A C T A
U N I V E R S I T A T I S
L O D Z J E N S I S
FOLIA MATHEMATICA 5, 1992
Zbigniew Jerzy Jakubowski, Jozef Kaminski
ON
OF
SOME
CLASSES
M O C A N U —B A Z I L E V I C
FUNCTIONS
In this paper we investigate some classes
by different types
/(I + Bz),
z
of relations
e A =■ {z: |z |< 1 }
complex values.
(1955)
the
where
of
functions generated
homography z -*• (1 + Az)/
parameters
A, B
may take
The main results concern certain families of ot-convex
Mocanu-Bazilevic functions
v i c
with
[14]).
(M o c a n u
(1969)
[78] ,
B a z i 1 e-
The results obtained are a continuation of the
considerations contained in
[40],
[41 ]
[55 ]
and
investigations are preceded
by a survey of various
[42 ]. The basic
classes
of
Ca-
ratheodory functions with positive real part.
1.
Let T
(1 .1 )
ON VARIOUS CLASSES OF CARATHEODORY FUNCTIONS
denote the well-known class of functions
p(z) =
1
+ p^z + ... + Pnzn + •••
holomorphic and satisfying the
A = {z: |z| < 1},
(1.2)
([17]).
f(z) = z +
condition
Re p(z) > 0 in the disc
As is known, many classes of functions
+ ... + anzn + ...,
are generated by functions belonging
others, the known classes
z e A,
to V .
Here
S*, Sc , T, C, U, R, K
belong,
of functions (of
the form (1 .2 )) starlike, convex, typically-real,
direction of the imaginary axis,
real axis, possessing
A, close-to-convex
f e R
convex
in
the
star like in the direction of the
a derivative with
(see e.g. [32]).
satisfy the condition
among
a
In
f'(z) =p(z),
positive real part in
particular,
z e A,
functions
p e T3.
There
also hold corresponding relations
functions.
for
the
remaining
In the investigations, the so-called classes
order a,
a e <0 ,
[107]).
So, let
1
), appeared
°Pa
system of symbols.
functions
f
by functions
In 1968
([91];
of
cf.
the form
S* - the
class
(1 .2 ) starlike of order a (zf'(z)/f(z) =
p e cf>a )>
ones are introduced.
early
the condition Re p(z) > a,
of functions of the form
z e A,
functions of
denote the family of functions
(1.1) satisfying in A
= p(z),
comparatively
of
classes of
Similarly,
the
(In the sequel, we
So, e.g. S*(A, B)
classes
S° and other
shall apply an analogous
stands
for
the
class
of
generated in the same way as starlike functions, but
p belonging to a fixed family ‘PiA, B)).
R. M.
G o e 1
([28])
investigated the class p ^]'
M > j, of functions (1 .1 ) satisfying the condition
(1.3)
|p(z) - M| < M,
zeA,
and certain two classes generated by
(cf. also [27]).
investigations were later extended by
[45] - 1969;
[43], [50],
[46],
[51],
[92], [102],
[105],
half-plane
Re w > a
M a c G r e g o r
[47] - 1970)
[76], [79],
[HO],
W.
J a n o w s k i
([44],
and other authors (e.g. [38],
[80],
[85],
[8 6 ], [87], [8 8 ],
[112]).
The
idea
to
by a disc appeared earlier
([6 6 ] - 1962;
considered certain implications
These
replace
in the papers by
[67] - 1963; [6 8 ] - 1964)
of
condition
the
(1.3)
who
with M = 1.
This particular case can also be found in later publications (e.g.
[57],
[93],
[96]).
In 1971 ([34])
the author introduced the class T
M of funcm,M
tions of the form (1 .1 ) which satisfy the conditions
(1.4)
|p(z) - m| < M,
z e A,
where the real numbers m, M satisfy the inequality
(1.5)
|1 - m| < M £ m.
He also introduced some classes of functions of the form (1.2) ge­
nerated by the family T m M ([3 5 ]).
