3. Local behaviour of maximum modulus of analytic in ball function

DOI: 10.24193/subbmath.2018.4.06

Sufficient conditions of boundedness of L-index and analog of Hayman’s Theorem for analytic functions in a ball

Andriy Bandura and Oleh Skaskiv

Abstract. We generalize some criteria of boundedness ofL-index in joint variables for analytic in an unit ball functions. Our propositions give an estimate maximum modulus of the analytic function on a skeleton in polydisc with the larger radii by maximum modulus on a skeleton in the polydisc with the lesser radii. An analog of Hayman’s Theorem for the functions is obtained. Also we established a connection between class of analytic in ball functions of boundedlj-index in every direction 1j, j ∈ {1, . . . , n} and class of analytic in ball of functions of boundedL-index in joint variables, whereL(z) = (l1(z), . . . , ln(z)), lj:Bⁿ→R⁺ is continuous function,1j= (0, . . . ,0, 1

|{z}

j−th place

,0, . . . ,0)∈Rⁿ+, z∈Cⁿ.

Mathematics Subject Classification (2010):32A05, 32A10, 32A30, 32A40, 30H99.

Keywords:Analytic function, unit ball, boundedL-index in joint variables, max- imum modulus, partial derivative, boundedL-index in direction.

1. Introduction

Recently, there was introduced a concept of analytic function in a ball in Cⁿ of bounded L-index in joint variables [8]. We also obtained criterion of boundedness of L-index in joint variables which describes a local behavior of partial derivatives on a skeleton in the polydisc and established other important properties of analytic functions in a ball of bounded L-index in joint variables. Those investigations used an idea of exhaustion of a ball inCⁿ by polydiscs.

The presented paper is a continuation of our investigations from [8]. We set the goal to prove new analogues of criteria of boundedness ofL-index in joint variables for analytic in a ball functions. Particular, we prove an estimate of maximum modulus on a greater polydisc by maximum modulus on a lesser polydisc (Theorems 3.1, 3.2) and obtain an analog of Hayman’s Theorem for analytic functions in a ball of bounded

L-index in joint variables (Theorems 4.1 and 4.2). For entire functions similar propo- sitions were obtained by A. I. Bandura, M. T. Bordulyak, O. B. Skaskiv [4, 5] in a case L(z) = (l1(z), . . . , ln(z)), z∈Cⁿ.Also A. I. Bandura, N.V. Petrechko, O. B. Skaskiv [6, 7] deduced same results for analytic in a polydisc functions. Hayman’s Theorem and its generalizations for different classes of analytic functions [1, 3, 5, 7, 12, 15, 20, 21]

are very important in theory of functions of bounded index. The criterion is helpful [1, 9] to investigate boundedness of index of entire solutions of ordinary or partial differential equations.

Note that the corresponding theorems for entire functions of bounded l-index and of boundedL-index in direction were also applied to investigate infinite products (see bibliography in [21, 1]). Thus, those generalizations for analytic in a ball functions are necessary to study L-index in joint variables of analytic solutions of PDE’s, its systems and multidimensional counterparts of Blaschke products. At the end of the paper, we present a scheme of application of Hayman’s Theorem to study properties of analytic solutions in the unit ball.

2. Main definitions and notations

We need some standard notations. Denote R⁺= (0,+∞),0= (0, . . . ,0), 1= (1, . . . ,1), 1_j= (0, . . . ,0, 1

|{z}

j−th place

,0, . . . ,0)∈Rⁿ+,

R= (r1, . . . , rn)∈Rⁿ+, z= (z1, . . . , zn)∈Cⁿ,|z|=q Pn

j=1|zj|².

ForA= (a1, . . . , an)∈Rⁿ, B= (b1, . . . , bn)∈Rⁿ we will use formal notations without violation of the existence of these expressions

AB= (a₁b₁,· · · , a_nb_n), A/B= (a₁/b₁, . . . , a_n/b_n), A^B =a^b₁¹a^b₂²·. . .·a^b_nⁿ, kAk=a1+· · ·+an,

and the notation A < B means that aj < bj, j ∈ {1, . . . , n}; the relation A ≤ B is defined similarly. For K = (k1, . . . , kn) ∈ Zⁿ+ denote K! = k1!·. . . ·kn!. The polydisc{z∈Cⁿ : |zj−z_j⁰|< r_j, j= 1, . . . , n}is denoted by Dⁿ(z⁰, R),its skeleton {z ∈ Cⁿ : |zj −z⁰_j| = rj, j = 1, . . . , n} is denoted by Tⁿ(z⁰, R), and the closed polydisc {z ∈ Cⁿ : |zj −z⁰_j| ≤ rj, j = 1, . . . , n} is denoted by Dⁿ[z⁰, R]. The open ball{z ∈Cⁿ : |z−z⁰|< r} is denoted by Bⁿ(z⁰, r),its boundary is a sphere Sⁿ(z⁰, r) ={z∈Cⁿ : |z−z⁰|=r},the closed ball{z∈Cⁿ : |z−z⁰| ≤r}is denoted byBⁿ[z⁰, r],Bⁿ =Bⁿ(0,1),D=B¹={z∈C: |z|<1}.

