Asymptotic estimates of entire functions of bounded $\mathbf{L}$-index in joint variables

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by  A. I. Bandura

1
ASYMPTOTIC ESTIMATES OF ENTIRE FUNCTIONS
OF BOUNDED -INDEX IN JOINT VARIABLES
L
A. I. BANDURA, O. B. SKASKIV
7
1
Abstract
0
2
A. I. Bandura, O. B. Skaskiv, Asymptotic estimates of entire functions of bounded L-index
n
a
in joint variables
J

8 In this paper, there are obtained growth estimates of entire in Cn function of bounded L-
2
indexinjointvariables. Theydescribethebehaviourofmaximum modulusof entirefunction

]
V on a skeleton in a polydisc by behaviour of the function L(z) = (l (z),...,l (z)), where for
1 n
C every j ∈ {1,...,n} l : Cn → R is a continuous function. We generalised known results
j +
.
h
of W. K. Hayman, M. M. Sheremeta, A. D. Kuzyk, M. T. Borduyak, T. O. Banakh and V.
t
a
O. Kushnir for a wider class of functions L. One of our estimates is sharper even for entire
m
[ in C functions of bounded l-index than Sheremeta’s estimate.

1
v
6
7
1 Introduction
2
8
0
. Let l : C → R be a fixed positive continuous function, where R = (0,+∞). An entire
1 + +
0 function f is said to be of bounded l−index [1] if there exists an integer m, independent of z,
7
1 such that for all p and all z ∈ C |f(p)(z)| ≤ max{|f(s)(z)|: 0 ≤ s ≤ m}. The least such integer m is
: lp(z)p! ls(z)s!
v
called the l−index of f(z) and is denoted by N(f,l). If l(z) ≡ 1 then we obtain the definition of
i
X
function of bounded index [2] and in this case we denote N(f) := N(f,1).
r
a
In 1970 W. J. Pugh and S. M. Shah [3] posed some questions about properties of entire
functions of bounded index. One of those questions is following: I. What are the growth properties
of functions of bounded index: (c) is it possible to derive the boundedness (or the unboundedness)
of the index from the asymptotic properties of the logarithm of the maximum modulus of f(z), i.e.,
lnM(r,f)?
W. K. Hayman [4] proved that entire function of bounded index has exponential type which
isnotgreaterthanN(f)+1.LaterA.D.Kuzyk andM.M. Sheremeta [1]obtainedgrowthestimate
of entire function of bounded l−index. M. M. Sheremeta [5], T. O. Banakh and V. O. Kushnir
[6] deduced analogical inequalities for analytic in a unit disc and in arbitrary complex domain
function of bounded l-index, respectively.
Clearly, the question of Shah and Pugh can be formulated for entire in Cn function: What
are the growth properties of functions of bounded L-index in joint variables? Is it possible to derive
2
the boundedness (or the unboundedness) of the L-index in joint variables from the asymptotic
properties of the logarithm of the maximum modulus of F(z) on a skeleton in a polydisc?
M. T. Bordulyak and M. M. Sheremeta [7] gave an answer to the question if L(z) =
(l (|z |),...,l (|z |)), and for every j ∈ {1,...,n} the function l : R → R is continuous.
1 1 n n j + +
In this paper we extend their results for L(z) = (l (z),...,l (z)), where l : Cn → R is a contin-
1 n j +
uous function for every j ∈ {1,...,n}. In some sense our results are new even in one-dimensional
case (see below Corollaries 2 and 3).
2 Notations and definitions
We need some standard notations. Let R = [0,+∞). Denote
+
0 = (0,...,0) ∈ Rn, 1 = (1,...,1) ∈ Rn, 2 = (2,...,2) ∈ Rn,
+ + +
e = (0,...,0, 1 ,0,...,0) ∈ Rn, [0,2π]n = [0,2π]×···×[0,2π].