Let fi denote the family of functions
(1 .6 )
q(z) = q ^ + ... + qnzn + ...
holomorphic and satisfying the condition |q(z)| < 1 in A.
In the investigations carried
lowing fact was applied:
=
out among other
things,
the
fol­
p e °Pm M if and only if
(1.7)
p(z)
(1 + Aq(z ) ) / ( 1
(1.8)
A = (M2 - m 2 + m)M_1,
+ B g (z )),
q 6 2,
B = (1 - m)M-1,
z e A ,
(A, B) e
where
(1.9)
Ex = {(A, B): -1 < A S 1,
-1 < B < A).
In papers [34]-[38], certain properties
of
the family T m M were
obtained as well as of the classes generated
by this family (e.g.
Rm,M' Tm,M)'
In 1973 W. J a n o w s k i
V (A, B), (A, B ) e D where
( [48],
(1.10)
-1 £ B < A),
D = {(A, B): -1 < A S 1,
directly by condition (1.7).
(A, B) g E 1
[49]) defined the class
It is obvious
has been extended because
that
the
the
points
condition
(A, B) e E2,
where
(1.11)
E 2 = {(A, B): -1 < A £ 1,
were added.
Therefore the class
B = -1},
VIA, B),
prises Carathéodory functions of order
(A, B) e D, also
a = (1 - A)/2.
com­
Moreover,
notice that, for example,
T i l , B) = *>[1/(1+B)],
([28]),
VIA, 0) = <p(A) =: ( p e V :
VIA, -A) = V (A) =:
A > 0, ([16]).
Obviously,
^(1, -1) = V.
concern the properties
of
|p(z) - 1| < A},
{pe V :( |p(z)
-
A > 0, ( [96] ),
l|/|p(z)
+ 1|) <
A},
The basic results of W. J a n o w s k i
the classes ‘PÍA, B) and S*(A, B). Many
other
problems connected with the classes V m ,M„ or *¡P(A, B) and the
families generated by them can be found in papers [1], [2], [3],
[5],
[6 ], [7], [8 ],
[1 0 ], [15], [25],
[29],
[39],. [41], [52],
[53], [54], [59], [60], [82], [83], [94], [108], [113].
In papers [34]
(1971) and
to replace in (1.4) the number
[38]
m
placing simultaneously condition
(1.12)
|1 - c| < M S Re c.
(1973),
by
(1.5)
the
an attempt was made
complex
number c, re­
by the inequality
Functions of the family V
M
thus
defined
have the form (1.7),
where instead of (1 .8 ) we have
(1.13)
A = (M2 - |c|2 + c)M-1,
It follows easily from (1.12)
B = (1 - c)M-1.
and
(1.13) that
Among other things, exact estimations
coefficients in this class
of
were obtained.
taneously been obtained by R. J.
v i n g s t o n
in [63].
J.
Z y g m u n t
This result has simul­
obtained
(e.g. [34],
of coefficients.
J.
of
the
A. E.
general
L icase,
in [106] generalizations of some re­
sults known earlier
and
and
and B are complex numbers, were
signifi­
A. S z y n a l ,
J. S z y n a l
and
cerning estimations
w i c z
absolute values of
L i b e r a
In the next years, the investigations
i.e. the case when A
cantly extended. So,
|A| £ 1, |B| < 1.
the
[35], [44],
W a n i u r s k i
things, the class
*Pn (A, B),
= 1 + Pnzn +...,
zeA,
[63], [107])
In [101]
con­
J. S t a n k i e-
investigated,
among other
|A| £ 1, |Bf £ 1, of functions p(z) =
n = 1, 2, ...
functions considered did not have
In both those cases, the
to belong to V.
Various clas­
ses of functions with complex parameters A, B were also investiga­
ted, e.g. in [42],
[56], [71], [111].
Notice also that the investigation of the class of functions
of the form (1.7) with complex parameters A, B can always be re­
duced to the case when either
6 C K R , | A | i 1,
0 < B £ 1).