For K = (k₁, . . . , k_n) ∈ Zⁿ+ and the partial derivatives of an analytic in Bⁿ functionF(z) =F(z₁, . . . , z_n) we use the notation

F^(K)(z) =∂^kKkF

∂z^K = ∂^k1+···+knF

∂z^k₁¹. . . ∂zn^kn

.

Let L(z) = (l₁(z), . . . , l_n(z)), where l_j(z) : Bⁿ → R+ is a continuous function such that

(∀z∈Bⁿ) : lj(z)> β/(1− |z|), j∈ {1, . . . , n}, (2.1)

whereβ >√

nis a some constant. For a polydisc A.I. Bandura, N.V. Petrechko and O.B. Skaskiv [6, 7] imposed the restriction (∀z ∈ Dⁿ(0,1)) : lj(z) > β/(1− |zj|), j∈ {1, . . . , n}.A similar condition is used in one-dimensional case by S.N. Strochyk, M.M. Sheremeta, V.O. Kushnir [22, 14, 21].

Note that if R ∈Rⁿ+, |R| ≤ β, z⁰ ∈Bⁿ and z ∈ Dⁿ[z⁰, R/L(z⁰)] then z ∈ Bⁿ (see Remark 1 in [8]).

An analytic function F: Bⁿ → C is said to be of bounded L-index (in joint variables),if there exists n0∈Z⁺such that for allz∈Bⁿ and for allJ ∈Zⁿ+

|F^(J)(z)|

J!L^J(z) ≤max

|F^(K)(z)|

K!L^K(z) : K∈Zⁿ+, kKk ≤n₀

. (2.2)

The least such integer n0 is called the L-index in joint variables of the function F and is denoted byN(F,L,Bⁿ) (see [8]). Entire and analytic in polydisc functions of boundedL-index in joint variables are considered in [4, 5, 6, 7, 10, 13, 19, 18, 16, 17].

ByQ(Bⁿ) we denote the class of functionsL, which satisfy (2.1) and the following condition

(∀R∈Rⁿ+,|R| ≤β, j∈ {1, . . . , n}) : 0< λ_1,j(R)≤λ_2,j(R)<∞, (2.3) where λ1,j(R) = inf

z⁰∈Bⁿinf

lj(z)/lj(z⁰) :z∈Dⁿ

z⁰, R/L(z⁰) , λ_2,j(R) = sup

z⁰∈Bⁿ

sup

l_j(z)/l_j(z⁰) :z∈Dⁿ

z⁰, R/L(z⁰) . Λ₁(R) = (λ_1,1(R), . . . , λ_1,n(R)), Λ₂(R) = (λ_2,1(R), . . . , λ_2,n(R)).

We need the following results.

Theorem 2.1 ([8]). LetL∈Q(Bⁿ). An analytic inBⁿ functionF has boundedL-index in joint variables if and only if for eachR∈Rⁿ+,|R| ≤β,there existn₀∈Z+,p₀>0 such that for everyz⁰∈Bⁿ there existsK⁰∈Zⁿ+,kK⁰k ≤n0, and

max

|F^(K)(z)|

K!L^K(z):kKk ≤n0, z∈Dⁿ

z⁰, R/L(z⁰)

≤p0

|F^(K0)(z⁰)|

K⁰!L^K0(z⁰). (2.4) DenoteL(z) = (ee l₁(z), . . . ,el_n(z)). The notationLLe means that there exist

Θ1= (θ1,j, . . . , θ1,n)∈Rⁿ+, Θ2= (θ2,j, . . . , θ2,n)∈Rⁿ+

such that∀z∈Bⁿ θ_1,jel_j(z)≤l_j(z)≤θ_2,jel_j(z) for eachj∈ {1, . . . , n}.

Theorem 2.2 ([8]). LetL∈Q(Bⁿ),LL, β|Θe 1|>√

n.An analytic inBⁿ functionF has bounded L-index in joint variables if and only ife F has boundedL-index in joint variables.

3. Local behaviour of maximum modulus of analytic in ball function

For an analytic inBⁿ functionF we put

M(r, z⁰, F) = max{|F(z)|:z∈Tⁿ(z⁰, r)},

where z⁰ ∈ Bⁿ, r ∈Rⁿ+. Then M(R, z⁰, F) = max{|F(z)|: z ∈ Dⁿ[z⁰, R]}, because the maximum modulus for an analytic function in a closed polydisc is attained on its skeleton.

The following proposition uses an idea about the possibility of replacing universal quantifier by existential quantifier in sufficient conditions of index boundedness [2]. To prove an analog of Hayman’s Theorem we need this theorem which has an independent interest.

Theorem 3.1. Let L ∈ Qⁿ, F : Bⁿ → C be analytic function. If there exist R⁰, R⁰⁰∈Rⁿ+, R⁰< R⁰⁰,|R⁰⁰|< β andp₁=p₁(R⁰, R⁰⁰)≥1such that for every z⁰∈Cⁿ

M R⁰⁰

L(z⁰), z⁰, F

≤p₁M R⁰

L(z⁰), z⁰, F

(3.1) thenF is of boundedL-index in joint variables.