j +
j−th place n−thtimes
For R = (r1,...,rn) ∈ Rn+, Θ|={z(}θ1,...,θn) ∈ [0,2π]n and K = (|k1,...,kn{)z∈ Zn+ let}us to denote
kRk = r + ···+ r , ReiΘ = (r eiθ1,...,r eiθn), K! = k ! · ... · k !. For A = (a ,...,a ) ∈ Cn,
1 n 1 n 1 n 1 n
B = (b ,...,b ) ∈ Cn, we will use formal notations without violation of the existence of these
1 n
expressions
|A| = (|a |,...,|a |), A±B = (a ±b ,...,a ±b ), AB = (a b ,··· ,a b ),
1 n 1 1 n n 1 1 n n
argA = (arga ,...,arga ),A/B = (a /b ,...,a /b ), AB = ab1ab2 ·...abn,
1 n 1 1 n n 1 2 n
and a notation A < B means that a < b for all j ∈ {1,...,n}; similarly, the relation A ≤ B is
j j
defined.
The polydisc {z ∈ Cn : |z −z0| < r , j = 1,...,n} is denoted by Dn(z0,R), its skeleton
j j j
{z ∈ Cn : |z − z0| = r , j = 1,...,n} is denoted by Tn(z0,R), and the closed polydisc
j j j
{z ∈ Cn : |z −z0| ≤ r , j = 1,...,n} is denoted by Dn[z0,R]. For a partial derivative of entire
j j j
function F(z) = F(z ,...,z ) we will use the notation
1 n
∂kKkF ∂k1+···+knf
F(K)(z) = = , where K = (k ,...,k ) ∈ Zn.
∂zK ∂zk1 ...∂zkn 1 n +
1 n
Let L(z) = (l (z),...,l (z)),where l (z) arepositive continuous functionsofvariablez ∈ Cn,
1 n j
j ∈ {1,2,...,n}.
An entire function F(z) is called a function of bounded L-index in joint variables, [9, 10] if
there exists a number m ∈ Z such that for all z ∈ Cn and J = (j ,j ,...,j ) ∈ Zn
+ 1 2 n +
|F(J)(z)| |F(K)(z)|
≤ max : K ∈ Zn, kKk ≤ m . (1)
J!LJ(z) K!LK(z) +
(cid:26) (cid:27)
3
If l = l (|z |) then we obtain a concept of entire functions of bounded L-index in sense of
j j j
definition in the papers [7, 8]. If l (z ) ≡ 1, j ∈ {1,2,...,n}, then the entire function is called a
j j
function of bounded index in joint variables [11, 12, 13].
The least integer m for which inequality holds is called L-index in joint variables of the
function F and is denoted by N(F,L).
For R ∈ Rn, j ∈ {1,...,n} and L(z) = (l (z),...,l (z)) we define
+ 1 n
l (z) R
λ (z ,R) = inf j : z ∈ Dn z0, , λ (R) = inf λ (z ,R),
1,j 0 (cid:26)lj(z0) (cid:20) L(z0)(cid:21)(cid:27) 1,j z0∈Cn 2,j 0
l (z) R
λ (z ,R) = sup j : z ∈ Dn z0, , λ (R) = sup λ (z ,R),
2,j 0 l (z0) L(z0) 2,j 2,j 0
(cid:26) j (cid:20) (cid:21)(cid:27) z0∈Cn
Λ (R) = (λ (R),...,λ (R)), Λ (R) = (λ (R),...,λ (R)).
1 1,j 1,n 2 2,1 2,n
By Qn we denote a class of functions L(z) which for every R ∈ Rn and j ∈ {1,...,n} satisfy
+
the condition
0 < λ (R) ≤ λ (R) < +∞. (2)
1,j 2,j
3 Auxiliary propositions
Proposition 1. Let L(z) = (l (z),...,l (z)), l : Cn → C and ∂lj be continuous functions in Cn
1 n j ∂zm
for all j, m ∈ {1,2,...,n}. If there exist numbers P > 0 and c > 0 such that for all z ∈ Cn and
every j,m ∈ {1,2,...,n}
1 ∂l (z)
j
≤ P (3)
c+|l (z)| ∂z
j (cid:12) m (cid:12)
(cid:12) (cid:12)
then L∗ ∈ Qn, where L∗(z) = (c+|l (z)|,...,c+(cid:12) |l (z)(cid:12)|).
