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, n .. () , D n (n 2 ) , . ABOUT INTEGRAL REPRESENTATIONS AND BOUNDARY PROPERTIES OF FUNCTIONS OF SEVERAL COMPLEX VARIABLES HOLOMORPHIC IN CIRCULAR DOMAINS N Nelaev A. V. (Moscow)
D n (n 2 ) classes considered by the author are distributed

On holomorphic functions in the limited convex circular domains

some boundary properties holomorphic functions of one complex variable. 10. 1926 [12] , , , , . - ( ) . 1933 .. .. [4]
112


.. -- -10, 2002, .112-113

. . U = {z : z < 1} (, .. ([9], . II, § 5) ­ . E U = { : < 1} ­ , . U f H1 (U ) z U
f ( 2i f (z ) = lim m - E

e ( ) d . z e ( z )

)

m

(*)

(z ) = p(z ) + iq (z ) - U («» ), p(z ) ­ U , E ( ) E: 1, E , E ( ) = 0, CE , q( z ) ­ p (z ) . (1) U. 1958 . .. [1] h , ( ). .. [11], I , h 1 ­ . , , D n (n 2 ) (., ., [5], [6], [7], [8]), D. , D .
113


2. (I)

20 . Rn = ( 1 , K , n ) (n - 1) - = { : 1 + K + n = 1, 1 > 0, K , n > 0} . 1 : 1 = 1 - 2 - L - n . , = { : 1 = 1 - 2 - L - n , ( 2 , K , n ) } , . . , D n, n 2 , ( D ), : 1) ,
n -1

= {0 < 2 < 1, 0 < 3 < 1 - 2 , K ,0 < n < 1 - 2 - L -

}

= { n: =

=1



n

R



( )

, , T , = 1, n ,

(1)

R



( ) (

, = 1, n

)




, det R ( ) 0 , T = { n:

1 = 1, K , n = 1}. 2)
D = . I {z n:


=1



n

r1 ( )z1 + L + rn ( )z n < 1 ,




(2)

r ( ) = R



( )

-1

( ). D, 1 () =1

(2) (1) ­ , D . (1) , , = 2, n , t = e

-it

t1 0 . (1) = ( , t , ) =
114



n

R



( )e

-it

, ,


.. -- -10, 2002, .112-115

u = u ( , t , z ) =

=1

(r1 ( )z
n

n

1

+L+ r

n

( )z

n

)

e

it



(
,
n

= 2, n ) , = 1 .


u = u ( , t , z ) =
=1
n

(r1 ( )z1
=1

+ L + rn ( )z n e

)

it



(3)

= ( , t ) = ^ ^
^ 1



R



( )

e

-it

^ ^ , = ( , t ) =

=1



r ( )e

it



,

= ( , t ) =

L ^ n 0 L 0

^ 1 2 L ^ n 2 1 ^ 2 L n ^ 2

1 ([5], [6]). f (z ) D (0 n - 1) D . k = 0,1, K , z D
f (z ) =

^ 1 n LL ^ n L n 1 ^ L n LL n ^ L n L

^ - 1 L ^ - n

^ 1 t2 L ^ n t2 1 ^ t2 L n ^ tn

1 ^
L

n ^

^ 1 tn LL ^ n L tn . 1 ^ L t2 LL n ^ L tn L

(n - k - 1)! (2 )n i

d d

t

( - u )nk- =1
k -1



n - k -1

f (
k

)

d , f
k -1

(4)



f 0 ( ) f ( ) ,

f k ( ) = (n - k ) f

( )

+



( )

,

k = 1, ,



d = d 2 L d n ,




d t =

2


0

dt 2 L dt n .
0

2

1. .. [3], f k ( ) (4) f k = L(nk-)k , n-1 [ f ] = Ln-1 [Ln-2 K[Ln-k [ f ]]K]
115


2. (I)


L p [ f (z )] = pf +

( L(no,)n

-1

[

f (z ) zj j =1 1) f ] f , L(n-1, n



n

z

j

-1

[f ]

Ln

-1

[f ]

).

3. D . . [13] n- (T ) , n = 2 . 30. f (z ) A(D ) , D , h (D ) ( > 0) ,
lim d
1

2. (4) f (z ) A(D ) C k (D f k (z ) A(D ) C (D ) .

)



d

2 t 0

~ f ( )



d < ,

~ ~ ~ ~ = ( 1 ,K, n ) , =

=1



n

R



( )

e i (

-t

) , = 1, n .

2. , D f (z ) h (D ) , , : 1) ( , t ) ( ) f ( ( , t , u )) f

=1

R1 ( )u
f (z

n

e

-it

,K,

=1



n

Rn ( )u e

-it



u < 1 ; 2)





)



,

..



d





d

2 t 0

~ f ( )

d < .

. (2) (3) , z D , u < 1 , z D u 1 . .. f (z ) D, f ( ( , t , u )) ( , t ) u 116


.. -- -10, 2002, .112-117

u < 1 .
2 0


d



,



~ f ( )









0 < < 1 . , . ([9],

. 15),
lim d
1



d

2 t


0

~ f (

)



d = d





lim 1

2 0

~ f ( )



d d t ,

(5)

1). . 2). . 1) 2) . . ([9], . 89) ( , t )
2

lim

1


0

~ f (

)



d =
2 t

2


0

~ lim f ( 1

)



d ,

(5)
lim d
1



d


0

~ f (

)



d = d





d

2 t 0

~ f ( )



d .

f (z ) h (D ) . .