Proof. At first, we assume that0< R⁰ <1< R⁰⁰.

Letz⁰∈Bⁿ be an arbitrary point. We expand a functionF in power series F(z) = X

K≥0

bK(z−z⁰)^K = X

k₁,...,k_n≥0

bk₁,...,k_n(z1−z⁰₁)^k1. . .(zn−z_n⁰)^kn, (3.2)

wherebK=bk₁,...,k_n= ^F(K)_K!^(z0).

Let µ(R, z⁰, F) = max{|b_K|R^K: K ≥ 0} be a maximal term of power series (3.2) and

ν(R) =ν(R, z⁰, F) = (ν₁⁰(R), . . . , ν_n⁰(R)) be a set of indices such that

µ(R, z⁰, F) =|b_ν(R)|R^ν(R),

kν(R)k=

n

X

j=1

νj(R) = max{kKk:K≥0, |bK|R^K =µ(R, z⁰, F)}.

In view of inequality (3.8) we obtain for any|R|<1− |z⁰|, µ(R, z⁰, F)≤M(R, z⁰, F).

Then for givenR⁰ andR⁰⁰ with 0<|R⁰|<1<|R⁰⁰|< β we conclude M(R⁰R, z⁰, F)≤X

k≥0

|bk|(R⁰R)^k ≤X

k≥0

µ(R, z⁰, F)(R⁰)^k

=µ(R, z⁰, F)X

k≥0

(R⁰)^k =

n

Y

j=1

1

1−r⁰_jµ(R, z⁰, F).

Besides,

lnµ(R, z⁰, F) = ln{|b_ν(R)|R^ν(R)}= ln

|b_ν(R)|(RR⁰⁰)^ν(R) 1 (R⁰⁰)^ν(R)

= ln{|b_ν(R)|(RR⁰⁰)^ν(R)}+ ln 1

(R⁰⁰)^ν(R)

≤lnµ(R⁰⁰R, z⁰, F)− kν(R)kln min

1≤j≤nr_j⁰⁰. This implies that

kν(R)k ≤ 1

ln min1≤j≤nr_j⁰⁰(lnµ(R⁰⁰R, z⁰, F)−lnµ(R, z⁰, F))

≤ 1

ln min1≤j≤nr⁰⁰_j



lnM(R⁰⁰R, z⁰, F)−ln(

n

Y

j=1

(1−r⁰_j)M(R⁰R, z⁰, F))





≤ 1

ln min_1≤j≤nr⁰⁰_j lnM(R⁰⁰R, z⁰, F)−lnM(R⁰R, z⁰, F)

− Pn

j=1ln(1−r_j⁰) min_1≤j≤nr_j⁰⁰

= 1

min_1≤j≤nr⁰⁰_j lnM(R⁰⁰R, z⁰, F) M(R⁰R, z⁰, F) −

Pn

j=1ln(1−r_j)

min_1≤j≤nr⁰⁰_j . (3.3) PutR= _L(z¹₀₎.Now letN(F, z⁰,L) be theL-index of the functionFin joint variables at point z⁰ i. e. it is the least integer for which inequality (2.2) holds at point z⁰. Clearly that

N(F, z⁰,L)≤ν 1

L(z⁰), z⁰, F

=ν(R, z⁰, F). (3.4)

But

M R⁰⁰/L(z⁰), z⁰, F

≤p1(R⁰, R⁰⁰)M R⁰/L(z⁰), z⁰, F

. (3.5)

Therefore, from (3.3), (3.4), (3.5) we obtain that∀z⁰∈Bⁿ N(F, z⁰,L)≤−P2

j=1ln(1−r⁰_j)

ln min{r⁰⁰₁, r₂⁰⁰} + lnp1(R⁰, R⁰⁰) ln min{r⁰⁰₁, r₂⁰⁰}.

This means that F has bounded L-index in joint variables, if 0 < R⁰ < 1 < R⁰⁰,

|R⁰⁰|< β.

Now we will prove the theorem for any 0 < R⁰ < R⁰⁰, |R⁰⁰| < β. From (3.1) with 0< R1< R2 it follows that

max

|F(z)|:z∈Tⁿ

z⁰, 2R⁰⁰ R⁰+R⁰⁰

R⁰+R⁰⁰ 2L(z⁰)

≤P1max

|F(z)|:z∈Tⁿ

z⁰, 2R⁰ R⁰+R⁰⁰

R⁰+R⁰⁰ 2L(z⁰)

.

DenotingL(z) =e _R^2L(z)0+R⁰⁰,we obtain max

(

|F(z)|:z∈Tⁿ z⁰, 2R⁰⁰ (R⁰+R⁰⁰)eL(z⁰)

!)

