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Approximation of Entire Function of Slow Growth in Several
Complex Variables
Mushtaq Shakir A. Hussein
(1, 2 )
(1)
and Aiman Abdali Jaffar
( 2)
Department of Mathematics, College of science, Mustansiriha University, Baghdad, Iraq.
Abstract
In the present paper, we study the polynomial approximation of entire function in Banach space ( B( p, q, k )
space, Hardy space and Bergman space). The coefficient characterizations of generalized type of entire function
of slow growth in several complex variables have been obtained in terms of the approximation errors.
Keyword: Entire function, generalized order, generalized type, approximation error.
1.Introduction
Let f  z  

a
n 0
n
z n be an entire function and M r , f   max f z  be its maximum modulus.
z r
The growth of f z  is measured in terms of its order ρ and type τ defined as under
ln ln M (r , f )

r
ln ln M (r , f )
lim sup

r 
r
lim sup
(1.1)
r 
(1.2)
for 0 < ρ < ∞. Various workers have given different characterizations for entire function of fast growth
(ρ = ∞). M. N. Seremeta [6] defined the generalized order and generalized type with the help of
general functions as follows.
Let L0 denoted the class of functions h satisfying the following conditions
(i) hx  is defined on [a,∞) and is positive, strictly increasing, differentiable and tend to ∞ as x   ,
(ii)
h1  1  x x
 1,
h x 
for every function  x  such that  x    as x   .
lim
x
Let Λ denoted the class of function h satisfying condition (i) and
hcx 
1
x h x 
lim
for every c > 0, that is, h(x) is slowly increasing.
For the entire function f z  and function  x    ,  x   L0 , the generalized order of an entire
function in the terms of maximum modulus is defined as
  ,  , f   lim sup
r 
 ln M r , f 
.
 ln r 
(1.3)
Further, for  x  L0 ,  1 x  L0 ,  x  L0 , generalized type of an entire function f of finite
generalized order ρ is defined as
  ,  , f   lim sup
r 
152
 ln M r , f 
.

   r  


(1.4)
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where 0 < ρ < ∞ is a fixed number.
Above relation were obtained under certain conditions which do not hold if    . To overcome
this difficulty, G. P. Kapoor and Nautiyal [4] defined generalized order  ( ; f ) of slow growth with
the help of general functions as follows
Let Ω be the class of functions hx  satisfying (i) and
(iv) there exists a  x  and x0 , K1 and K 2 such that
0  K1 
d (h( x))
 K 2   for all x  x0 .
d ( (log x))
Let  be the class of functions h(x) satisfying (i) and
(v)
d (h( x))
 K , 0 < K < ∞.
d (log x)
lim
x
Kapoor and Nautiyal [4] showed that class Ω and  are contained in Λ. Further,      and
they defined the generalized order   ; f  for entire function f z  of slow growth as
  ; f   lim sup
r 
where  x  either belongs to Ω or to  .
 (ln M (r , f ))
,
 (ln r )
Let f ( z1 , z 2 ,, z n ) be an entire function, z  ( z1 , z 2 ,, z n )  C n . Let G be a full region in Rn
(positive hyper octant). Let GR  C n denoted the region obtained from G by a similarity
transformation about the origin, with ratio of similitude R. let d t (G) sup zG z
z  z1
t
t1
t
t
, where
t
z 2 2  z n n , and let G denoted the boundary of the region G. Let
f ( z )  f ( z1 , z 2 , , z n ) 
t  t1  t 2    t n ,
be

 at1tn z1t1  zntn 
t1 ,t2 ,,tn 0
the
power

a
t 0
series
t
zt ,
expansion
of
the
function
f (z ) .
Let
M G ( R, f )  max f ( z ) . To characterize the growth of f, order (  G ) and type ( G ) of f are defined
z  GR
as [2]
ln ln M G ( R, f )
,
R
ln R
ln M G ( R, f )
 G  lim sup
.
R
R G
 G  lim sup
For the entire function, f ( z ) 

a
t 0
t
z t , A. A. Gol'dberg [3, Th .1] obtained the order and type in
terms of the coefficients of its Taylor expansion as
 G  lim

