Analytic in an unit ball functions of bounded $L$-index in direction

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by  A. I. Bandura

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ANALYTIC IN THE UNIT BALL FUNCTIONS OF
BOUNDED L-INDEX IN DIRECTION
BANDURA ANDRIY1 AND SKASKIV OLEH2
1 Department of Higher Mathematics, Ivano-Frankivsk National Technical University of Oil
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and Gas, 15 Karpatska st., Ivano-Frankivsk, 76019, Ukraine
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E-mail: [e-mail]
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p 2 Department of Theory of Functions and Probability Theory, Ivan Franko National Univer-
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S sity of Lviv, 1 Universytetska st., Lviv, 79000, Ukraine

6 [e-mail]
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] Abstract
V
C We propose a generalisation of analytic in a domain function of bounded index, which
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h was introduced by J. G. Krishna and S. M. Shah [14]. In fact, analytic in the unit ball
t
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function of bounded index by Krishna and Shah is an entire function. Our approach allows
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us to explore properties of analytic in the unit ball functions.
[

3 We proved the necessary and sufficient conditions of bounded L-index in direction for
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analytic functions. As a result, they are applied to study partial differential equations and
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6 get sufficient conditions of bounded L-index in direction for analytic solutions. Finally, we
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4 estimated growth for these functions.
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.
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Keywords: analytic function, unit ball, bounded L-index in direction, growth estimates,
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5 partial differential equation, several complex variables
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: MSC[2010] 32A10, 32A17, 35B08
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a 1 Introduction
B. Lepson [19] introduced a class of entire functions of bounded index. He raisedthe problem
to characterise entire functions of bounded index. An entire function f is said to be of bounded
index if there exists an integer N > 0 that
|f(n)(z)| |f(j)(z)|
(∀z ∈ C)(∀n ∈ {0,1,2,...}): ≤ max |f(z)|, : 1 ≤ j ≤ N . (1)
n! j!
n o
The least such integer N is called the index of f.
Afterwards, S.Shah[21]andW.Hayman[13]independentlyprovedthateveryentirefunction
ofboundedindexisafunctionofexponentialtype. Namely, itsgrowthisatmostthefirstorderand
normal type. Further, W. Hayman showed that an entire function is of bounded value distribution
if and only if its derivative is of bounded index. An entire function f is said to be of bounded
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value distribution if for every r > 0 there exists a fixed integer p(r) > 0 such that the equation
f(z) = w has never more than p(r) roots in any disc of radius r and for any w ∈ C. The functions
of bounded index have been used in the theory value distribution and differential equations (see
bibliography in [21]).
T. Lakshminarasimhan [18] generalised a bounded index. He introduced entire functions of
L-bounded index, where L(r) is a positive continuous slowly increasing function. D. Somasun-
daram and R. Thamizharasi [25]-[26] continued his investigations of entire functions of L-bounded
index. They studied growth properties and characterisations of these functions.
B. C. Chakraborty, Rita Chanda and Tapas Kumar Samanta [10]-[12] introduced bounded
index and L-bounded index for entire functions in Cn. They found a necessary and sufficient
condition for an entire function to be of L-bounded index and proved some interesting properties.
J. Gopala Krishna and S. M. Shah [14] studied of the existence and analytic continuation of
the local solutions of partial differential equations. They introduced an analytic in a domain (a
nonempty connected open set) Ω ⊂ Cn (n ∈ N) function of bounded index for α = (α ,...,α ) ∈
1 n
Rn. Namely, let Ω = {z = (z ,...,z ) ∈ Ω: z > 0 (j ∈ {1,...,n})}, that is a subset of all
+ + 1 n j
points of Ω with positive real coordinates. We say that an analytic in Ω function F is a function
of bounded index (Krishna-Shah bounded index or F ∈ B(Ω,α)) for α = (α ,...,α ) ∈ Ω in
1 p +
domain Ω if and only if there exists N = N(α,F) = (N ,...,N ) ∈ Zn such that inequality
1 n +
αmT (z) ≤ max{αpT (z): p ≤ N},
m p
is valid for all z ∈ Ω and for every m ∈ Zn, where αm = αm1 ···αmn, T (z) = |F(m)(z)|/m!,
+ 1 n m
F(m)(z) = ∂kmkF be kmk-th partial derivative of F, F(0,...,0) = F, m! = m !···m !, kmk =
∂z1m1···∂znmn 1 n
m +...m , m = (m ,...m ) ∈ Zn.