≤P1max (

|F(z)|:z∈Tⁿ z⁰, 2R⁰⁰ (R⁰+R⁰⁰)eL(z⁰)

!) ,

where 0< _R0^2R+R⁰⁰⁰ <1< _R^2R0+R⁰⁰⁰⁰.Taking into account the first part of the proof, we conclude that the function F has bounded L-index in joint variables. By Theoreme 2.2, the functionF is of boundedL-index in joint variables.

Also the corresponding necessary conditions are valid.

Theorem 3.2. LetL∈Q(Bⁿ).If an analytic in Bⁿ functionF has boundedL-index in joint variables then for anyR⁰, R⁰⁰∈Rⁿ+, R⁰< R⁰⁰,|R⁰⁰| ≤β,there exists a number p1=p1(R⁰, R⁰⁰)≥1 such that for everyz⁰∈Bⁿ inequality (3.1)holds.

Proof. LetN(F,L) =N <+∞.Suppose that inequality (3.1) does not hold i.e. there existR⁰, R⁰⁰,0<|R⁰|<|R⁰⁰|< β,such that for eachp_∗≥1 and for somez⁰=z⁰(p_∗)

M R⁰⁰

L(z⁰), z⁰, F

> p∗M R⁰

L(z⁰), z⁰, F

. (3.6)

By Theorem 2.1, there exists a number p₀=p₀(R⁰⁰)≥1 such that for everyz⁰∈Bⁿ and some K⁰ ∈Zⁿ+,kK⁰k ≤N, (i.e.n₀ =N, see proof of necessity of Theorem 2.1 in [8]) one has

M R⁰⁰

L(z⁰), z⁰, F^(K0)

≤p0|F^(K0)(z⁰)|. (3.7) We put

b1=p0





n

Y

j=2

λ^N_2,j(R⁰⁰)



(N!)ⁿ⁻¹





N

X

j=1

(N−j)!

(r₁⁰⁰)^j





r₁⁰⁰r₂⁰⁰. . . r_n⁰⁰ r⁰₁r₂⁰. . . r⁰_n

^N ,

b2=p0

ⁿ Y

j=3

λ^N_2,j(R⁰⁰)

(N!)^{n−2 N}

X

j=1

(N−j)!

(r⁰⁰₂)^j

r⁰⁰₂. . . r_n⁰⁰ r⁰₂. . . r⁰_n

N 1, 1

(r₁⁰)^N

, . . .

b_n−1=p0λ^N_2,n(R⁰)N!





N

X

j=1

(N−j)!

(r⁰⁰_n−1)^j





r_n−1⁰⁰ r⁰⁰_n r_n−1⁰ r⁰_n

N

max

1, 1

(r₁⁰. . . r_n−2⁰ )^N

,

bn=p0





N

X

j=1

(N−j)!

(r_n⁰⁰)^j



 r_n⁰⁰

r_n⁰ N

max

1, 1

(r⁰₁. . . r⁰_n−1)^N

and

p∗= (N!)ⁿp0

r₁⁰⁰r⁰⁰₂. . . r_n⁰⁰ r⁰₁r⁰₂. . . r⁰_n

N

+

n

X

k=1

bk+ 1.

Letz⁰=z⁰(p_∗) be a point for which inequality (3.6) holds andK⁰be such that (3.7) holds and

M R⁰

L(z⁰), z⁰, F

=|F(z^∗)|, M r⁰⁰

L(z⁰), z⁰, F^(J)

=|F^(J)(z^∗_J)|

for everyJ ∈Zⁿ+,kJk ≤N.We apply Cauchy’s inequality

|F^(J)(z⁰)| ≤J!

L(z⁰) R⁰

^J

|F(z^∗)| (3.8)

for estimate the difference

|F^(J)(z_J,1^∗ , z_J,2^∗ , . . . , z_J,n^∗ )−F^(J)(z₁⁰, z_J,2^∗ , . . . , z_J,n^∗ )|

=

Z z_J,1^∗ z⁰₁

∂^kJk+1F

∂z₁^j1+1∂z₂^j2. . . ∂zn^jn

(ξ, z_J,2^∗ , . . . , z_J,n^∗ )dξ

≤

∂^kJk+1F

∂z^j₁¹⁺¹∂z^j₂². . . ∂zn^jn

(z_(j^∗

1+1,j2,...,jn))

r₁⁰⁰

l1(z⁰). (3.9) Taking into account (z₁⁰, z_J,2^∗ , . . . , z^∗_J,n)∈Dⁿ[z⁰,_L(z^R000)],for allk∈ {1, . . . , n},

|z^∗_J,k−z_k⁰|= r_k⁰⁰

lk(z⁰), l_k(z₁⁰, z_J,2^∗ , . . . , z^∗_J,n)≤λ_2,k(R⁰⁰)l_k(z⁰) and (3.8) withJ =K⁰,by Theorem 2.1 we have

|F^(J)(z⁰₁, z^∗_J,2, . . . , z_J,n^∗ )|

≤J!l^j₁¹(z₁⁰, z_J,2^∗ , . . . , z_J,n^∗ )Qn

k=2l^j_k^k(z₁⁰, z_J,2^∗ , . . . , z^∗_J,n)

K⁰!L^K0(z⁰) p₀|F^(K0)(z⁰)|

≤ J!L^J(z⁰)Qn

k=2λ^j_2,k^k (R⁰⁰) K⁰!L^K0(z⁰) p₀K⁰!