(e  G  G )1 G  lim sup t
t 
where d t (G)  max r t ;
r G
r t  r1t1 r2t2  rntn .
153
t 
1 G
t ln t
 ln at
.
1 t
[ at d t (G)]
(1.5)
 , (0 < 
G
< ∞)
(1.6)
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for an entire function of several complex variables f z  

a
t
t 0
z t , and functions  x   ,
 x  L , Seremeta [6, Th .1] proved that
( t )
 [ln M G ( R, f )]
.
  lim sup
 lim sup
R 
k 
1
 (ln R)
 [ ln  at d t (G ) ]
0
(1.7)
t
Further, for  x  L0 ,  1 x  L0 ,  x  L0 , Seremeta [6, Th .1] proved that
t
  lim sup
R 
( )

 [ln M G ( R, f )]
.
 lim sup
t 
 [( ( R))  ]
 [( {e1  [ at d t (G )]1 t })  ]
(1.8)
where 0 < ρ < ∞ is a fixed number.
And let H q denote the Bergman space of functions f z  satisfying the condition
1
q
1

f H '   
f z  d 1d 2  d n    ,
q
z

G
A

where z  ( z1 , z2 , , zn ) , d j  dx j dy j , z j  x j  iy j , j  1,2, , n. and A is the area of G.
For q = ∞, let f
H 
 f
H
 sup{ f z  , z U } . Then H q and H q are Banach space for q ≥ 1. In
analogy with spaces of functions of one variable, we call H q and H q the Hardy and Bergman spaces
respectively.
The function f z  analytic in U belong to the space B(p, q, k), where 0  p  q   , and
0  k   , if
1k
f
p ,q ,k
1

  (1  R) k (1 p1 q ) 1 M qk,G ( R, f ) dR 
0

.
And
f
p ,q ,
 sup{(1  R)1 p1 q M q,G ( R, f ) ; 0  R  1}   .
It is known [1] that B(p, q, k) is a Banach space for p > 0 and q, k  1 , otherwise it is a Freachet
space. Further , we have
Let Pm  { p : p 
a
t m
t
q
H q  H q  B( , q, q), 1  q   .
2
z t } be the class polynomials of degree at most m and let X denote one of the
Banach spaces defined. Then we defined error of an entire function f on the region G as
E t ( f )  E t ( f , G, X )  inf{ f  p X : p  Pm } .
Vakarchuk and Zhir [5] obtained the characterizations of generalized order and generalized type of
f z  in terms of the errors E t ( f ) defined above. These characterizations do not hold good when α =
β = γ. i.e. for entire functions of slow growth. In this paper we have tried to fill this gap. We define the
generalized type  ( ; f ) of an entire function f z  having finite generalized order as
 ( ; f )  lim sup
r 
Where  x  either belongs to Ω or to  .
 (ln M G ( R, f ))
.
[ (ln R)]
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2. Main Results
Theorem 2.1: Let  x    , then the entire function f z  of generalized order ρ, 1 < ρ < ∞, is of
generalized type τ if and only if

 
 (ln M G ( R, f ))
  lim sup

lim
sup
,
1 t
R
t 
[ (ln R)]
{ [ 1 ln  at d t (G )  ]} 1
t
(2.1)
provided dF ( x ; ,  ) d ln x  O(1) as x   for all τ, 0 < τ < ∞.
Proof. Let
 (ln M G ( R, f ))
 .
[ (ln R)]
We suppose τ < ∞. Then for every ε > 0,  R  
 (ln M G ( R, f ))
      , R  R  .
[ (ln R)]
(or)
ln M G ( R, f )  ( 1{ [ (ln R)] }) .
Choose R  Rt  to be the unique root of the equation

1
t
F [ln R ;  , ] .
ln R

lim sup
R
(2.2)
Then
t
t 1
1
ln R   1[(  ( ))1 (  1) ]  F [ ; ,   1] .