1 n 1 n +
For entire functions in two variables, M. Salmassi [20] generalised bounded index and proved
three criteria of index boundedness. Besides, he researched a system of partial differential equa-
tions and found conditions of bounded index for entire solutions.
To consider the functions of nonexponential type A. D. Kuzyk and M. M. Sheremeta [17]
introducedaboundedl-index, replacing |f(p)(z)| on |f(p)(z)| in(1), wherel : R → R isacontinuous
p! p!lp(|z|) + +
function. Besides, they proved that growth of entire function of L-bounded index is not higher
than a normal type and first order.
Afterwards, S. M. Strochyk and M. M. Sheremeta [27] considered bounded l-index for func-
tions, that are analytic in a disc. Later T. O. Banakh, V. O. Kushnir and M. M. Sheremeta
generalised this term for analytic in arbitrary complex domain G ⊂ C functions ([1], [15] – [16]).
Yu. S. Trukhan and M. M. Sheremeta got sufficient conditions of bounded l-index for infinite
products, which are analytic in the unit disc. In particular, they researched Blaschke product and
Naftalevich-Tsuji product ([23], [28] – [32]).
M. T. Bordulyak and M. M. Sheremeta ([8] – [9]) defined a function of bounded L-index
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in joint variables, where L = L(z) =(l (z ),...,l (z )), l (z ) are positive continuous functions,
1 1 n n j j
j ∈ {1,...n}. If L(z) ≡ 1 ,..., 1 and Ω = Cn then a Bordulyak-Sheremeta’s definition
α1 αn
matches with a Krishna -(cid:16)Shah’s defi(cid:17)nition. If n = 2 and L(z) ≡ (1,1) then a Bordulyak-
Sheremeta’s definition matches with a Salmassi’s definition [20].
Methods for investigation of analytic functions in Cn are divided into several groups. One
group is based on the study of function F as analytic in each variable separately. Other methods
ariseinthestudyofslicefunctionthatisanalyticfunctionsofonevariableg(τ) = F(a+bτ),τ ∈ C.
This is a restriction of the analytic function F to arbitrary complex lines {z = a+bτ : τ ∈ C}, a,
b ∈ Cn.
Using the first approach, M. T. Bordulyak and M. M. Sheremeta [8] proved many properties
and criteria of bounded L-index for entire functions in Cn. They got sufficient conditions of
bounded L-index for entire solutions of some systems of partial differential equations. However,
this approach did not allow to find an equivalent to a criterion of bounded L-index by the estimate
ofthelogarithmicderivative outsidezero set. Inparticular, effortstoexplore L-index boundedness
for some important classes of entire functions (for example, infinite products with ”plane” zeros)
were unsuccessful by technical difficulties.
For the reasons given above, there was a natural problem to consider and to explore an entire
in Cn function of bounded L-index by a second approach.
Applying this method, we proposed a new approach to introduce an entire in Cn function of
bounded L-index in direction [3] – [7]. In contrast to the approach proposed by M.T. Bordulyak
and M. M. Sheremeta, our definition is based on directional derivative. It allowed to generalize
more results from C to Cn and find new assertions because a definition contains a directional
derivative and it has influence on the L-index.
This success gives possibility of generalisation of bounded L-index in direction for analytic in
a ball functions. Besides, analytic in a domain function of bounded index by Krishna and Shah is
an entire function. It follows from necessary condition of l-index boundedness for analytic in the
r
unit disc function ([22],Th.3.3, p.71): l(t)dt → ∞ as r → 1. In this paper, we proved criteria of
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L-index boundedness in direction, which describe a maximum modulus estimate on a larger circle
R
by maximum modulus on a smaller circle, an analogue of Hayman Theorem, a maximum modulus
estimate on circle by minimum modulus on circle, an estimate of logarithmic directional derivative
outside zero set and an estimate of counting function of zeros. They helped to get conditions on
partial differential equation which provide bounded L-index in direction for analytic solutions.