A
or B are
real
It is worth noticing that in all classes
form (1.1)
of
(e.g.
(A, B) e
functions of the
considered earlier, the values p(z), z e A, always be­
longed to some convex
set
(halfplane, disc).
pertied of such functions
the other hand, in all
term of subordination
in [1 0 1 ]).
general pro­
can be found, for instance, in [30]. On
the
on (1.7) was obligatory.
Some
cases,
Therefore,
the
in
definition pattern based
these
definitions,
and its properties can be applied
the
(e.g. as
2. ON THE CLASS M q (A, B)
In 1969
P. T.
M o c a n u
( [78])
introduced the class Ma ,
0 < a £ 1, of functions of the form (1.2) satisfying in the disc A
the conditions:
(2 .1 )
f(z)f'(z)z_ 1 *
(2.2)
Re J(f, z, a) > 0,
0
,
where
(2.3)
J(f, z, a) = (1 - a) ^
f (z)
The class
Ma
proved to be interesting to many mathematicians
([22], [26], [58],
It was proved
longing to
[69], [72], [73] - [75], [77], [103],
in
Ma
+ a(l + Zl"AZ\).
f (z)
[74], among
are,
B a z i l e v i c
for
a >
other
,
0
[104]).
things, that functions
elements
be­
of the known class of
functions, [14].
The consecutive investigations developed in several directions.
In particular, in papers
[42],
[55], [61],
[9i] ,
[97] -
[4], [9], [18], [20], [21], [23], [40] -
[62], [76], [81],
[100],
[109],
[84],
condition
[89],
(2.2)
[90],
[94],
underwent various
modifications.
Let (A, B) e C
the conditions
2
be
a couple of complex
numbers
satisfying
‘A t B
I®I
(2.4)
* 1,
|A - B| £ 1 - Re (AB),
if |B| = 1 ,
A = - |A|B,
Denote by
Ma (A, B)
then
|A| £ 1.
the class of functions (1.2)
satisfying con­
dition (2 .1 ) as well as the condition
<2-5) Jl£' *■ «> ‘ rr w fr
for some
q e n,
where
(2.4).
It follows from (2.4)
Obviously,
a 2 0,
and
Ma (l, -1) = Ma ,
z * *•
and (A, B)
satisfies assumptions
(2.5) that Re J(f, z, a) > 0 in A.
Mq (1, -1)
=
S*,
M 1 (l,
-1)
= sc ,
Mq (A, B) = S * (A, B).
In paper [42],
demonstrated.
the formulae
It
some
properties of the families Ma (A, B) were
was also shown that the function
f
defined by
A-B
z(l + B 2 )
f(Z) ='
belongs to
ze
Az
,
Mq (A, B),
for no values
of
,
3
and
a > 0.
z e A,
if
B * 0,
z e A,
if
B = 0
that it does
not belong
Next, the functions
fk ,
to
Ma (A, b )
k = 1, 2, de­
fined by the formula
1 + a? 14
J(fk , z, o) = ----z e A,
1 + Bz*
belong to
Ma (A, B)
for any
(f(0) = f'(0) - 1 = 0),
a £ 0.
Moreover, they turned out to
be functions realizing the extrema of certain functionals.
The next sections of the present paper are a natural continua­
tion
of
[42].
The results obtained are generalizations
of
the
respective results obtained earlier by different authors.
In the sequel, unless otherwise stated, we
the couple (A, B), conditions (2.4) hold.
assume
that, for
3. ON SOME RELATION BETWEEN HARDY CLASSES AND THE CLASS Ma (A, B)
As is well known, a function
the Hardy class HX (o < X < +«)
f holomorphic in
if
A
belongs to
2n
it x
lim s |f(re )| dt < +».
r-»l" °
Denote also by H
the class
hic in A (e.g. [24] , p. 2).
of functions bounded
and holomorp­
It was shown in [55], among other things, that if f e M a (A, B),
(A, B) e D
where D is the set
ned by (1.10), then
where
f
X = X(a, A, B).
and
f' belong to certain Hardy classes H*
belongs
were determined.