L(z⁰) R⁰

^K0

|F(z^∗)|

= p0J!L^J(z⁰)Qn

k=2λ^j_2,k^k (R⁰⁰)

(R⁰)^K0 |F(z^∗)|. (3.10)

From inequalities (3.9) and (3.10) it follows that

∂^kJk+1F

∂z₁^j1+1∂z₂^j2. . . ∂z^jnⁿ

(z_(j^∗

1+1,j₂,...,j_n))

≥ l1(z⁰) r₁⁰⁰

n|F^(J)(z_j^∗)| − |F^(J)(z₁⁰, z_J,2^∗ , . . . , z_J,n^∗ )|o

≥ l₁(z₁⁰)

r₁⁰⁰ |F^(J)(z^∗_j)| −p₀J!L^{(j1+1,j2,...,jn)}(z⁰)Qn

k=2λ^j_2,k^k (R⁰⁰)

r⁰⁰₁(R⁰)^K0 |F(z^∗)|.

Then

|F^(K0)(z_K^∗0)| ≥l1(z⁰) r₁⁰⁰

∂^kK0k−1f

∂z₁^k10−1∂z₂^k02. . . ∂zn^k0n

(z^∗_(k0

1−1,k⁰₂,...,k⁰_n))

−p0(k₁⁰−1)!k₂⁰!. . . k⁰_n!L^K0(z⁰)Qn i=2λ^k

0 i

2,i(R⁰⁰) r₁⁰⁰(R⁰)^K0 |F(z^∗)|

≥ l₁²(z⁰) (r⁰⁰₁)²

∂^kK0k−2f

∂z₁^k01−2∂z₂^k02. . . ∂zn^k0n

(z^∗_(k0

1−2,k⁰₂,...,k⁰_n))

−p₀(k₁⁰−2)!k₂⁰!. . . k⁰_n!L^K0(z⁰)Qn

i=2λ^k_2,i⁰ⁱ(R⁰⁰) (r⁰⁰₁)²(R⁰)^K0 |F(z^∗)|

−p0(k⁰₁−1)!k₂⁰!. . . k⁰_n!L^K0(z⁰)Qn i=2λ^k

0 i

2,i(r⁰⁰_i) r₁⁰⁰(R⁰)^K0 |F(z^∗)|

. . .

≥ l₁^k01(z⁰) (r⁰⁰₁)^k01

∂^kK0k−k01f

∂z^k₂⁰². . . ∂z^kn⁰ⁿ

(z_(0,k^∗ 0 2,...,k⁰_n))

− p0

(R⁰)^K0L^K0(z⁰)

n

Y

i=2

λ^k_2,i⁰ⁱ(R⁰⁰)

!

k₂⁰!. . . k⁰_n!

k⁰₁

X

j₁=1

(k₁⁰−j1)!

(r₁⁰⁰)^j1 |F(z^∗)|. . .

≥l^k

0 1

1 (z⁰) (r₁⁰⁰)^k10

l^k

0 2

2 (z⁰) (r⁰⁰₂)^k02

∂^{kK0k−k01−k02}f

∂z^k₃⁰³. . . ∂z^kn⁰ⁿ

(z_(0,0,k^∗ 0 3,...,k_n⁰))

−l^k₁⁰¹(z⁰)p₀L^{(0,k20,...,k0n)}(z⁰) (r⁰⁰₁)^k01(R⁰)^K0

n

Y

i=3

λ^k

0 i

2,i(R⁰⁰)

!

k₃⁰!. . . k⁰_n!

k⁰₂

X

i₂=1

(k⁰₂−j₂)!

(r₂⁰⁰)^j2 |F(z^∗)|

− p0

(R⁰)^K0L^K0(z⁰)

n

Y

i=2

λ^k_2,i⁰ⁱ(R⁰⁰)

!

k₂⁰!. . . k⁰_n!

k⁰₁

X

j1=1

(k₁⁰−j1)!

(r₁⁰⁰)^j1 |F(z^∗)|

. . .

≥

L(z⁰) R⁰⁰

|F(z^∗₀)| − |F(z^∗)|

b

X

i=1

˜bi, (3.11)

where in view of the inequalitiesλ2,i(R⁰⁰)≥1 andR⁰⁰≥R⁰ we have

˜b1= p0

(R⁰)^K0L^K0(z⁰)

n

Y

i=2

λ^k

0 i

2,i(R⁰⁰)

!

k⁰₂!. . . k_n⁰!

k₁⁰

X

j1=1

(k⁰₁−j1)!

(r⁰⁰₁)^j1

=

L(z⁰) R⁰⁰

^K0R⁰⁰ R⁰

^K0 p0

n

Y

i=2

λ^k_2,i⁰ⁱ(R⁰⁰)

!

k⁰₂!. . . k_n⁰!

k⁰₁

X

j1=1

(k⁰₁−j1)!