Finally, we describe the growth of analytic in B function of bounded L-index in direction.
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Remark. We investigate analytic functions in the unit ball instead the ball of arbitrary
radius.
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2 Main definition and properties functions of bounded L-
index in direction
Let b = (b ,...,b ) ∈ Cn be a given direction, B = {z ∈ Cn : |z| < 1}, B = {z ∈ Cn :
1 n n n
|z| ≤ 1}, L : B → R be a continuous function that for all z ∈ B
n + n
β|b|
L(z) > , β = const > 1,b 6= 0. (2)
1−|z|
For a given z ∈ B we denote S = {t ∈ C : z +tb ∈ B }.
n z n
Remark 1. Notice that if η ∈ [0,β], z ∈ B , z+t b ∈ B and |t−t | ≤ η then z+tb ∈ B .
n 0 n 0 L(z+t0b) n
Indeed, we have |z +tb| = |z +t b+(t−t )b| ≤ |z +t b|+|(t−t )b| ≤ |z +t b|+ η|b| <
0 0 0 0 0 L(z+t0b)
|z +t b|+ β|b| = 1.
0 β|b|
1−|z+t0b|
Analytic in B function F(z) is called a function of bounded L-index in a direction b ∈ Cn,
n
if there exists m ∈ Z that for every m ∈ Z and every z ∈ B the following inequality is valid
0 + + n
1 ∂mF(z) 1 ∂kF(z)
≤ max : 0 ≤ k ≤ m , (3)
m!Lm(z) ∂bm k!Lk(z) ∂bk 0
(cid:12) (cid:12) (cid:26) (cid:12) (cid:12) (cid:27)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) n (cid:12) (cid:12) (cid:12)
where ∂0F(z) = F(z), ∂F(z) =(cid:12) ∂F(z(cid:12))b = hgrad F, bi,(cid:12)∂kF(z) =(cid:12) ∂ ∂k−1F(z) , k ≥ 2.
∂b0 ∂b ∂zj j ∂bk ∂b ∂bk−1
j=1
(cid:16) (cid:17)
TheleastsuchintegermP= m (b)iscalledtheL-index in direction b of the analytic function
0 0
F(z) and is denoted by Nb(F,L) = m0. If n = 1, b = 1, L = l, F = f, then N(f,l) ≡ N1(f,l) is
called the l-index of function f.
In the case n = 1 and b = 1 we have definition of analytic in the unit disc function of
bounded l-index [27].
Now we state several lemmas that contain the basic properties of analytic in the unit ball
functions of bounded L-index in direction. Let l (t) = L(z + tb), g (t) = F(z + tb) for given
z z
z ∈ Cn.
Lemma 1. If F(z) is an analytic in Bn function of bounded L-index Nb(F,L) in direction b ∈
Cn, then for every z0 ∈ B the analytic function g (t), t ∈ S , is of bounded l -index and
n z0 z0 z0
N(gz0,lz0) ≤ Nb(F,L).
Proof. Let z0 ∈ B be a fixed point and g(t) ≡ g (t), l(t) ≡ l (t). Since for every p ∈ N
n z0 z0
∂pF(z0 +tb)
g(p)(t) = , (4)
∂bp
then by the definition of bounded L-index in direction b for all t ∈ S and for all p ∈ Z we
z0 +
obtain
|g(p)(t)| 1 ∂pF(z0 +tb) 1 ∂kF(z0 +tb)
= ≤ max :
p!lp(t) p!Lp(z0 +tb) ∂bp k!Lk(z0 +tb) ∂bk
(cid:12) (cid:12) n (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
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|g(k)(t)|
0 ≤ k ≤ Nb(F,L) = max : 0 ≤ k ≤ Nb(F,L) .
k!lk(t)
o n o
From here, g(t) is a function of bounded l-index and N(g,l) ≤ Nb(F,L). Lemma 1 is proved.