 
(2.3)
By Cauchy's inequality,
at d t (G )  R
t
M G ( R, f )
 exp{ t ln R  ( 1{ [ (ln R)] })}
By using (2.5) and (2.6), we get
at d t (G)  exp{ t F 
t

F}
or

 1
ln( at d t (G))
1 t
t
1
  1{[(  ( ))1 (  1) ]}


or
(  )
t
   
{ [ 1 ln( at d t (G ))
1 t
]} 1
.
Now proceeding to limits, we obtain
(  )
t
  lim sup
t 
{ [ 1 ln( at d t (G ))
1 t
Inequality (2.4) obviously holds when τ = ∞.
Conversely, let
(  )
t
lim sup
t 
{ [ 1 ln  at d t (G ) 
1 t
]} 1
 .
Suppose σ < ∞. Then for every ε > 0 and for all t  N   , we have
155
]} 1
.
(2.4)
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(  )
t
{ [ 1 ln  at d t (G ) 
1 t
at d t (G) 
]} 1
  
1
exp{(   1)
at d t (G) R
 Re
t
t

The inequality
t
F [  ; 1 ,   1]
t

 F [
 1

t
; 1 ,  1]
(2.5)
1
2
(2.6)
 1.
(2.7)

 
.
is fulfilled beginning with some t  m(R) . Then

 at dt (G) R 

t m ( R ) 1
1

t
t m ( R ) 1
2
t
We now express m® in terms of R. From inequality (2.6),
2 R  exp{(
t 1
 1
) F [ ; ,   1]} ,

 
m( R)  E[  1{ ( (ln R  ln 2))  1}] .
 x   R x exp{( 1 ) x F[ x ; 1 ,   1]}. Let
we
can
take
We
consider
the
dF[ x ; 1 ,   1]
 x 
 1
x 1
 ln R  (
) F[ ; ,   1] 
 0.
 x 

 
d ln x
As x   , by the assumption of the theorem, for finite  (0    ) ,
dF[ x ;  ,   1] d ln x is bounded. So there is a A > 0 such that for x  x1 we have
dF [ x ; 1 ,   1]
 A.
d ln x
function
(2.8)
(2.9)
A  ln 2 . It is then obvious that inequalities (2.6) and (2.7) hold for
t  m1 ( R)  E[  1{ ( (ln R  ln 2))(  1) }]  1 .
m0
We
let
designate
the
We can take
number max( N ( ) , E[ x1 ]  1) . For R  R1 (m0 ) we have  (m0 )  (m0 )  0 . From (2.9) and (2.8)
it follows that  (m1 ( R))  (m1 ( R))  0 . We hence obtain that if for R  R1 (m0 ) we let x * ( R)
designate the point where  ( x* ( R)) 
max
m0  x  m1 ( R )
 ( x) , then
m0  x* ( R)  m1 ( R) and x* ( R)   1{ ( (ln R  a( R)))  1} .
Where
 A  a( R) 
dF [ x ; 1 ,   1]
d ln x
x  x* ( R )
 A.
Further
max
m0  t  m1 ( R )
( at d t (G ) R 
t
max
 ( x) 
R 
1
{ ( (ln R  a ( R )))  1 }
 1
1
e  { ( (ln Ra ( R ))) } (ln Ra ( R ))
 exp{a( R)  1{ ( (ln R  a( R)))  1}} 
m0  x  m1 ( R )
 exp{ A 1{ ( (ln R  A))  1}}.
It is obvious that (for R  R1 (m0 ) )
M G ( R, f ) 

a
t 0
t
156
d t (G ) R
t


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a

t 0
t
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d t (G) R 
t
m1 ( R )


a
at d t (G) R 
t
t
t  m1 ( R ) 1
t m0 1
 O( R )  m1 ( R)
m0
d t (G) R
t
( at d t (G) R )  1
t
max
m0  t  m1 ( R )
M G ( R, f )(1  o(1))  exp{( A  o(1)) 1[ ( (ln R  A))  1 ]}
 (ln M G ( R, f ))   [ (ln R  A)] 1   [ (ln R  A)] .
We then have
 [( A  o(1)) 1 ln M G ( R, f )]
   .
[ (ln R  A)]
Since  (x)     , now proceeding to limits we obtain
lim sup
R
 (ln M G ( R, f ))
 .
[ (ln R)]
(2.10)
From inequality (2.4) and (2.10), we get the required the result.
Now we prove
Theorem 2.2: Let  x    , then a necessary and sufficient condition for an entire function
f z   B( p, q, k ) to be of generalized type τ having finite generalized order ρ, 1 < ρ < ∞ is
(  )
t
  lim sup
t 
[{ 1 ln(( E t ( B( p, q, k )) d t (G ))
1 t
)}](  1)
.
(2.11)
Proof. First we consider the space B( p, q, k ) , q = 2, 0 < ρ < ∞ and k ≥ 1. Let f ( z)  B( p, q, k ) be
of generalized type τ with generalized order ρ. Then from the Theorem 2.1, we have
(  )
t
lim sup
t 
{ [ 1 ln( at d t (G ))
1 t
]} 1
 .
(2.12)
.
(2.13)
For a given ε > 0, and all t  m  m( ) , we have
at d t (G) 
Let g t ( f , z ) 
1
exp{(   1)
F [  ; 1 ,   1]}
t
t