(r⁰⁰₁)^j1 ≤

L(z⁰) R⁰⁰

^K0 b1,

˜b₂= p₀

(R⁰)^K0L^K0(z⁰) ⁿ

Y

i=3

λ^k_2,iⁱ⁰(R⁰⁰)

k⁰₃!. . . k_n⁰! (r⁰⁰₁)^k01

k⁰₂

X

j₂=1

(k₂⁰−j₂)!

(r₂⁰⁰)^j2 ≤ L(z⁰)

R⁰⁰ K⁰

b₂, . . .

˜b_n−1= p₀

(R⁰)^K0L^K0(z⁰)λ^k_2,nⁿ⁰ (R⁰⁰) k⁰_n!

(r⁰⁰₁)^k01. . .(r⁰⁰_n−2)^kn−20 ×

×

k⁰_n−1

X

jn−1=1

(k_n−1⁰ −j_n−1)!

(r⁰⁰_n−1)^jn−1 ≤

L(z⁰) R⁰⁰

^K0 b_n−1,

˜bn= p0

(R⁰)^K0L^K0(z⁰) 1

(r₁⁰⁰)^k10. . .(r_n−1⁰⁰ )^k0n−1

k⁰_n

X

jn=1

(k⁰_n−jn)!

(r⁰⁰_n)^jn ≤

L(z⁰) R⁰⁰

^K0 bn. Thus, (3.11) implies that

|F^(K0)(z_K^∗0)| ≥

L(z⁰) R⁰⁰

^K0

|F(z^∗)|







|F(z^∗₀)|

|F(z^∗)|−

n

X

j=1

b_j





 . But in view of (3.6) and a choice ofp∗ we have

|F(z₀^∗)|

|F(z^∗)| ≥p∗>

n

X

j=1

bj. Thus, (3.7) and (3.8) imply

|F^(K0)(z^∗_K0)| ≥

L(z⁰) R⁰⁰

^K0

|F(z^∗)|





 p_∗−

n

X

j=1

bj







≥

L(z⁰) R⁰⁰

^K0





 p_∗−

n

X

j=1

bj







|F^(K0)(z⁰)|(R⁰)^K0 K⁰!L^K0(z⁰)

≥

r⁰₁. . . r⁰_n r₁⁰⁰. . . r_n⁰⁰

N



 p∗−

n

X

j=1

bj







|F^(K0)(z^∗_K0)|

p0(n!)ⁿ . Hence, we havep_∗≤p0

r⁰₁...r⁰_n r⁰⁰₁...r⁰⁰_n

N

(N!)ⁿ+Pn

j=1bj,but this contradicts the choice of

p_∗.

4. Analogue of Theorem of Hayman for analytic in a ball function of bounded L-index in joint variables

Theorem 4.1. Let L∈Q(Bⁿ). An analytic functionF in Bⁿ has bounded L-index in joint variables if and only if there existp∈Z+ andc∈R+ such that for eachz∈Bⁿ

max

|F^(J)(z)|

L^J(z) : kJk=p+ 1

≤c·max

|F^(K)(z)|

L^K(z) : kKk ≤p

. (4.1)

Proof. LetN=N(F,L,Bⁿ)<+∞.The definition of the boundedness ofL-index in joint variables yields the necessity withp=N andc= ((N+ 1)!)ⁿ.

We prove the sufficiency. ForF ≡0 theorem is obvious. Thus, we suppose that F 6≡0.Denote β= (^√β_n, . . . ,^√β_n).

Assume that (4.1) holds,z⁰∈Bⁿ, z∈Dⁿ[z⁰,_L(z^β0)].For allJ ∈Zⁿ+,kJk ≤p+ 1, one has

|F^(J)(z)|

L^J(z⁰) ≤Λ^J₂(β)|F^(J)(z)|

L^J(z) ≤c·Λ^J₂(β) max

|F^(K)(z)|

L^K(z) : kKk ≤p

≤c·Λ^J₂(β) max

Λ^−K₁ (2)|F^(K)(z)|

L^K(z⁰) :kKk ≤p

≤BG(z), (4.2)

whereB=c·max{Λ^K₂ (β) : kKk=p+ 1}max{Λ^−K₁ (β) : kKk ≤p},and G(z) = max

|F^(K)(z)|

L^K(z⁰) : kKk ≤p

. We choose

z⁽¹⁾= (z₁⁽¹⁾, . . . , z_n⁽¹⁾)∈Tⁿ(z⁰, 1 2β√

nL(z⁰)) and

z⁽²⁾= (z₁⁽²⁾, . . . , z_n⁽²⁾)∈Tⁿ(z⁰, β L(z⁰)) such thatF(z⁽¹⁾)6= 0 and

|F(z⁽²⁾)|=M β

L(z⁰), z⁰, F

6= 0. (4.3)

These points exist, otherwise ifF(z)≡0 on skeleton Tⁿ

z⁰, 1 2β√

nL(z⁰)

or Tⁿ

z⁰, β L(z⁰)

then by the uniqueness theorem F ≡0 in Bⁿ. We connect the points z⁽¹⁾ and z⁽²⁾ with plane

α=











z2=k2z1+c2, z3=k3z1+c3, . . . z_n=k_nz₁+c_n, where

ki= z_i⁽²⁾−z_i⁽¹⁾ z₁⁽²⁾−z₁⁽¹⁾

, ci =z_i⁽¹⁾z⁽²⁾₁ −z⁽²⁾_i z₁⁽¹⁾ z⁽²⁾₁ −z⁽¹⁾₁

, i= 2, . . . , n.