An equation (4) implies a following proposition.
Lemma 2. If F(z) is an analytic in B function of bounded L-index in direction b ∈ Cn then
n
Nb(F,L) = max{N(gz0,lz0) : z0 ∈ Bn}.
However, maximum can be calculated on the subset A with points z0, which has property
{z0 +tb : t ∈ S ,z0 ∈ A} = B . So the following assertion is valid.
z0 n
Lemma 3. If F(z) is an analytic in B function of bounded L-index in direction b ∈ Cn and j
n 0
is chosen with bj0 6= 0 then Nb(F,L) = max{N(gz0,lz0) : z0 ∈ Cn,zj00 = 0} and if nj=1bjk 6= 0
then Nb(F,L) = max{N(gz0,lz0) : z0 ∈ Cn, nj=1zj0 = 0}. P
Proof. We prove that for every z ∈ B thePre exist z0 ∈ Cn and t ∈ S with z = z0 + tb and
n z0
z0 = 0. Put t = z /b , z0 = z −tb , j ∈ {1,2,...,n}. Clearly, z0 = 0 for this choice.
j0 j0 j0 j j j j0
However, a point z0 may not be contained in B . But there exists t ∈ C that z0 +tb ∈ B .
n n
Let z0 ∈/ B and |z| = R < 1. Therefore, |z0 + tb| = |z − zj0b + tb| = |z + (t − zj0)b| ≤
n 1 bj0 bj0
|z|+|t− zj0|·|b| ≤ R +|t− zj0|·|b| < 1. Thus, |t− zj0| < 1−R1.
bj0 1 bj0 bj0 |b|
In second part we prove for every z ∈ B there exist z0 ∈ Cn and t ∈ S that z = z0 +tb
n z0
and n z0 = 0. Put t = Pnj=1zj and z0 = z −tb , 1 ≤ j ≤ n. Thus, the following equality is
j=1 j Pnj=1bj j j j
valid n z0 = n (z −tb ) = n z − n b t = 0.
P j=1 j j=1 j j j=1 j j=1 j
Lemma 3 is proved.
P P P P
Note that for a given z ∈ B we can pick uniquely z0 ∈ Cn and t ∈ S such that n z0 = 0
n z0 j=1 j
and z = z0 +tb.
P
Remark 2. If for some z0 ∈ Cn {z0 +tb : t ∈ C} B = ∅ then we put N(g ,l ) = 0.
n z0 z0
T
Lemmas 1–3 imply the following proposition.
Theorem 1. An analytic in B function F(z) is a function of bounded L-index in direction
n
b ∈ Cn if and only if there exists number M > 0 such, that for every z0 ∈ B function g (t)
n z0
is of bounded l -index with N(g ,l ) ≤ M < +∞, as a function of one variable t ∈ S , and
z0 z0 z0 z0
Nb(F,L) = max{N(gz0,lz0) : z0 ∈ Bn}.
Proof. Necessity follows from Lemma 1.
We prove sufficiency.
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Since N(g ,l ) ≤ M there exists max{N(g ,l ) : z0 ∈ B }. We denote this maximum by
z0 z0 z0 z0 n
Nb(F,L) = max{N(gz0,lz0) : z0 ∈ Bn} < ∞. Suppose that Nb(F) is not L-index in direction b of
function F(z). So there exists n∗ > Nb(F,L) and z∗ ∈ Bn
1 ∂n∗F(z∗) 1 ∂kF(z∗)
> max ,0 ≤ k ≤ Nb(F,L) . (5)
n∗!Ln∗(z∗) ∂bn∗ k!Lk(z∗) ∂bk
(cid:12) (cid:12) (cid:26) (cid:12) (cid:12) (cid:27)
(cid:12) (cid:12) (cid:12) (cid:12)
But we have g (t) = F(z0 +tb), g(p)(t) = ∂pF(z0+tb). We can rewrite (5) as
z0 z0 ∂bp
g(n∗)(0) |g(k)(0)|
z∗ > max z∗ : 0 ≤ k ≤ Nb(F,L) .
n∗!ln∗(0) k!lk (0)
(cid:12) z∗ (cid:12) (cid:26) z∗ (cid:27)
(cid:12) (cid:12)
Itcontradicts thatalllz0-indices N(gz0,lz0)areboundedbynumber Nb(F).Thus Nb(F)isL-index
in direction b of function F(z). Theorem 1 is proved.