We shall
Ma (A, B), (A, B) e D, possesses
the fol­
lowing property corresponding to the known theorem for
([70]).
defi­
Also, for f e Ma (A, B), a > 0, (A, B) e D,
the Hardy classes to which f"
now show that the class
of couples of real numbers,
M (1, -1),
THEOREM 3.1. (i) There exists an f e M Q (A,B), (A,B)eD such that
f'V H^
for any value of X > 0.
(ii)
There exists an
for no value
Proof,
of
X>0,
f e Ma (A, B),
f*n * £ h \
(A,
B) e D,
such
that,
n = 3, 4, 5, ...
(i) In [65], a function g was constructed
holomorphic in A, continuous and univalent
A, g(0) = 0 and such that
(3.1)
lim |g' (re.it,) | = +<°
in
the
which is
closed
disc
r-»l"
for almost all
number b > 0
(3.2)
t s < 0, 20).
It
is clear
such that |g(z)| < b
9 i (z) = abT^'
that
for z e A.
there exists
a
Take the functions
z 6 A'
and
1 + Ag^z)
P1(Z)
if
1 + Bg^z)
(A, B) e Ex,
(3.3)
1 +
P 2 (z) = 1 + —
-— A gx (z)
if
(A, B) e E 2
(cf. (1.9) and (1.11)).
A s g 1 (0) = 0,
where (A, B) e E^,
(3.4)
lim
r-l'
|g1 (z)| < 1,
k = 1, 2.
z e A,
It
zf'(z)
* . = P. (z),
r k (z)
x
f£
on
B)
(3.1)-(3.3) that
<0 ,
211).
defined by
(A, B) e E^,
z e A,
k = 1, 2.
k = 1, 2.
We shall prove
H^ for any X > 0.
Assume that there exists
tion
fk
Pk e V (A,
from
a.e.
(fv (0) = 0 ) ,
k.
Obviously, f^ e Mq (A, B),
that
follows
|P^(re.it,) | = +»
Next, consider the functions
(3.5)
therefore
f e h\
X > 0,
X > 0
such that f^ e H*. Each func­
has a radial limit in almost any direction.
r
Moreover, as
f^ t 0,
therefore also
fk (elt) i 0
on any set
of
positive measure
fact that
f
(e.g. [24], p. 17).
Thus
and f' belong to certain
and from (3.5) that
lim P'(re:''t )
r+l”
it
Hardy
follows
classes
from the
(cf. [55])
exists almost everywhere. This
contradicts (3.4).
(ii)
Consider the functions fk defined by the condition
J(fk , z, a) = Pk (z),
k =
where
Pk
1
,
2
by
k = 1, 2.
sults of [55], in
fk 4 HX
z e A,
,
are defined
(A, B) e Ek ,
(fk (0) = fk (0) - 1 = 0 ) ,
the
(3.3).
Obviously,
fk e Ma (A,
Taking into account the
same way as in
for any X > 0.
(i),
appropriate
we
can
prove
Hence and from the fact that if
0 < X < 1, then f e HX/(1_X) (e.g. [24], p.
88
B),
),
re­
that
f' e h \
we obtain
(ii).
4. ON p-VALENT FUNCTIONS
As is well known, a function f
disc A
if
possesses
it is holomorphic in A
p
is said to be
and
roots in A for some wQ
the
Let A, B, a
p 2 1
and
J(f, z, a)
of the family
be
f(z)
and, for any complex
w, the number of solutions of the equation
not exceed p (e.g. [32], vol. I, p. 89).
in the definition
p-valent in the
equation
defined
f(z) = w
in
= w
o
number
in A does
the same way as
Ma (A, B) (see (2.3), (2.4)), and
an arbitrarily fixed positive integer.