It is easy to check thatz⁽¹⁾ ∈αandz⁽²⁾∈α.LetG(ze 1) =G(z)|α be a restriction of the functionGontoα.

For every K ∈ Zⁿ+ the function F^(K)(z)

_α is analytic function of variable z1

and ˜G(z⁽¹⁾₁ ) = G(z⁽¹⁾)

_α 6= 0 becauseF(z⁽¹⁾)6= 0. Hence, all zeros of the function F^(K)(z)

_αare isolated as zeros of a function of one variable. Thus, zeros of the function G(z˜ 1) are isolated too. Therefore, we can choose piecewise analytic curveγontoαas following

z=z(t) = (z1(t), k2z1(t) +c2, . . . , knz1(t) +cn), t∈[0,1],

which connect the pointsz⁽¹⁾, z⁽²⁾ and such thatG(z(t))6= 0 and Z 1

0

|z₁⁰(t)|dt≤ 2β

√nl₁(z⁰₁).

For a construction of the curve we connectz₁⁽¹⁾ andz₁⁽²⁾ by a line z^∗₁(t) = (z₁⁽²⁾−z₁⁽¹⁾)t+z₁⁽¹⁾, t∈[0,1].

The curveγcan cross pointsz1at which the functionG(ze 1) = 0.The number of such pointsm=m(z⁽¹⁾, z⁽²⁾) is finite. Let (z_1,k^∗ ) be a sequence of these points in ascending order of the value |z⁽¹⁾₁ −z^∗_1,k|, k∈ {1,2, ..., m}. We choose

r < min

1≤k≤m−1{|z_1,k^∗ −z_1,k+1^∗ |,|z_1,1^∗ −z⁽¹⁾₁ |,|z_1,m^∗ −z₁⁽²⁾|, 2β²−1 2π√

nβl₁(z⁰)}.

Now we construct circles with centers at the points z^∗_1,k and corresponding radii r⁰_k< ₂^r_k such thatG(ze ₁)6= 0 for allz₁ on the circles. It is possible, becauseF 6≡0.

Every such circle is divided onto two semicircles by the linez^∗₁(t). The required piecewise-analytic curve consists with arcs of the constructed semicircles and segments of linez₁^∗(t), which connect the arcs in series between themselves or with the points z₁⁽¹⁾, z₁⁽²⁾. The length ofz₁(t) in C(but notz(t) inCⁿ!) is lesser than

β/√ n

l₁(z⁰) + 1 2√

nβl₁(z⁰)+πr≤ 2β

√nl₁(z⁰). Then

Z 1 0

|z_s⁰(t)|dt=|ks| Z 1

0

|z₁⁰(t)|dt≤ |zs⁽²⁾−zs⁽¹⁾|

|z₁⁽²⁾−z₁⁽¹⁾|

√ 2β nl1(z⁰)

≤ 2β²+ 1 2√

nβls(z⁰) 2√

nβl1(z⁰) 2β²−1

√ 2β

nl1(z⁰) ≤ 2β(2β²+ 1) (2β²−1)√

nls(z⁰), s∈ {2, . . . , n}.

Hence,

Z 1 0

n

X

s=1

ls(z⁰)|z_s⁰(t)|dt≤ 2β(2β²+ 1)√ n

2β²−1 =S. (4.4)

Since the functionz=z(t) is piece-wise analytic on [0,1], then for arbitraryK∈Zⁿ+, J ∈Zⁿ+,kKk ≤p,either

|F^(K)(z(t))|

L^K(z⁰) ≡ |F(J)(z(t))|

L^J(z⁰) , (4.5)

or the equality

|F^(K)(z(t))|

L^K(z⁰) = |F^(J)(z(t))|

L^J(z⁰) (4.6)

holds only for a finite set of pointst_k ∈[0; 1].

Then for function G(z(t)) as maximum of such expressions ^|F(J)_LJ(z^(z(t))|₀₎ by all kJk ≤ptwo cases are possible:

1. In some interval of analyticity of the curve γ the function G(z(t)) identically equals simultaneously to some derivatives, that is (4.5) holds. It means that G(z(t)) ≡ ^|F(J)_LJ(z^(z(t))|0) for some J, kJk ≤ p. Clearly, the function F^(J)(z(t)) is analytic. Then |F^(J)(z(t))| is continuously differentiable function on the in- terval of analyticity except points where this partial derivative equals zero

|F^(j1,j2)(z₁(t), z₂(t))| = 0. However, there are not the points, because in the opposite caseG(z(t)) = 0.But it contradicts the construction of the curveγ.