From Lemma 3 the following condition is enough in Theorem 1: there exists M < +∞ that
an inequality holds N(g ,l ) ≤ M for every z0 ∈ Cn with n z0 = 0.
z0 z0 j=1 j
Since Lemma 3 and 1 there is a natural question: what is the least set A that the following
P
equality is valid Nb(F,L) = maxN(gz0,lz0).
z0∈A
Below we prove propositions that give a partial answer to this question. A solution is partial
because it is unknown whether our sets are the least which satisfy the mentioned equality.
Theorem 2. Let b ∈ Cn be a given direction, A be an arbitrary set in Cn with {z + tb : t ∈
0
S , z ∈ A } = B . Analytic in B function F(z) is of bounded L-index in direction b ∈ Cn if and
z 0 n n
only if there exists M > 0 that for all z0 ∈ A function g (t) is of bounded l -index N(g ,l ) ≤
0 z0 z0 z0 z0
M < +∞, as a function of variable t ∈ Sz0. And Nb(F,L) = max{N(gz0,lz0) : z0 ∈ A0}.
Proof. By Theorem 1, analytic in B function F(z) is of bounded L-index in direction b ∈ Cn if
n
and only if there exists number M > 0 such that for every z0 ∈ B function g (t) is of bounded
n z0
l -index N(g ,l ) ≤ M < +∞, as a function of variable t ∈ S . But for every z0 + tb by
z0 z0 z0 z0
properties of set A there exist z0 ∈ A and t ∈ B
0 0 z0
e
e z0 +teb = z0 +tb.
For all p ∈ Z we have e e
+
(g (t))(p) = (g (t))(p).
z0 z0
e
But t is dependent of t. Therefore, a condition g (t) is of bounded l -index for all z0 ∈ B is
z0 e z0 n
equivalent to a condition g (t) is of bounded l -index for all z0 ∈ A .
z0 z0 0
e e e
Remark 3. An intersection of arbitrary hyperplane H = {z ∈ Cn : hz,ci = 1} and set Bb =
e n
{z + 1−hz,cib: z ∈ B }, where hb,ci =6 0, satisfies conditions of Theorem 2.
hb,ci n
7
We prove that for every w ∈ B there exist z ∈ H Bb and t ∈ C such that w = z +tb.
n n
hw,ci−1
Choosing z = w + 1−hbhw,c,icib ∈ H Bbn, t = hb,ciT, we obtain
T
1−hw,ci hw,ci−1
z +tb = w + b+ b = w.
hb,ci hb,ci
Theorem 3. Let A be an everywhere dense set in B . Analytic in B function F(z) is of bounded
n n
L-index in direction b ∈ Cn if and only if there exists number M > 0 that for every z0 ∈ A
function g (t) is of bounded l -index N(g ,l ) ≤ M < +∞, as a function of t ∈ S , and
z0 z0 z0 z0 z0
Nb(F,L) = max{N(gz0,lz0) : z0 ∈ A}.
Proof. The necessity follows from Theorem 1 (in this theorem same condition is satisfied for all
z0 ∈ B , and we need this condition for all z0 ∈ A, that A∩B = B ).
n n n
Now we prove a sufficiency. Since A has been everywhere dense in B , for every z0 ∈ B
n n
there exists a sequence (zm), that z(m) → z0 as m → +∞ and z(m) ∈ A for all m ∈ N. But
F(z + tb) is of bounded l -index for all z ∈ A ∩ B as a function of t. Therefore, by bounded
z n
l -index there exists M > 0 that for all z ∈ A, t ∈ C, p ∈ Z
z +
(p) (k)
|g (t)| |g (t)|
z z
≤ max : 0 ≤ k ≤ M .