­ 2) ­ ,

1. 1 , , f k (z ) D f k (z ) h1 (D ) . , f k (z ) h1 (D ) . f (z ) D 1 1 f [m ] (z ) , f [m ] (z ) = f 1 - z1 , K , 1 - z n , m m

{

}

m = 2, 3, K , D, f k[m ] (z ) C (D ) . (4) f [m ] (z )
117


2. (I)

m , (4) f k (z ) h1 (D ) . (4) . 40. , k = n - 1 (4)

f (z ) =

(2 ) i
n

1

d





d

t

=1



f

n -1 ( ) d , -u

(6)

, u. . 3. f (z ) D f n-1 (z ) h1 (D ) . E ( , t , ) ­ , ( , t ) ( , t ) = E ( , t , ) d ( E ( , t , ) ­ E ( , t , ) ). z D
f (z ) = lim
m

=1

(2 )n i

1

d d

t

E ( , t ,



m e ,t ( ) (u ) d ,t ) - u e

f

n -1

( )

(7)

( ), «» , t (u ) = p , t (u ) + iq , t (u ) u < 1 , p ,t (u ) ­ , E ( , t , ) ( , t ) , q , t (u ) ­ ,
p ,t (u ) .

. , 2, , z D , u < 1 ,
z D u 1 , (6) : 1 (8) f (z ) = d f ( ( , t , u )) d t . (2 )n -1 n -1
118


.. -- -10, 2002, .112-119

, f n-1 (z ) h1 (D ) , , 2, ( , t ) f n-1 ( ( , t , u )) 1 (*). , ( , t
f
n-1

u < 1 , ,

)

( ) m f n-1 ( ) e ,t 1 ( ( , t , u )) = lim - u e ,t (u ) d . m 2i E ( ,t , )

(9)

f

n -1

, ( ( , t , u )) u < 1

= 1 . (8) (9)
( ) m f n -1 ( ) e ,t f (z ) = d lim (u ) d d t . (10) m - u e ,t (2 )n i E ( , t , )
1

,
e ,t ( ) d d t (u ) d - u e ,t E ( , t , ) ­ . , (10) .
f
n -1

( )

m


J=

d





f

n -1

( (

, t , u )) d t -

,t
,t

- d





f n-1 ( ) e 1 - u e 2i E ( ,t , )



( ) m d d t (u )





d





f

n -1

( ) m f n -1 ( ) e ,t 1 ( ( , t , u )) - d d t . 2i E (t , ) - u e ,t (u ) ,

, (*) .. .. ([9], . 106) m :
119


2. (I)

f (z ) = e

- m ( z

)1

2i

E



f ( ) e m ( ) d + e -z
(u ) 1

- m ( z

)1

2i

CE



f ( ) e m ( ) d . -z


f
n -1

( (
,t

, t , u )) = e
(u ) 1

- m

,t

2i E (t , ) ,

f

n -1

( )

e -u

m

,t

( )

d +

+e

- m

2i

CE ( ,t ,



f
)

n -1

( )

e -u

m

,t

( )

.

d

J , :
J d e
- m
,t

(u ) 1

2i

CE ( , t ,



f
)

n -1

( )

e -u

m

,t

( )

d d t

1 d e 2

- mp

,t

(u )

d

t

CE ( , t ,



f
)

n -1

( )

e

m

,t

( )

-u
=1
,t

d .

u d = min - u .
J 1 d 2d



d

t

CE ( , t ,



f
)

n -1

( )

e

- mp

(u )
,t

d

( e

m

( )

, .. ( , t

)

CE ( , t , ) , E ( , t , ) = 0 . 0 < p , t (u ) = Re[ , t (u )] < 1 , ,
1) e ,t 0 m , ( f n-1 ( ) m) ; - mp (u ) 2) e ,t 1 , .. m vf n-1 ( ) , . , ,
120
- mp

p ,t ( )

(u )


.. -- -10, 2002, .112-121



d





d

t

CE ( , t ,



f

)

n -1

( )

e

- mp

,t

(u )

d

m , .. J 0 m . . 4. , 2 3, [1] [11]. . 1. .. // . ­ 1958. ­ . 120, 5. ­ . 935 ­ 938. 2. .. . . ­ : «» (. ). ­ 1990. ­ 248 . 3. .. . .: «» . ­ 1991. ­ 200 . 4. .., .. Carleman'a // . . ­ 1933. ­ . 40, 2. ­ . 144 ­ 149. 5. .. // . ­ ., . ­ 1985. ­ 107 . ­ . 20.08.1985, 6123-85. 6. .. // . . . « ». .: «» . ­ 1996. ­ . 75 ­ 92. 7. .. n // . . . « ». .: «» . ­ 1998. ­ . 89 ­ 105. 8. .. // . . : . . /
121


2. (I)

9. 10. 11. 12. 13.

. .. . ­ ­ : « », 2002. ­ . 9. ­ . 2. ­ . 584 ­ 596. .. . ­ .-.: , 1950. ­ 336 . .. // . ­ 1958. ­ . 120, 5. ­ . 976 ­ 979. .. ­ // . . . ­ ., 1962. ­ . 110.- . 133 ­ 139. Carleman T. Les functions quasianalytiques. ­ Paris, 1926. ­ 116 p. Opial Z., Siciak J. Integral formulas for functions holomorphic in convex n-circular domains // Zesz. Nauk. Univ. Jagiell. ­ 1963. ­ V. 9, 77. ­ P. 67 ­ 75.

122