1 n
(cid:12) (cid:12)
Proof. Clearly, the function L∗(z) is positive and continuous. For given z ∈ Cn, z0 ∈ Cn we define
an analytic curve ϕ : [0,1] → Cn
ϕ (τ) = z0 +τ(z −z0), j ∈ {1,2,...,n},
j j j j
where τ ∈ [0,1]. It is known that for every continuously differentiable function g of real variable τ
the inequality d|g(τ)| ≤ |g′(τ)| holds except the points where g′(τ) = 0. Using restrictions of this
dt
lemma, we establish the upper estimate of λ (z ,R) :
2,j 0
c+|l (z)| R
λ (z ,R) = sup j : z ∈ Dn z0, =
2,j 0 c+|l (z0)| L (z0)
(cid:26) j (cid:20) 1 (cid:21)(cid:27)
= sup exp ln(c+|l (z)|)−ln(c+|l (z0)|) =
j j
z∈Dn z0, R
h L1(z0)i(cid:8) (cid:8) (cid:9)(cid:9)
1 d(c+|l (ϕ(τ))|) R
=sup exp j : z ∈ Dn z0, ≤
c+|l (ϕ(τ))| L (z0)
(cid:26) (cid:26)Z0 j (cid:27) (cid:20) 1 (cid:21)(cid:27)
4
1 n |ϕ′ (τ)| ∂l (ϕ(τ))
≤ sup exp m j dτ ≤
c+|l (ϕ(τ))| ∂z
z∈Dnhz0,L1R(z0)i( (Z0 mX=1 j (cid:12)(cid:12) m (cid:12)(cid:12) ))
(cid:12) (cid:12)
1 n (cid:12) (cid:12)
≤ sup exp P|z −z0 |dτ ≤
m m
z∈Dnhz0,L1R(z0)i( (Z0 mX=1 ))
n n
Pr P
j
≤ sup exp ≤ exp r .
j
c+|l (z0)| c
z∈Dnhz0,L1R(z0)i( (mX=1 m )) mX=1 !
n
Hence, for all R ≥ 0 λ (R) = sup λ (z0,η) ≤ exp P r < ∞. Using d|g(t)| ≥ −|g′(t)|
2,j 2,j c j dt
z0∈Cn (cid:18) m=1 (cid:19)
Pn
it can be proved that for every η ≥ 0 λ (R) ≥ exp −P r > 0. Therefore, L∗ ∈ Qn.
1,j c j
(cid:18) m=1 (cid:19)
P
For estimate of growth of entire functions of bounded L-index in joint variables we will use
the following theorem which describes local behaviour of these entire functions.
Theorem 1 ([9, 10]). Let L ∈ Qn. An entire function F is of bounded L-index in joint variables
if and only if there exist numbers R′, R′′, 0 < R′ < e < R′′, and p = p (R′,R′′) ≥ 1 such that for
1 1
every z0 ∈ Cn
R′′ R′
max |F(z)|: z ∈ Tn z0, ≤ p max |F(z)|: z ∈ Tn z0, . (4)
L(z0) 1 L(z0)
(cid:26) (cid:18) (cid:19)(cid:27) (cid:26) (cid:18) (cid:19)(cid:27)
At first we prove the following lemma.
Lemma 1. If L ∈ Qn, then for every j ∈ {1,...,n} and for every fixed z∗ ∈ Cn |z |l (z∗+z e ) →
j j j j
∞ as |z | → ∞.
j
(m)
Proof. On the contrary, if there exist a number C > 0 and a sequence z → ∞ such that
j
|z(m)|l (z∗ +z(m)e ) = k ≤ C, i. e. |z(m)| = km . Then
j j j j m j lj(z∗+zj(m)ej)
1 l z∗ +z(m)e − kmeiargzj(m)ej = |zj(m)|l (z∗) → +∞, j → +∞,
lj(z∗ +zj(m)ej) j j j lj(z∗ +zj(m)ej)! km j
that is λ (Ce ) = +∞ and L ∈/ Qn.
2,j j
4 Estimates of growth of entire functions
By Kn we denote a class of positive continuous functions L(z) for which there exists c > 0
such that for every R ∈ Rn and j ∈ {1,...,n}
+
l (ReiΘ2)
j
max ≤ c.