Denote by
MP (A, B)
the family of functions f
(4.1)
fp (z) = zp + bp+ 1 zp + 1 + bp+ 2 zp + 2 + ...
holomorphic in A and such that f (z )f '(z)z1_2p f 0,
P
P
d.2)
i j u p,
„i,
for z e A and for some
q e £2.
Obviously, M^(A, B) = M 0 (A, B).
(A, B) e D
was investigated in [41].
The
The
class
class
MP (A,
Mp (l,
B)
where
-1)
was
introduced in [40].
The families
Mp (l, -1)
and
MP (1,
-1)
are
well-known subclasses of p-valent starlike and p-valent convex
functions, respectively, investigated in [31].
Notice that, for
MP (A, B) c Mp (1, -1).
all
(A, B)
satisfying
Therefore functions belonging
are, in particular, p-valent starlike [40].
introduced class
M^(A / B)
THEOREM 4.1.
If
(2.4),
is
The
we
to
have
MP (A, B)
structure of the
described by the following
[Ma/p*A » B)]P
denotes
the
set of the p-th
powers of functions of the class Ma/p (A, b ), then
(4.3)
m £(A,
B) •- [Ma/p(A, B)]p .
Proof.
For any function f
e MP (A, B), consider the func­
tion
p
“VT? =
1
, z e A.
Then condition (4.3) follows from the following identity in A:
i J(fp , z, a) = J(f, z, J).
By Theorem 4.1, it is obvious that
p-th powers
MP (A, B)
consists
of the
of starlike univalent functions belonging to Mq (A, B)
= S*(A, B). Simultaneously, MP (A, B), p > 1, is not identical with
the set of the p-th
powers
M 1 (A, B) = sc (A, B).
convex function
of
of
class
functions
univalent
functions
It seems to be interesting that
MP (A, B)
function belonging to the class
THEOREM 4.1
convex
is
p-valent
the p-th power of some a-convex
Ma <A / B) for a = 1/p.
also enables one to obtain the properties
B) corresponding to certain properties
of
a
Ma (A, B).
The
following
Lemmas
of
are
of the
univalent
well-known.
LEMMA 4.1
[42].
If 0 $ P S a, then Ma (A, B) c Mg(A, B).
LEMMA 4.2
[42].
If the function
longs to the class
in
Ma (A < B), then
f
of the
form
(1.2)
be­
(4.5)
|a3 - Xa2 | S 2|* ~ ^
max(l, s),
Xe C,
where
s = (1 + a )"2 |2X(1 + 2a)(A - B) + Ba 2
+ (5B - 3A)a + 2B - A| .
The estimations in (4.4)
exact.
Using Lemmas 4.1, 4.2
prove, for instance,
and (4.5) in the class
as well as conditions
THEOREM 4.2.
If
0 £ |3 £ a,
THEOREM 4.3.
If
f
then
Ma (A, B) are
(4.3),
we
can
MP (A, B) c M?(A, B).
a
p
of the form (4.1) belongs
to
M^(A,
B),
then, for any X e C,
(4.8)
|bp + 2 - Xb2+1| £ p 2 2|A - B |} max(lf u)
where
u = (p + ot)~2 12p2 X(p + 2a) (A - B) - p3(A - B)
+ p2(B - 2aA + 2aB) + pa(3B - A) + Ba2 |.
Estimation (4.6) in M^(A, B) is exact.
5. ON k-SYMMETRIC FUNCTIONS
Let Ma (A, B, k)
class of functions
(5.1)
(k - a fixed positive
fk
integer)
denote
the
of the form
fk (z) = z + bk+ 1 zk + 1 + b 2 k+1 z2 k + 1 + ...,
z e A,
belonging to Ma (A, B).
Notice that Ma (l, -1, k)
is
the
class
symmetric functions, introduced in [19].
classes
Ma (A, B)
and
M^iA, B, k)
of
The
allow
one
a-convex
k-fold
definitions
of the
easily to get the
following relation between theses classes:
THEOREM 5.1.