t
a
j 0
j
z j be the t th partial sum of the Taylor series of the function f (z ) . Following
[5, p.1396], we get
E t ( B( p,2, k ); f )  B1 k [( t  1)k  1; k (1 p  1 2)]{

(a
j  t 1
j
d j (G)) 2 }1 2 (2.14)
where B(a, b) (a, b  0) denotes the beta function. By using (2.13), we have
E t ( B( p,2, k ); f ) 
B1 k [( t  1)k  1; k (1 p  1 2)]
exp{(   1)
t 1

F[
t 1

;  ,   1]}
1

{
j  t 1
where
 j ( ) 
exp{
t 1

(   1)[ 1{(
exp{  (   1)[ {(
1
j
Set
157
(
t 1

 
(
j

 
) 1 (  1)
)
) 1 (  1)
)
}]}
}]}

.
2
j
( )}1 2 ,
(2.15)
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 ( )  exp{ 
(   1)

 ( 1 ) 1 (  1)
[ {(
)
}]} .
 
1
Since  (x) is increasing and j  t  1 , we get
 j ( )  exp{
 ( t 1 1 (  1)
(   1)[ {(
)
}]}  
 
(( t  1)  j
j ( t 1)
1

( ) .
(2.16)
Since  ( )  1 , we get from (2.15) and (2.16),
B1 k [( t  1)k  1; k (1 p  1 2)]
E t ( B( p,2, k ); f ) 
(1   ( )) [exp{(   1)
2
12
t 1

[ {(
1
(
t 1

 
) 1 (  1)
)
. (2.17)
}]}]
For t  m , (2.17) yields
  
 ( t 1 )
{ ( (1 1 )(  1) {ln(( E t d t )
1 t
)  ln(
B1 k [( t 1) k 1;k (1 p 1 2 )] 1 t
t
})}(  1)
)
(1 2 ( ))1 2
Now
B[( t  1)k  1; k (1 p  1 2)] 
(( t  1)k  1)(k (1 p  1 2))
(( t  1 2  1 p)k  1)
.
Hence
B[( t  1)k  1; k (1 p  1 2)] ~

e
e
[( t 1) k 1]
( t 1) k 3 2
[( t  1)k  1]
[( t 1 21 p ) k 1]
(1 p  1 2)
[( t  1 2  1 p)k  1]
( t 1 2 1 p ) k  3 2
.
Thus
1 ( t 1)
{B[( t  1)k  1; k (1 p  1 2)]}
 1.
(2.18)
)}](  1)
(2.19)
Now proceeding to limits, we obtain
(  )
t
  lim sup
t 
[{ 1 ln(( E t d t (G ))
1 t
For reverse inequality, by [5, p.1398], we have
at 1 B1 k [( t  1)k  1; k (1 p  1 2)]  E t ( B( p,2, k ); f ) .
(2.20)
Then for sufficiently large t , we have
(  )
t
[{ 1 ln(( E t d t (G ))
1 t
)}](  1)

 ( t )
[{ 1 {ln(( at 1 d t (G))