p!lp(t) k!lk(t)
( z )
After substitution instead of z a sequence z(m) ∈ A and z(m) → z0, for each m ∈ N the
following inequality holds
(p) (k)
|g (t)| |g (t)|
zm ≤ max zm : 0 ≤ k ≤ M
p!lp (t) k!lk (t)
zm ( zm )
In other words, we have
1 ∂pF(zm +tb) 1 ∂kF(zm +tb)
≤max :
p!Lp(zm +tb) ∂bp k!Lk(zm +tb) ∂bk
(cid:12) (cid:12) (cid:26) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) 0(cid:12) ≤ k ≤ M}. (cid:12) (cid:12) (6)
(cid:12) (cid:12) (cid:12) (cid:12)
But F is an analytic in B function and L is a positive continuous. In (6) we calculate a
n
limit m → +∞ (zm → z0). We have that for all z0 ∈ B , t ∈ S , m ∈ Z
n z0 +
1 ∂pF(z0 +tb) 1 ∂kF(z0 +tb)
≤ max :
p!Lp(z0 +tb) ∂bp k!Lk(z0 +tb) ∂bk
(cid:12) (cid:12) (cid:26) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) 0(cid:12) ≤ k ≤ M}. (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Since this inequality F(z0 +tb) is of bounded L(z0 +tb)-index too, as a function of t, for every
given z0 ∈ B . Applying Theorem 1 we get a needed conclusion. Theorem 3 is proved.
n
Since Remark 3 and Theorem 3 the following corollary is true.
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Corollary 1. Let b ∈ Cn be a given direction, A be a set in Cn and its closure is A = {z ∈ Cn :
0 0
hz,ci = 1} Bb, where hc,bi =6 0, Bb = {z + 1−hz,cib: z ∈ B }. Analytic in B function F(z) is
n n hb,ci n n
of bounded L-index in direction b ∈ Cn if and only if there exists number M > 0 such that for all
T
z0 ∈ A function g (t) is of bounded l -index N(g ,l ) ≤ M < +∞, as a function of variable
0 z0 z0 z0 z0
t ∈ Sz0. And Nb(F,L) = max{N(gz0,lz0) : z0 ∈ A0}.
Proof. Since Remark 3 in Theorem 2 we can take an arbitrary hyperplane B = {z ∈ Bn : hz,ci =
0
1}, where hc,bi =6 0. Let A be an everywhere dense set in B , A = B . Repeating considerations
0 0 0 0
of Theorem 3, we obtain a needed conclusion.
Indeed, the necessity follows from Theorem 1 (in this theorem same condition is satisfied for
all z0 ∈ Cn, and we need this condition for all z0 ∈ A , that A ∩B = {z ∈ B : hz,ci = 1}).
0 0 n n
To prove the sufficiency, we use a density of the set A . Obviously, for every z0 ∈ B there
0 0
exists a sequence z(m) → z0 and z(m) ∈ A . But g (t) is of bounded l -index for all z ∈ A as a
0 z z 0
function of t. Since conditions of Corollary 1, for some M > 0 and for all z ∈ A , t ∈ C, p ∈ Z
0 +
the following inequality holds
(p) (k)
g (t) |g (t)|
z z
≤ max : 0 ≤ k ≤ M .
p!lp(t) k!lk(t)
z ( z )
Substituting an arbitrary sequence z(m) ∈ A, z(m) → z0 instead of z ∈ A0, we have
(p) (k)
|g (t)| |g (t)|
z(m) ≤ max z(m) ,
p!lp (t) k!lk (t) : 0 ≤ k ≤ M
z(m) ( z(m) )
i.e.
1 ∂pF(z(m)+tb) 1 ∂kF(z(m)+tb)
≤max : 0 ≤ k ≤ M .
Lp(z(m) +tb) ∂bp k!Lk(z(m)+tb) ∂bk
(cid:12) (cid:12) (cid:26) (cid:12) (cid:12) (cid:27)
(cid:12) (cid:12) (cid:12) (cid:12)
However, F is an a(cid:12)nalytic in B fu(cid:12)nction, L is a positive co(cid:12)ntinuous. So w(cid:12)e calculate a limit as
(cid:12) n (cid:12) (cid:12) (cid:12)
m → +∞ (zm → z). For all z0 ∈ B , t ∈ S , m ∈ Z we have
0 z0 +
1 ∂pF(z0 +tb) 1 ∂kF(z0 +tb)
≤ max : 0 ≤ k ≤ M}.