Ma (A, B, k)
A function
fk
of
the
form
(5.1)
if and only if the function f of the form
belongs to
f(z) = [fk (z1/k)]k = z + a 2 z2 + a 3 z3 +
belongs to the family
Mp(A, B)
where
The Theorem stated above permits
properties of the class
-known properties
of the class
p = ak.
one
Ma (A, B, k)
z e A,
to
formulate
corresponding
Ma (A, B).
to
certain
the well-
In particular, Theorem
5.1 and Lemma 4.2 imply
THEOREM 5.2.
the class
(5.2)
If a function
f^
of
M^fA, B, k), then, for any
Ib 2 k+l “ Xbk+ll *
2k(1
+
2 ak)
the form (5.1) belongs to
X e C,
maX (1' V)
where
v = (1 + ak)”2 |Ba2 k 2 + (3B - A)ak + 2(2X - l)a(A - B)
+ B + 2X ~
-1
(A - B) |.
For each X e C, estimation (5.2) is exact.
The following corollary is a direct implication
5.1 and 5.2 as well as of Lemma 4.2:
COROLLARY 5.1.
If a function
of
Theorems
f^ of the form (5.1) belongs to
Ma (A, B, k), then
1bk+lI 2 k(î + ak)'
1b 2 k+l
I
5
2k
( i V 2 ak) max(1' v)
where
v = (1 + ak)“ 2 |i(B - A) + Ba 2 k 2 + ak(3B - A)
+ 2a(B - A) + B|.
6
. ON CERTAIN CLASSES OF FUNCTIONS OF TWO COMPLEX VARIABLES
As before, let A c C
be the unit disc, (A, B) e C
satisfying assumptions (2.4),
a 2 0.
Denote by
2
U c C
- a couple
2
a
fixed
bounded complete Reinhardt domain with its centre at the origin
(e.g. [13]). We shall also apply the following notation:
HU - the family of holomorphic functions f: U-»C,f(0, 0) = 1,
i2U - the family of holomorphic functions w: U -+C,w(0, 0) = 0,
IwfZj^, z2) | < 1
for
(z^, z2) e U,
L - the differential-functional operator
values
defined
on
Hu with
Lf(zx , z2) = f (z^, z2) + z1 f^lz1, z2 ) + z2 f2 (zlf z2).
We shall consider the family
M^(A, B) of functions f
6
Hu for
which
(6.1)
t(zlt z2 )Lf(zir z2 ) # 0,
(6 .2 )
(1
where
Lf(z,, Zj)
L 2 f(z., Z_ )
- a) — —
+ a -----±
f(zx , z2 )
Lf(z1, z2)
L2f = L(Lf),
w e i2U ,
Obviously, if A = 1,
valent to the inequality
B = -1,
Lf(Z.,
Re[(1
Therefore
■
I.
~1)
[64]
is the family considered
(see also [33]).
Z1
in
plane
z^ = 0.
Let F^ and F 2
by
(zl' Z2 > e U P.
L i c z-
The class M^(A, B) is also a
M^(l, -1), M^(l, -1 ) introdu­
[12].
the intersection
projection of the intersection
equi­
L 2 f(Z., z7)
Zy)
B a v r i n
Denote by
U.
then condition (6.2) is
generalization of the known classes
ced by
6
~i(zx, z2 ) + “ ™Lf(Zj ,1 z2 )J > °'
M^d,
b e r s k i
(z^ z2)
1 + Aw (z, , z~.)
^
-- i_
1 + Bw(z1, z2) '
U fl {z2 = 0} and by
un{z
1
= k z 2),
keC,
be functions of one
Z 2 - the
onto
variable
the
with
values
F 1 (z1) = z1 f(z1,
^ 2 (z2 )
0
),
z2 ^ (^z2 ' ^2 ^f
zx e Z1#
z2
Applying the method used in [12],
^
^2
we
can give the following
terpretation of functions belonging to
THEOREM 6.1.
only if
A function f e HU
*
in­
(A, B).
