1k and z0∈Cn∖GR(F), Cauchy’s integral formula can be written in the following form(49)lnFz0J=J!2πin∫Tnz0,R/Lz0lnFz1mz-z0J-1m+1dz,where m is such that jm=1.

If z0∈Cn∖GR(F) and z′∈ZF⊂GRF, then for every j∈{1,…,n}(50)zj0-zj′≥rjljz′≥rjλ1,jRljz0>rjλ1,jR2ljz0.Let us consider the set A=⋃z0∈Cn∖GR(F)Tn(z0,RΛ1(R)/2L(z0)). We want to find the greatest radius R∗∈R+n such that GR∗(F)∩A=∅:(51)RLz′-RΛ1R2Lz0≥RLz′-RΛ1R2Λ1RLz′=R2Lz′.Thus, for R∗=R/3A⊂Cn∖GR∗(F). Using (45), we obtain that for every z0∈Cn∖GR(F)(52)lnFz0J≤12πn∫Tnz0,RΛ1R/2Lz0lnFz1mz-z0J-1m+1dz≤12πn∫Tnz0,RΛ1R/2Lz02Lz0RΛ1RJ-1m+1p113Rlmzdz≤12πn∫Tnz0,RΛ1R/2Lz02Lz0RΛ1RJ-1m+1p113Rλ2,m0.5RΛ1Rlmz0dz=rmλ1,mR22πn2Lz0RΛ1RJ+1p113Rλ2,m0.5RΛ1R∫Tnz0,RΛ1R/2Lz0dz=0.5rmλ1,mRp113Rλ2,m0.5RΛ1R2Lz0RΛ1RJ≤CRLJz0,where C(R)=0.5p11/3RmaxJ∈J{λ2,m0.5RΛ1Rrmλ1,m(R)2/RΛ1(R)J}. Thus, we proved that inequality (28) is valid.

For n=1 Theorem 9 implies the following corollary.

Corollary 11.

Let l∈Q,f be an entire in C function, n(r,z0,f) a number of zeros of the f in the disc |z-z0|≤r/l(z0). If the function f satisfies the following conditions:

for every r>0 there exists p1=p1(r)>0 such that for all z∈C∖Gr(f)(53)f′zfz≤p1lz,

for every r>0 and r′≥0 exists p2=p2(r,r′)≥1 that for all z0∈C such that z∈C:z-z0=r/lz0∖Gr′(f)=⋃iCi≠∅, where the sets Ci are connected disjoint sets, and either (a) maximinz∈Cifz≤p2miniminz∈Cifz, or (b) maximaxz∈Cifz≤p2minimaxz∈Ci|f(z)|, or (c) |f(z∗)|=maximaxz∈Ci|f(z)|, |f(z∗∗)|=miniminz∈Ci|f(z)|, and z∗, z∗∗ belong to the same set Ci0

for every r>0 there exists n∗(r)>0 such that for all z0∈Cn(r,z0,f)≤n∗(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 Cn(n≥2), it is required because Dn[z0,R/L(z0)]∖GR′(F) is a multiply connected domain, when Dn[z0,R/L(z0)] contains zeros of the function F.

We need some notations from [1]. Let b∈Cn∖{0} be a given direction. For a given z0∈Cn we denote gz0(t)≔F(z0+tb). If one has gz0(t)≠0 for all t∈C, then Grb(F,z0)≔∅; if gz0(t)≡0, then Grb(F,z0)≔{z0+tb:t∈C}. And if gz0(t)≢0 and ak0 are zeros of the function gz0(t), then Grb(F,z0)≔⋃k{z0+tb:|t-ak0|≤r/L(z0+ak0b)}, r>0. Let (54)GrbF=⋃z0∈CnGrbF,z0.

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 z∈Cn∖Grj1j(F),j∈{1,2,…,n}. However, Grj1j(F)⊂GR(F), where R=(r1,…,rn). Thus, condition (1) in Theorem 10 is weaker than the corresponding assumption in Theorem 8 from [1].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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