1 t
)  ln( B
 t k
[( t  1)k  1; (1 p  1 2)])}}](  1)
 ( t )
[ { 1 {ln(( at d t (G ))
1 t
)  ln( B
 t k
[( t  1)k  1; (1 p  1 2)])}}] (  1)
.
By applying limits and from (2.12), we obtain
(  )
t
lim sup
t 
[{ 1 ln(( E t d t (G ))
158
1 t
)}](  1)
 .
(2.21)
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From (2.19) and (2.21), we obtain the required relation
(  )
t
lim sup
t 
[{ 1 ln(( E t d t (G))
1 t
)}](  1)
 .
(2.22)
In the second step, we consider the spaces B( p, q, k ) for 0  p  q, q  2, and q, k  1. Gvaradze
[1] showed that, for p  p1 , q  q1 and k  k1 , if at least one of the inequalities is strict, then the
strict inclusion B( p, q, k )  B( p1 , q1 , k1 ) holds and the following relation is true:
f
p1 ,q1 ,k1
 21 q1 q1 [k (1 p  1 q)]1 k 1 k1 f
p ,q ,k
.
For any function f ( z)  B( p, q, k ) , the last relation yields
E t ( B( p1 , q1 , k1 ); f )  21 q1 q1 [k (1 p  1 q)]1 k 1 k1 E t ( B( p, q, k ); f ).
(2.23)
For the general case B( p, q, k ), q  2, we prove the necessity of condition (2.13).
Let f ( z)  B( p, q, k ) be an entire transcendental function having finite generalized order  ( ; f )
whose generalized type is defined by (2.12). Using the relation (2.13), for t  m we estimate the
value of the best polynomial approximation as follows
E t ( B( p, q, k ); f )  f  g t ( f )
1
 (  (1  R) ( k (1 p1 q )1 M qk,G dR)1 k .
p ,q ,k
0
Now
f
q

a
t
zt
q
 ( at r t ) q  (r

a
t 1
k  t 1
k
)q .
Hence
E t ( B( p, q, k ); f ) dt (G)  B [( t  1)k  1; k (1 p  1 2)]

a
1k

k  t 1
B1 k [( t  1)k  1; k (1 p  1 q)]
(1   ( ))[exp{
t 1

(   1)[ {(
1
(
t 1

 
) 1 (  1)
)
k
. (2.24)
}]}]
For t  m , (2.24) yields
  
 ( t 1 )
{ ( (1 1 )(  1) {ln (( E t d t (G ))
1 t
t
)  ln(
B1 k [( t 1) k 1;k (1 p 1 2 )] 1 t
(1 ( ))
)
})}(  1)
Since  ( )  1 , and    , proceeding to limits and using (2.18), we obtain
(  )
t
  lim sup
t 
[ { 1 ln(( E t d t (G))
1 t
)}](  1)
.
For the reverse inequality, let 0  p  q  2 and k , q  1 . By (2.23), where p1  p, q1  2, and
k1  k , and the condition (2.13) is already proved for the space B( p,2, k ) , we get
(  )
t
lim sup
t 
[{ 1 ln(( E t ( B( p, q, k ); f ) d t (G ))
1 t
)}](  1)
(  )
t
 lim sup
t 
[{ 1 ln(( E t ( B( p,2, k ); f ) d t (G ))
Now let 0  p  2  q . Since we have
159
1 t
)}](  1)
 .
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M 2,G ( R, f )  M q,G ( R, f ), 0  R  1,
therefore
E t ( B( p, q, k ); f )  a t 1 B1 k [( t  1)k  1; k (1 p  1 q)].
(2.25)
Then for sufficiently large t , we have
(  )
t
[{ 1 ln(( E t d t (G ))
1 t
)}](  1)

(  )
t
[ { 1 {ln(( a t 1 d t (G ))
1 t
 t k
)  ln( B
[( t  1)k  1; k (1 p  1 q)])}}] (  1)
(  )
t

[{ 1 {ln(( a t d t (G))
1 t
)  ln( B
 t k
[( t  1)k  1; k (1 p  1 q)])}}](  1)
.
By applying limits and from (2.12), we obtain
lim sup
t 
 ( t )
[{ 1 ln(( E t d t (G))
1 t
)}](  1)
 ( t )
 lim sup
[{ 1 ln(( a t d t (G))
t 
1 t
)}](  1)
 .
Now we assume that 2  p  q . Set q1  q, k1  k , and 0  p1  2 in the inequality (2.23), where
p1 is an arbitrary fixed number. Substituting p1 for p in (2.25), we get
E t ( B( p, q, k ); f )  a t 1 B1 k [( t  1)k  1; k (1 p1  1 q)] .
(2.26)
Using (2.26) and applying the same analogy as in the previous case 0  p  2  q , for sufficiently
large t , we have
(  )
t
[{ 1 ln(( E t d t (G ))
1 t
)}](  1)

(  )
t
[{ 1 {ln(( a t 1 d t (G))
1 t
)  ln( B
 t k
[( t  1)k  1; k (1 p1  1 q)])}}](  1)
(  )
t