Θ1,Θ2∈[0,2π]n lj(ReiΘ1)
5
If L(z) = (l (|z |,...,|z |),...,l (|z |,...,|z |)) then L ∈ Kn. It is easy to prove that |ez|+1 ∈
1 1 n n 1 n
Q1 \K1, but ee|z| ∈ K1 \Q1, z ∈ C. Besides, if L ,L ∈ Kn then L +L ∈ Kn and L L ∈ Kn.
1 2 1 2 1 2
For simplicity, let us to write K ≡ K1 and M(F,R) = max{|F(z)|: z ∈ Tn(0,R)}.
Theorem 2. Let L ∈ Qn ∩Kn. If an entire function F has bounded L-index in joint variables,
then
n rj
lnM(F,R) = O min min l (R(j,σ ,t)eiΘ)dt as kRk → ∞, (5)
j n
σn∈SnΘ∈[0,2π]n j=1 Z0 !
X
r0, if σ (k) < j,
k n
where σ is a permutation of {1,...,n}, R(j,σ ,t) = (r′,...,r′), r′ = t, if k = j,
n n 1 n k 



r , if σ (k) > j,
k n
k ∈ {1,...,n}, R0 = (r10,...,rn0) is sufficiently large radius, Sn is a set ofall permutations of

{1,...,n}.
Proof. Let R > 0, Θ ∈ [0,2π]n and a point z∗ ∈ Tn(0,R+ 2 ) be a such that
L(ReiΘ)
2
|F(z∗)| = max |F(z)| : z ∈ Tn 0,R+ .
L(ReiΘ)
(cid:26) (cid:18) (cid:19)(cid:27)
Denote z0 = z∗R . Then
R+2/L(ReiΘ)
z∗R z∗2/L(ReiΘ) 2
|z0 −z∗| = −z∗ = = and
R+2/L(ReiΘ) R+2/L(ReiΘ) L(ReiΘ)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) z∗R ((cid:12)R+(cid:12)2/L(ReiΘ))eiarg(cid:12)z∗R
L(z0) = L (cid:12) = L (cid:12) (cid:12) (cid:12) = L(Reiargz∗).
(cid:12) (cid:12) (cid:12) (cid:12)
R+2/L(ReiΘ) R+2/L(ReiΘ)
(cid:18) (cid:19) (cid:18) (cid:19)
Since L ∈ Kn we havethat cL(z0) = cL(Reiargz∗) ≥ L(ReiΘ) ≥ 1L(z0).Weconsider two skeletons
c
Tn(z0, 1 ) and Tn(z0, 2 ). By Theorem 1 there exists p = p (1,c2) ≥ 1 such that (4) holds
L(z0) L(z0) 1 1 c
with R′ = 1, R′′ = c2, i.e.
c
2 2
max |F(z)|: z∈Tn 0,R+ =|F(z∗)|≤max |F(z)|: z∈Tn z0, ≤
L(ReiΘ) L(ReiΘ)
(cid:26) (cid:18) (cid:19)(cid:27) (cid:26) (cid:18) (cid:19)(cid:27)
c2 1
≤ max |F(z)|: z ∈ Tn z0, ≤ p max |F(z)|: z ∈ Tn z0, ≤
L(z0) 1 cL(z0)
(cid:26) (cid:18) (cid:19)(cid:27) (cid:26) (cid:18) (cid:19)(cid:27)
e
≤ p max |F(z)| : z ∈ Tn 0,R+ (6)
1 L(ReiΘ)
(cid:26) (cid:18) (cid:19)(cid:27)
A function ln+max{|F(z)|: z ∈ Tn(0,R)} is a convex function of the variables lnr , ..., lnr
1 n
(see [14], p. 138 in Russian edition or p. 84 in English translation). Hence, the function admits a
representation
ln+max{|F(z)|: z ∈ Tn(0,R)}−ln+max{|F(z)|: z ∈ Tn(0,R+(r0 −r )e )} =
j j j
6
rj A (r ,...,r ,t,r ,...,r )
j 1 j−1 j+1 n
= dt (7)
t
Zrj0
for arbitrary 0 < r0 ≤ r < +∞, where the function A (r ,...,r ,t,r ,...,r ) is a positive
j j j 1 j−1 j+1 n
non-decreasing in variable t ∈ (0;+∞), j ∈ {1,...,n}.