Lp(z0 +tb) ∂bp k!Lk(z0 +tb) ∂bk
(cid:12) (cid:12) (cid:26) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Therefore, F(z0 + t(cid:12)b) is of bound(cid:12)ed L(z0 + tb)-index as(cid:12)a function of(cid:12)t at each z0 ∈ Bn. By
(cid:12) (cid:12) (cid:12) (cid:12)
Theorem 3 and Remark 3 F is of bounded L-index in direction b.
Remark 4. Let H = {z ∈ Cn : hz,ci = 1}. The condition hc,bi =6 0 is essential. If hc,bi = 0 then
for all z0 ∈ H and for all t ∈ C the point z0 +tb ∈ H because hz0 +tb,ci = hz0,ci+thb,ci = 1.
Thus, this line z0 +tb does not describe points outside a hyperplane H.
We consider F(z ,z ) = exp(−z2+z2), b = (1,1), c = (−1,1). On a hyperplane −z +z = 1
1 2 1 2 1 2
function F(z ,z ) takes a look
1 2
F(z0 +tb) = F(z0 +t,z0 +t) = exp(−(z0 +t)2 +(1+z0 +t)2) =
1 2 1 1
9
= exp(1+2z0 +2t).
1
Using definition of l-index boundedness and evaluating corresponding derivatives it is easy to
prove that exp(1+2z0 +2t) is of bounded index with l(t) = 1 and N(g,l) = 4.
1
Thus, F is of unbounded index in direction b. On the contrary, we assume Nb(F) = m and
calculate directional derivatives
∂pF
= 2p(−z +z )pexp(−z +z ), p ∈ N.
∂bp 1 2 1 2
By definition of bounded index, an inequality holds ∀ p ∈ N ∀z ∈ Cn
2p|−z +z |p|exp(−z +z )| ≤ max 2k|−z +z |k|exp(−z +z )|. (7)
1 2 1 2 1 2 1 2
0≤k≤m
Let p > m and |−z +z | = 2. Dividing equation (7) by 2p|exp(−z +z )|, we get 22p ≤ 22m. It
1 2 1 2
is impossible. Therefore, F(z) is of unbounded index in direction b.
Using calculated derivatives it can be proven that function F(z ,z ) is of bounded L-index
1 2
in direction b with L(z1,z2) = 2|−z1 +z2|+1 and Nb(F,L) = 0.
Now we consider another function
∞
F(z) = (1+hz,di) (1+hz,ci·2−j)j, c 6= d.
j=1
Y
The multiplicity of zeros for function F(z) increases to infinity. Below in this paper, we will state
Theorem 12. By that theorem, unbounded multiplicity of zeros means that F(z) is of unbounded
L-index in any direction b (hb,ci =6 0) and for any positive continuous function L.
We select b ∈ Cn that hb,di = 0. Let H = {z ∈ Cn : hz,di = −1}. But for z0 ∈ H we have
∞
F(z0 +tb) = (1+hz0,di+thb,di) (1+hz0,ci2−j +thb,ci2−j)j ≡ 0.
j=1
Y
Thus, F(z0 +tb) is of bounded index as a function of variable t.
Theorem 4. Let (r ) be a positive sequence such that r → 1 as p → ∞, D = {z ∈ Cn : |z| = r },
p p p p
∞
A be an everywhere dense set in D (i.e. A = D ) and A = A . Analytic in B function F(z)
p p p p p n
p=1
is of bounded L-index in direction b ∈ Cn if and only if therSe exists number M > 0 that for all
z0 ∈ A function g (t) = F(z0 +tb) is of bounded l -index N(g ,l ) ≤ M < +∞, as a function
z0 z0 z0 z0
of variable t ∈ Sz0, where lz0(t) ≡ L(z0 +tb). And Nb(F,L) = max{N(gz0,lz0) : z0 ∈ A}.