2. In some interval of analyticity of the curve γ the function G(z(t)) equals si- multaneously to some derivatives at a finite number of points tk, that is (4.6) holds. Then the pointstk divide interval of analyticity onto a finite number of segments, in which of them G(z(t)) equals to one from the partial derivatives, i. e. G(z(t)) ≡ ^|F_L^(J)J^(z(t))|(z⁰) for some J, kJk ≤ p. As above, in each from these segments the functions|F^(J)(z(t))|, andG(z(t)) are continuously differentiable except the pointstk.

The inequality

d

dt|f(t)| ≤

df(t) dt

holds for complex-valued functions of real argument outside a countable set of points.

In view of this fact and (4.2) we have d

dtG(z(t))≤maxn 1 L^J(z⁰)

d

dtF^(J)(z(t))

: kJk ≤po

≤maxnXⁿ

s=1

∂^kJk+1F

∂z₁^j1. . . ∂zs^js+1. . . ∂z^jnⁿ

(z(t))

|z_s⁰(t)|

L^j(z⁰) : kJk ≤po

≤maxnXⁿ

s=1

∂^kJk+1F

∂z^j₁¹. . . ∂zs^js+1. . . ∂zn^jn

(z(t))

l_s(z⁰)|z_s⁰(t)|

l₁^j1(z⁰). . . l^js¹⁺¹(z⁰). . . l^jnⁿ(z⁰) : kJk ≤po

≤Xⁿ

s=1

ls(z⁰)|z_s⁰(t)|

maxn|F^(j)(z(t))|

L^J(z⁰) : kJk ≤p+ 1o

≤Xⁿ

s=1

ls(z⁰)|z⁰_s(t)|

BG(z(t)).

Therefore, (4.4) yields

lnG(z⁽²⁾) G(z⁽¹⁾) =

Z 1 0

1 G(z(t))

d

dtG(z(t))dt ≤B

Z 1 0

n

X

s=1

l_s(z⁰)|z_s⁰(t)|dt≤S·B.

Using (4.3), we deduce M

β

L(z⁰), z⁰, F

≤G(z⁽²⁾)≤G(z⁽¹⁾)e^SB.

Sincez⁽¹⁾∈Tⁿ(z⁰,_2β^√_nL(z¹ ₀₎),the Cauchy inequality holds

|F^(J)(z⁽¹⁾)|

L^J(z⁰) ≤J!(2β√ n)^kJkM

1 2β√

nL(z⁰), z⁰, F

. for allJ ∈Zⁿ+.Therefore, forkJk ≤pwe obtain

G(z⁽¹⁾)≤(p!)ⁿ(2β√ n)^pM

1 2β√

nL(z⁰), z⁰, F

,

M β

L(z⁰), z⁰, F

≤e^SB(p!)ⁿ(2β√ n)^pM

1 2β√

nL(z⁰), z⁰, F

.

Hence, by Theorem 3.1 the functionF has boundedL-index in joint variables.

The following result was also obtained for other classes of holomorphic functions in [21, 11, 7].

Theorem 4.2. Let L∈Q(Bⁿ). An analytic functionF in Bⁿ has bounded L-index in joint variables if and only if there exist c ∈ (0; +∞) and N ∈ N such that for each z∈Bⁿ the inequality

N

X

kKk=0

|F^(K)(z)|

K!L^K(z) ≥c

∞

X

kKk=N+1

|F^(K)(z)|

K!L^K(z). (4.7)

Proof. Let ¹_β < θ_j < 1, j ∈ {1, . . . , n}, Θ = (θ₁, . . . , θ_n). If the function F has bounded L-index in joint variables then by Theorem 2.2 F has bounded L-indexe in joint variables, where Le = (el1(z), . . . ,eln(z)),elj(z) = θjlj(z), j ∈ {1, . . . , n}. Let Ne =N(F,L,e Bⁿ). Therefore,

max

|F^(K)(z)|

K!L^K(z): kKk ≤Ne

= max

(Θ^K|F^(K)(z)|

K!eL^K(z) :kKk ≤Ne )

≥

n

Y

s=1

θ^N_s^emax

|F^(K)(z)|

K!eL^K(z):kKk ≤Ne

≥

n

Y

s=1

θ^N_s^e|F^(J)(z)|

J!eL^J(z) =

n

Y

s=1

θ^N_s^e−js|F^(J)(z)|

J!L^J(z) for allJ ≥0and

∞

X

kJk=N+1e

|F^(J)(z)|

J!L^j(z) ≤max

|F^(K)(z)|

K!L^K(z):kKk ≤Ne ^∞

X

kJk=Ne+1

θ_s^js−Ne

=

n

Y

i=1

θs

1−θs

max

|F^(K)(z)|

K!L^K(z):kKk ≤Ne

≤

n

Y

i=1

θs

1−θs Ne

X

kKk=0

|F^(K)(z)|

K!L^K(z). Hence, we obtain (4.7) withN =Ne and

c=

n

Y

i=1

θs

1−θ_s.