Using (6) we deduce
2
lnp ≥ lnmax |F(z)|: z ∈ Tn 0,R+ −
1 L(ReiΘ)
(cid:26) (cid:18) (cid:19)(cid:27)
e
−lnmax |F(z)| : z ∈ Tn 0,R+ =
L(ReiΘ)
(cid:26) (cid:18) (cid:19)(cid:27)
n 1+ n e
= lnmax |F(z)|: z ∈ Tn 0,R+ k=j k −
L(ReiΘ)
( P !)
j=1
X
1+ n e
−lnmax |F(z)|: z ∈ Tn 0,R+ k=j+1 k =
L(ReiΘ)
( P !)
n rj+2/lj(ReiΘ) 1 1 1 2
= A r + ,...,r + ,t,r + ,...,
t j 1 l (ReiΘ) j−1 l (ReiΘ) j+1 l (ReiΘ)
j=1 Zrj+1/lj(ReiΘ) (cid:18) 1 j−1 1
X
n
2 1 1 1
r + dt≥ ln 1+ A r + ,...,r + ,r ,
n l (ReiΘ) r l (ReiΘ)+1 j 1 l (ReiΘ) j−1 l (ReiΘ) j
n (cid:19) j=1 (cid:18) j j (cid:19) (cid:18) 1 j−1
X
2 2
r + ,...,r + (8)
j+1 l (ReiΘ) n l (ReiΘ)
1 n (cid:19)
By Lemma 1 the function r l (ReiΘ) → +∞ (r → +∞). Hence, for j ∈ {1,...,n} and r ≥ r0
j j j i i
1 1 1
ln 1+ ∼ ≥ .
r l (ReiΘ)+1 r l (ReiΘ)+1 2r l (ReiΘ)
(cid:18) j i (cid:19) j j j j
Thus, for every j ∈ {1,...,n} inequality (8) implies that
1 1 2 2
A r + ,...,r + ,r ,r + ,...,r + ≤
j 1 l (ReiΘ) i−1 l (ReiΘ) j j+1 l (ReiΘ) n l (ReiΘ)
(cid:18) 1 j−1 i+1 n (cid:19)
≤ 2lnp r l (ReiΘ).
1 j j
Let R0 = (r0,...,r0), where every r0 is above chosen. Applying (7) n-th times consequently we
1 n j
obtain
lnmax{|F(z)|: z ∈ Tn(0,R)} = lnmax{|F(z)|: z ∈ Tn(0,R+(r0 −r )e )}+
1 1 1
r1 A (t,r ,...,r )
+ 1 2 n dt = lnmax{|F(z)|: z ∈ Tn(0,R+(r0 −r )e +(r0 −r )e )}+
t 1 1 1 2 2 2
Zr10
r1 A (t,r ,...,r ) r2 A (r0,t,r ...,r )
+ 1 2 n dt+ 2 1 3 n dt = lnmax{|F(z)|: z ∈ Tn(0,R0)}+
t t
Zr10 Zr20
n rj A (r0,...,r0 ,t,r ,...,r )
+ j 1 j−1 j+1 n dt ≤ lnmax{|F(z)|: z ∈ Tn(0,R0)}+
t
j=1 Zrj0
X
7
n rj
+2lnp l (r0eiθ1,...,r0 eiθj−1,teiθj,r eiθj+1,...,r eiθn)dt ≤
1 j 1 j−1 j+1 n
j=1 Zrj0
X
≤ lnmax{|F(z)|: z ∈ Tn(0,R0)}+
n rj
+2lnp l (r0eiθ1,...,r0 eiθj−1,teiθj,r eiθj+1,...,r eiθn)dt ≤
1 j 1 j−1 j+1 n
j=1 Z0
X
n rj
≤ (1+o(1))2lnp l (r0eiθ1,...,r0 eiθj−1,teiθj,r eiθj+1,...,r eiθn)dt.