belongs
to
M^(A, B)
if and
1
F.^
is
holomorphic
in
Z1 ,
1 + Aw (z
0)
F x (0)
=
F£ (0)
-
1
=
0,
ZI1 F 1 (Z1 )F1 (Z1 > * °'
J(Fi' zi' a) = r + Bw(z|! o)'
2°
in Z2,
For any fixed
k e C,
the
F 2 (0) = F'(0) - 1 = 0 ,
J(F2 ,
(here
z
2,
where
function
F2
z2 1 F 2 (z2 )F^(z2)
1 + Aw(kz2 , z2 )
a) = i + Bw(kz2, z2 )'
zk , a),
zi e V
k = 1, 2,
z2 e Z2'
is
w e nUholomorphic
0,
w e n
are defined as in (2.3)).
Next, we shall present certain properties of
ted with some theorems concerning
B ) connec­
the class Ma (A, B) of functions
of one variable.
THEOREM 6.2.
Proof.
If
0 S, 0 S a,
then
MU
(A, B) c Mqp (A, B) .
a
Take
a function f e M^(A, B) for |B| < 1 and an
o
o
arbitrarily fixed point (z^, z2) e U. It follows from the proo
o
perties of U that, for any C e A, also the point (Cz^, Cz2) e u.
We construct a function
C e A.
g
of the variable
g(£) = CfKz^ C°2 ),
Then the following identity holds in the disc A:
Lf (£z. , c? )
L2f K z . , cS,)
(1 - a) ---------- — + a ----------- — = J(g, C, a).
f(Czx, C°2)
Lf(C°lf Cz2)
Hence and from the definition
that, for |B| < 1,
s = -3^-M,
1
(see [42]).
-
of
B) it follows
where
P = - LA- - B l ,
|S|
This means that
Hence
class
|J(g, C, a ) - s | < p
i
-
|B|2
g e Ma (A, B),
Therefore, according to Lemma 4.1,
0 S 0 £ a.
the
also
|Bf < 1
(cf. [42]).
g e M^IA, B),
|B| < 1,
Lf(C?.,
CZ,)
ICl - P) ----------—
L2 f ( C z , ,
+ $ ------i----—
f(Cz1# C§2 )
Hence, because of the arbitrary choice
we obtain that, for each
- s| < p.
Lf(C°lf Cz2 >
of the point
(z^, z2) e U,
the
(z^, §2) e U,
following
condition
is satisfied:
Lf(Z., Zj)
111
-
L 2 f(z,, z,)
z'i
91
This implies that
Lti^,
<»■
f e Mg (A, B) for |B| < 1.
It can be noticed that
f e M^(A, B)
for
|B| = 1 if and only
if condition (6 .1 ) holds and if
Lf(zn, z,)
L 2 f(z., z,)
1 - A.
^
+ a ———-— ---- Ł-] > ---— i
f(zL, z2)
Lf(zx, z2)J
2
Re [ ( 1 - a)
-1 < Aj S 1
(cf. 1421).
where
“ IA|,
if
A = |A|B,
|A|
IA|,
if
A = - |A|B,
|A| 2
< 1,
A.^ =
The proof of Theorem 6.2
for
the case when
1,
|B| = 1
can thus be constructed in a similar way as for |B| < 1.
Our next step is
tions of
M^(A, B).
to show some integral representation of func­
The proof of the Theorem given below
can
be
carried out similarly as in [64].
THEOREM 6.3.
A function
f e M^(A, B),
a > 0, if and only if
it possesses the following representation:
1
i i
S [g(tzx, tz2 )]ata
o
.
f(zx, z2 ) =
where
-1
dt}“
g e M^(A, B),
Finally, we apply the following notation:
Y
_ Re(AB) + 1A - BI IBI
2
-
IBI 2
IBI 2
v'= Re(AB) - IA - BlIBI - IbI2
2 IB | 2
K = K(ot, A, B; r)
where
K =
e lA lr
for a « 0, B = 0
(1 + |B |r)Y (1 - |B |r )y/
for o = 0, B # 0
[♦(I,
for o > 0, B = 0
i + I,
a
[Fl(i'
l A L r)]«
a
Zi' i1'
1 + o; "lB lr'lB lr)]a
for a>0, BifO
(a)
In these formulae, i>(a, c; z) = 2
n zn
is
a
degenerate hy-
n-0 <c>n
pergeometric function, whereas
F 1 (a, b, b', c; x, y) =
T.