[{ 1 {ln(( a t d t (G ))
1 t
)  ln( B
 t k
[( t  1)k  1; k (1 p  1 q)])}}](  1)
By applying limits and using (2.12), we obtain
(  )
t
lim sup
t 
[{ 1 ln(( E t d t (G ))
1 t
)}](  1)
 .
From relation (2.19) and (2.21), and the above inequality, we obtain the required relation (2.22).
Theorem 2.3: Assuming that the condition of Theorem 2.2 are satisfied and  ( ) is a positive
number, a necessary and sufficient condition for a function f ( z )  H q to be an entire function of
generalized type  ( ) having finite generalized order  is that
(  )
t
lim sup
t 
[{ 1 ln(( E t ( H q ; f ) d t (G ))
160
1 t
)}](  1)
  ( ).
(2.27)
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Proof.
Let f ( z ) 
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
a
t 0
t
z t be an entire transcendental function having finite generalized order ρ
and generalized type τ. Since
t
lim
t 
at  0
(2.28)
f ( z)  B( p, q, k ) , where 0  p  q   and q, k  1 . From relation (1.8), we get
E t ( B(q 2 , q, q); f )   q E t ( H q ; f ), 1  q   .
(2.29)
where  q is a constant independent of t and f. In the case of Hardy space H  ,
E t ( B( p, , ); f )  E t ( H  ; f ),
1 p  .
(2.30)
Since
(  )
t
 ( ; f )  lim sup
t 
[{ 1 ln(( E t ( H q ; f ) d t (G ))
1 t
)}](  1)
(  )
t
 lim sup
t 
[{ 1 ln(( E t (( B(q 2 , q, q); f ) d t (G ))
1 t
)}](  1)
,
1  q  .
Using estimate (2.30) we prove inequality (2.31) in the case q   .
For the reverse inequality
(2.31)
 ( ; f )   ,
(2.32)
we use the relation (2.13), which is valid for t  m , and estimate from above, the generalized type τ
of an entire transcendental function f (z ) having finite generalized order ρ, as follows. We have
E t ( H q ; f )  f  gt

a

j  t 1

Hq
j

1
[exp{(   1)
t 1

[ {(
1
(
t 1

 
) 1 (  1)
)

j
j  t 1
}]}]
( )
Using (2.16),
E t ( H q ; f )  f  gt

Hq
1
d t (G) (1   ( ))[exp{(   1)
t 1

[ {(
1
1
 (1   ( ))[exp{(  1)
E t ( H q ; f ) d t (G )
(
t 1

 
t 1

) 1 (  1)
)
[ {(
1
.
}]}]
(
t 1

 
) 1 (  1)
)
}]}
This yields
  
(
[{ 1 [ln(( E t ( H q ; f ) d t (G))
t 1

1 t 1
)
)  ln((1   ( ))
1 t 1
)]}](  1)
.
(2.33)
Since  ( )  1 and by applying the properties of the function α, passing to the limit as t   in
(2.33), we obtain inequality (2.32). thus we have finally
161
 ( )   .
(2.34)
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This prove Theorem 2.2.
Remark : An analog of Theorem 2.3 for the Bergman space follows from (1.8) for 1  q   and
from Theorem 2.2 for q = ∞.
References
[1] M.I.Gvaradze, On the class of spaces of analytic functions, Mat. Zametki, (21)(2)(1994), 141150.
[2] B.A.Fuks, Introduction to the theory of analytic functions of several complex variables, Trasl.
of Math. Monographs. Vol.8, Amer. Math. Soc., Providence, R.I., 1963.
[3] A.A.Gol'dberg, Elemntary remarks on the formulas for defining order and type of functions of
several variables, Akad. Nauk Armjan. SSR. Dokl., 29(1959), 145-152.
[4] G.P.Kapoor and A.Nautiyal, Polynomial approximation of an entire function of slow growth,
J.Approx.Theory, (32)(1981), 64-75.
[5] S.B.Vakarchuk and S.I.Zhir, On some problems of polynomial approximation of entire
transcendental functions, Ukrainian Mathematical Journal, (54)(9)(2002), 1393-1401.
[6] M.N.Seremeta, On the connection between the growth of the maximum modulus of an entire
function and the moduli of the coefficients of its power series expansion, Amer. Math. Soc.
Transl., (2)(88)(1970), 291-301.
162
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