1 j 1 j−1 j+1 n
j=1 Z0
X
The function lnmax{|F(z)|: z ∈ Tn(0,R)} is independent of Θ. Thus, the following estimate
holds
lnmax{|F(z)|: z ∈ Tn(0,R)} =
n rj
=O min l (r0eiθ1,...,r0 eiθj−1,teiθj,r eiθj+1,...,r eiθn)dt , as kRk → +∞.
j 1 j−1 j+1 n
Θ∈[0,2π]n j=1 Z0 !
X
Itisobviouslythatsimilarequalitycanbeprovedforarbitrarypermutationσ oftheset{1,2,...,n}.
n
Thus, estimate (5) holds. Theorem 2 is proved.
Corollary 1. If L ∈ Qn ∩ Kn, min l (ReiΘ) is non-decreasing in each variable r , k ∈
j k
Θ∈[0,2π]n
{1,...,n}, entire function F has bounded L-index in joint variables then
n rj
lnmax{|F(z)|: z ∈ Tn(0,R)} = O min l (R(j)eiΘ)dt as kRk → ∞,
j
Θ∈[0,2π]n j=1 Z0 !
X
where R(j) = (r ,...,r ,t,r ,...,r ).
1 j−1 j+1 n
Note that Theorem 2 is new too for n = 1 because we replace the condition l = l(|z|) by the
condition l ∈ K, i.e. there exists c > 0 such that for every r > 0 max l(reiθ2) ≤ c. Particularly,
θ1,θ2∈[0,2π] l(reiθ1)
the following proposition is valid.
Corollary 2. If l ∈ Q∩K and an entire in C function f has bounded l-index then
r
lnmax{|f(z)|: |z| = r} = O min l(teiΘ)dt as r → ∞.
(cid:18)θ∈[0,2π]Z0 (cid:19)
W. K. Hayman, A. D. Kuzyk, M M. Sheremeta, V. O. Kushnir and T. O. Banakh [4, 1, 6]
improved an estimate (5) by other conditions on the function l for a case n = 1. M. T. Bordulyak
and M. M. Sheremeta [7] deduced similar results for entire functions of bounded L-index in joint
variables, if l = l (|z |), j ∈ {1,...,n}. Using their method we will generalise the estimate for
j j j
l : Cn → R .
j +
Let us to denote a+ = max{a,0}, u (t) = u (t,R,Θ) = l (tReiΘ), where a ∈ R, t ∈ R ,
j j j rm +
j ∈ {1,...,n}, r 6= 0.
m
8
Theorem 3. Let L(ReiΘ) be a positive continuously differentiable function in each variable r ∈
k
[0,+∞), k ∈ {1,...,n}, Θ ∈ [0,2π]n. If an entire function F has bounded L-index N = N(F,L)
in joint variables then for every Θ ∈ [0,2π]n and for every R ∈ Rn (r 6= 0) and S ∈ Zn
+ m +
|F(K)(ReiΘ)| |F(K)(0)|
lnmax : kKk ≤ N ≤ lnmax : kKk ≤ N +
K!LK(ReiΘ) K!LK(0)
(cid:26) (cid:27) (cid:26) (cid:27)
rm n r τ n k (−u′(τ))+
+ max j (k +1)l ReiΘ + max j j dτ, (9)
j j
Z0 kKk≤N(Xj=1 rm (cid:18)rm (cid:19)) kKk≤NXj=1 lj rτmReiΘ 
 (cid:16) (cid:17) 
If, in addition, there exists C > 0 such that the function Lsatisfies inequalities
(−(u (t,R,Θ))′)+
sup max max max j t ≤ C, (10)
R∈Rn+t∈[0,rm]Θ∈[0,2π]n1≤j≤n rrmj lj2(rmt ReiΘ)
rm n r τ
max j l ReiΘ dτ → +∞ as kRk → +∞ (11)
j
Θ∈[0,2π]nZ0 j=1 rm (cid:18)rm (cid:19)
X
then
lnmax{|F(z): z ∈ Tn(0,R)}
lim ≤ (C +1)N +1. (12)
kRk→+∞ max rm n rj l τ ReiΘ dτ
Θ∈[0,2π]n 0 j=1 rm j rm
(cid:16) (cid:17)
R P
And if r (−(u (t,R,Θ))′)+/(r l2( t ReiΘ)) → 0 and (11) holds as kRk → +∞ uniformly
m j t j j rm
for all Θ ∈ [0,2π]n, j ∈ {1,...,n}, t ∈ [0,r ] then
m
lnmax{|F(z): z ∈ Tn(0,R)}
lim ≤ N +1. (13)
kRk→+∞ max rm n rj l τ ReiΘ dτ
Θ∈[0,2π]n 0 j=1 rm j rm
(cid:16) (cid:17)
R P
Proof. Let R ∈ R \ {(cid:3)}, Θ ∈ [0,2π]n. Then there exists at least one r 6= 0. Denote α = rj ,
m j rm
j ∈ {1,...,n} and A = (α ,...,α ). We consider a function
1 n
|F(K)(AteiΘ)|
g(t) = max : kKk ≤ N , (14)
K!LK(AteiΘ)
(cid:26) (cid:27)
where At = (α t,...,α t), AteiΘ = (α teiθ1,...,α teiθn).