m „«n
m,n-U
m+n
m____n
(c)m+n
.„mini
n
denotes a hypergeometric function of two variables ( [ll], pp. 219,
237). Let
ur = {(rz^, rz2): (z1# z2) e U},
THEOREM 6.4.
0
< r <
1
If f e M q (A, B),
(0, 1).
for any
(z^, z2) e Ur ,
,
-K(o, A, B; -r) £ If( ,
Proof.
number.
then,
re
Take
Let
rQ ,
z2)| S K(o, A, B; r).
0 < rQ < 1,
be
an
arbitrarily
next an arbitrarily fixed point (fj,
?2
fixed
)e 0
.
If
o
the number p
e Up
and
the domain
satisfies the inequality rQ < p <
(z-^p-1, z2 P_1) e U*
U
that also
It
1
, then (z^, z.,) e
follows from the properties of
(Cz^p *, C?2p 1) e U
holds
for
any
C e A. Next, consider the function
0(C) = Cf(CZjp-1, Cz2 p-1),
Notice that
t
is a holomorphic
0(0) = 4>'(0) - 1 = 0
f e M^(A, B).
function
and, moreover, that
lowing conditions are satisfied: 4>(C)* ( O C
of the variable C, e A,
in
1
the disc A the fol­
f
0
,
1 + AwlÇz.p-1, Çz.p'1)
J(4>, Ç, a) = ----------------------,
1 + Bw( Ç ^ p -1, Çz 2 p-1)
This implies that
|f(z)|,
<J> e Ma (A '
Applying
we
the
iîU .
estimations
for
f e Ma (A, B), derived in [42], we obtain
-|Ç|K(a, A, B; -|Ç|) S ICffC^p"1, C,Z2 f>~l)\
S |C|K(a, A, B; |C|)•
Putting Ç = p
and letting p
tend to rQ , we shall get
-K(a, A, B; -rQ ) S |f(Sj, §2)| S K(a, A, B; rQ ).
The above inequalities
Theorem 6.4
because
are equivalent
of the arbitrary
to
the
choice
proposition
of
of
(z^, z2 ) e Ü
o
and ro ,
0
< ro <
REMARK.
1
.
It follows from Theorem 6.3 that
Lf(zx, z2 ) = [f(Zx, Z2]
Hence and from Theorem 6.4
|tf(z1, z2 )|,
one
1 -1
I
“ [g(zx, z2)]a ,
g e M^(A, B).
can derive estimations for
f e M^(A, B),
a £ 1.
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Chair of Special Functions
University of Łódź
Faculty of Applied Physics and Mathematics
Technical University of Gdańsk
Zbigniew Jerzy Jakubowski, Jozef Kamiński
0 PEWNYCH KLASACH FUNKCJI MOCANU-BAZYLEWICZA
W pracy zbadano kilka klas funkcji
z homografią z -*■ (1 + Az)/(1 + Bz),
generowanych przez różnego typu związki
z e A » {z: |z| < 1},
szczono możliwość przyjmowania przez parametry A, B
sadnicze rezultaty
wicza
(M o c a n u
dotyczą pewnych rodzin
(1969) [78],
wyniki stanowią kontynuację
[55] i [42].
wartości
a-wypukłych
B a z i l e v i ć
czym
dopu­
zespolonych. Za­
funkcji Mocanu-Bazyle(1955) [14]). Otrzymane
wcześniejszych prac, a w szczególności [40], [41],
Podstawowe badania poprzedzono przeglądem
Caratheodory ego
przy
o części rzeczywistej dodatniej.
różnych
klas
funkcji