Proof. Theorem 1 implies the necessity of this theorem.
Sufficiency. It is easy to prove {z + tb : t ∈ S , z ∈ A} = B . Further, we repeat
z n
considerations with proof of sufficiency in Theorem 3 and obtain a needed conclusion.
10
3 Auxiliary class Qn
b
The positivity and continuity of function L and condition (2) are not enough to explore the
behaviour of entire function of bounded L-index in direction. Below we impose the extra condition
that function L does not vary as soon.
For η ∈ [0,β], z ∈ B , t ∈ S such that z +t b ∈ B we define
n 0 z 0 n
L(z +tb) η
b
λ (z,t ,η,L) = inf : |t−t | ≤ ,
1 0 L(z +t b) 0 L(z +t b)
(cid:26) 0 0 (cid:27)
λb(z,η,L) = inf{λb(z,t ,η,L) : t ∈ S }, λb(η,L) = inf{λb(z,η,L) : z ∈ B }, and
1 1 0 0 z 1 1 n
L(z +tb) η
b
λ (z,t ,η,L) = sup : |t−t | ≤ ,
2 0 L(z +t b) 0 L(z +t b)
(cid:26) 0 0 (cid:27)
λb(z,η,L) = sup{λb(z,t ,η,L) : t ∈ S }, λb(η,L) = sup{λb(z,η,L) : z ∈ B }.
2 2 0 0 z 2 2 n
b b b
If it will not cause misunderstandings, then λ (z,t ,η) ≡ λ (z,t ,η,L), λ (z, t , η) ≡
1 0 1 0 2 0
b b b b b b b b b
λ (z,t ,η,L), λ (z,η) ≡ λ (z,η,L), λ (z,η) ≡ λ (z,η,L), λ (η) ≡ λ (η,L), λ (z,η) ≡ λ (η,L).
2 0 1 1 2 2 1 1 2 2
By Qb,β(Bn) we denote the class of all functions L for which the following condition holds
for any η ∈ [0,β] 0 < λb(η) ≤ λb(η) < +∞. Let D ≡ B1, Q (D) ≡ Q (D).
1 2 β 1,β
The following lemma suggests possible approach to compose function with Qn.
b
Lemma 4. Let L : B → R be a continuous function, m = min{L(z) : z ∈ B }. Then L(z) =
n + n
β|b| · L(z) ∈ Qn(B ) for every b ∈ Cn \{0}, α ≥ 1.
m (1−|z|)α b n
e
Proof. Using definition Qn we have ∀z ∈ B ∀t ∈ S
b n 0 z
b
λ (z,t ,η,L) =
1 0
L(z +tb) (1−|z +t b|)α ηm(1−|z +t b|)α
0 0
= inf (1−|z +tb|)α · L(z +t b) e: |t−t0| ≤ β|b|L(z +t b) ≥
(cid:26) 0 0 (cid:27)
L(z +tb) ηm(1−|z +t b|)α
0
≥ inf : |t−t | ≤ ×
L(z +t b) 0 β|b|L(z +t b)
(cid:26) 0 0 (cid:27)
1−|z +t b| α ηm(1−|z +t b|)α
0 0
inf : |t−t | ≤
1−|z +tb| 0 β|b|L(z +t b)
(cid:26)(cid:18) (cid:19) 0 (cid:27)
Since Remark 1 the first infimum is not less than some constant K > 0 which is independent from
z and t . Besides, we have ∀z ∈ B and ∀t ∈ S m ≤ 1. Thus, for the second infimum the
0 n z L(z+t0b)
following estimates are valid
1−|z +t b| α ηm(1−|z +t b|)α
0 0
inf : |t−t | ≤ ≥
1−|z +tb| 0 β|b|L(z +t b)
(cid:26)(cid:18) (cid:19) 0 (cid:27)
1−|z +t b| α η(1−|z +t b|)α 1−|z +t b| α
0 0 0
≥ inf : |t−t | ≤ = .
1−|z +tb| 0 β|b| 1−|z +t∗b|
(cid:26)(cid:18) (cid:19) (cid:27) (cid:18) (cid:19)