1 n 1 n
Since the function |F(K)(AteiΘ)| is continuously differentiable by real t ∈ [0,+∞), outside the
K!LK(AteiΘ)
zero set of function |F(K)(AteiΘ)|, the function g(t) is a continuously differentiable function on
[0,+∞), except, perhaps, for a countable set of points.
Therefore, using the inequality d |g(r)| ≤ |g′(r)| which holds except for the points r = t such
dr
that g(t) = 0, we deduce
d |F(K)(AteiΘ)| 1 d d 1
= |F(K)(AteiΘ)|+|F(K)(AteiΘ)| ≤
dt K!LK(AteiΘ) K!LK(AteiΘ)dt dtK!LK(AteiΘ)
(cid:18) (cid:19)
1 n |F(K)(AteiΘ)| n k u′(t)
≤ F(K+ej)(AteiΘ)α eiθj − j j ≤
K!LK(AteiΘ) j K!LK(AteiΘ) l (AteiΘ)
(cid:12) (cid:12) j
(cid:12)Xj=1 (cid:12) Xj=1
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
9
≤ n |F(K+ej)(AteiΘ)| α (k +1)l (AteiΘ)+ |F(K)(AteiΘ)| n kj(−u′j(t))+ (15)
(K +ej)!LK+ej(AteiΘ) j j j K!LK(AteiΘ) lj(AteiΘ)
j=1 j=1
X X
For absolutely continuous functions h , h , ..., h and h(x) := max{h (z) : 1 ≤ j ≤ k}, h′(x) ≤
1 2 k j
max{h′(x) : 1 ≤ j ≤ k}, x ∈ [a,b] (see [5, Lemma 4.1, p. 81]). The function g is absolutely
j
continuous, therefore, from (15) it follows that
d |F(K)(AteiΘ)|
g′(t) ≤ max : kKk ≤ N ≤
dt K!LK(AteiΘ)
(cid:26) (cid:18) (cid:19) (cid:27)
n αj(kj +1)lj(AteiΘ)|F(K+ej)(AteiΘ)| |F(K)(AteiΘ)| n kj(−u′j(t))+
≤ max + ≤
kKk≤N( (K +ej)!LK+ej(AteiΘ) K!LK(AteiΘ) lj(AteiΘ) )
j=1 j=1
X X
n n k (−u′(t))+
≤ g(t) max α (k +1)l (AteiΘ) + max j j =
j j j
kKk≤N( ) kKk≤N( lj(AteiΘ) )!
j=1 j=1
X X
= g(t)(β(t)+γ(t)),
whereβ(t) = max n α (k +1)l (AteiΘ) ,γ(t) = max n kj(−u′j(t))+ .Thus,
kKk≤N j=1 j j j kKk≤N j=1 lj(AteiΘ)
d lng(t) ≤ β(t)+γ(t) annd o n o
dt P P
t
g(t) ≤ g(0)exp (β(τ)+γ(τ))dτ, (16)
Z0
because g(0) 6= 0. But r A = R. Substituting t = r in (16) and taking into account (14), we
m m
deduce
|F(K)(ReiΘ)| |F(K)(0)|
lnmax : kKk ≤ N ≤ lnmax : kKk ≤ N +
K!LK(ReiΘ) K!LK(0)
(cid:26) (cid:27) (cid:26) (cid:27)
rm n n k (−u′(τ))+
+ max α (k +1)l (AτeiΘ) + max j j dτ,
Z0 kKk≤N(j=1 j j j ) kKk≤N(j=1 lj(AτeiΘ) )!
X X
i.e. (9) is proved. Denote β(t) = n α l (AteiΘ). If, in addition, (10)-(11) hold then for some
j=1 j j
K∗, kK∗k ≤ N and K, kKk ≤ N,
P
e
γ(t) = e nj=e1 kj∗lj(−(Aut′je(itΘ)))+ ≤ n k∗ (−u′j(t))+ ≤ n k∗ ·C ≤ NC and
β(t) Pnj=1αjlj(AteiΘ) j=1 jαjlj2(AteiΘ) j=1 j
X X
β(t) n Pα (k˜ +1)l (AteiΘ) n α k˜ l (AteiΘ) n
=e j=1 j j j = 1+ j=1 j j j ≤ 1+ k˜ ≤ 1+N.
β(t) P nj=1αjlj(AteiΘ) P nj=1αjlj(AteiΘ) j=1 j
X
P P
Buet |F(AteiΘ)| ≤ g(t) ≤ g(0)exp t(β(τ)+γ(τ))dτ and r A = R. Then we put t = r and
0 m m
obtain
R
lnmax{|F(z): z ∈ Tn(0,R)} = ln max |F(ReiΘ)| ≤ ln max g(r ) ≤
m
Θ∈[0,2π]n Θ∈[0,2π]n
rm
≤ lng(0)+ max (β(τ)+γ(τ))dτ ≤
Θ∈[0,2π]nZ0
10
rm
≤ lng(0)+(NC +N +1) max β(τ)dτ =
Θ∈[0,2π]nZ0
rm n e
= lng(0)+(NC +N +1) max α l (AτeiΘ)dτ =
j j
Θ∈[0,2π]nZ0 j=1
X
rm n r τ
= lng(0)+(NC +N +1) max j l ReiΘ dτ.
j
Θ∈[0,2π]nZ0 j=1 rm (cid:18)rm (cid:19)
X
Thus, we conclude that (12) holds. Estimate (13) can be deduced by analogy. Theorem 3 is
proved.
We will write u(r,θ) = l(reiθ). Theorem 3 implies the following proposition for n = 1.
Corollary 3. Let l(reiθ) be a positive continuously differentiable function in variable r ∈ [0,+∞)
for every θ ∈ [0,2π]. If an entire function f has bounded l-index N = N(f,l) and there exists
C > 0 such that lim max (−u′r(r,θ))+ = C then
l2(reiΘ)
r→+∞θ∈[0,2π]
lnmax{|f(z): |z| = r}
lim ≤ (C +1)N +1.
r
r→+∞ max l(τeiθ)dτ
0
θ∈[0,2π]
R
Remark 1. Our result is sharper than known result of Sheremeta which is obtained in a case
n = 1, C 6= 0 and l = l(|z|). Indeed, corresponding theorem [5, p. 83] claims that
lnmax{|f(z): |z| = r}
lim ≤ (C +1)(N +1).
r
r→+∞ l(τ)dτ
0
Obviously, that NC +N +1 < (C +1R)(N +1) for C 6= 0 and N 6= 0.
Estimate (13) is sharp. It is easy to check for function F(z ,z ) = exp(z z ), l (z ,z ) =
1 2 1 2 1 1 2
|z |+1, l (z ,z ) = |z |+1. Then N(F,L) = 0 and lnmax{|F(z)|: z ∈ T2(0,R)} = r r .
2 2 1 2 1 1 2
References
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Notes, 39 (1986) (1), 3–8.
[2] B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions
of bounded index, Proc. Sympos. Pure Math., Amer. Math. Soc.: Providence, Rhode Island,
2 (1968), 298–307.
[3] W. J. Pugh and S. M. Shah, On the growth of entire functions of bounded index, Pacific J.
Math. 33 (1970) (1), 191–201.
[4] W. K. Hayman, Differential inequalities and local valency, Pacific J. Math. 44 (1973) (1),
117–137.