Internat. J. Math. & Math. Sci.
(1987) 79-88
Vol. i0 No.
ON BAZILEVIC FUNCTIONS
KHALIDA I. NOOR and SUMAYYA A. AL-BANY
Mathematics Department,-Girls College of Education
(Science Section) Sitteen Road, Malaz, Riyadh,
Saudi Arabia.
(Received January 23, 1985 and in revised form June 28, 1986)
Let B() be the class of Bazilevic functions of type B(>O).
ABSTRACT.
zf’ (z)
A function fgB(B) if it is analytic in the unit disc E and Re
f
(z)g (z)
I_B>O,
We generalize the class B(8) by taking g
Archlength, differk(k>2).
where g is a starlike function.
to be a
function of radius
Hankel determinant and some other problems are solved
ence of coefficient,
for
rotation at most
For k=2, we obtain some of these results for
generalized class.
this
the class B()
of Bazilevic functions of type
KEY WORDS AND PHRASES.
Bazilevic
rotation; Hankel determinant,
functions,
functions of bounded boundary
close-to-convex functions,
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES 30A 32,
i.
radius of s-convexity.
30A34.
INTRODUCTION.
Bazilevic
[i] introduced a class of analytic function f defined by the
For zgE:
following relation.
E
let
-ai
z
f(z)
l+a
(h() ai)
2
where a is real,
2
g
l+ai
l+a
B
2
(i.i)
() dE
0
B
> O, Re h(z)>O and g belongs to the class S
functions.
Such functions,
we have for
zeE
Re
he showed,
zf’ (z)
fl-8(z)gS(z
This class of Bazilevic functions of
note this
B
i
class of functions by B(B).
are univalent
[I].
of starlike
With a=O in
(1.2)
> 0
type
(I.i),
B
was
considered in
We notice that if
have the class K of close-to-convex functions.
B
[2].
=i in
We de-
(1.2), we
We need the following defi-
nations.
Definition I.i
A function f analytic in E belongs to the vlass V of functions with
k
I, f’(z) # 0, such that for
0, f’(0)
boundary rotation, if f(0)
bounded
z
9e
eE, 0<r<l
K.I. NOOR and S. A. AL-BANY
80
(zf’ (z))
Re
<
d@
f"-(z/
k>2
k
(1.3)
For k=2, we obtain the class C of convex functions.
It is known [3] that
for 2<k<4,_ V
consists entirely of univalent functions.
The class V
has
k
k
been studied by many authors, see [3], [4], [5] etc.
Definition 1.2
Let f be analytic in E and f(O)=O, f’(0)=l.
to
the class R
Then f is said to belong
of functions with bounded radius rotation,
k
if z--re
eE,
O<r<l
2
zf’ (z)
IRe -fCz)IdO<k
(1.4)
k>2
0
It is clear that feV
zf’eR k.
We also note that
We now give the following generalized form of the class B(8).
Definition 1.3
k
if and only if
Let f be analytic in E and f(0)=l, f’(O)=l.
class
Bk(8)
8>0
if there exists a
zf
Re
f
We notice that, when
8=1,
=1_ 8
(z)
(z) g 8 (z)
Bk(1)
duced and discussed in [6].
geRk; k>2_
T
Also B
k’
Then f belongs to the
such that
> O,
zeE
(1.5)
a class of analytic
2(8)
,
R2=S
B(8) and
B2(I
functions intro-
K, the class
of close-to-convex functions.
2.
PRELIMINARIES
We shall give here the results needed to prove our main theorems in
the preceeding section.
Lemma 2.1 [3].
Let feV
k.
Then there exist two starlike functions
zeE
k
i
k
4
i
(sl(z)/z) -+
f’ (z)
(S2(z)/z)
SI,S 2
k>2_
such that for
(2.1)
2
Lemma 2.2
Let H be analytic in E,
k
H(z)=(
Then,
(i)
and
for z
1__
2
+
IH(O) I<i
i
)h l(z)
(_k4
and be defined as
i
)h2(z) Rehi(z)>O’
i=1,2,k>2.
re
IH(z)
12-< l-(k2-1)r
2
l-r
(2.2)
BAZILEVIC FUNCTIONS
81
2T
(ii)
i
IH’ (z)Id@<
0
k
l-r
(2 3)
2
[6] and, for k=2,
This result is known
we obtain
Pommerenke’s result [7]
for functions of positive real parts
Lemma 2. 3
be univalent in E.
i
with
there exists a z
Then:
Let S
(i)
,IZll
I
=r such that for all z,
IZ-Zll IS l(z) I<_
2r
l-r
Izl
=r
2
see
2
(2.4)
[8]
and
r
(ii)
(l+r)
<--IS1 (z) I<
2
r
(2.5)
[9]
see
(l-r) 2’
Definition 2.1.
z+ Z
Let f be analytic in E and be given by f(z)
qth Hankel determinant of f is defined for q>l,
a
a
n
a
n+l
n>l
2
a z
n
n
Then the
n+q-1
an+ 1
(2.6)
H (n)
q
an+q- 1
a
n+2q-2
Definition 2.2.
Let z
f(z)
z+
be a non-zero complex number
1
Y,
a z
n
n=2
n
Then, with
A0(n,zl,f)
an;
we define for j>l,
Aj (n,zl,f)
Aj_l(n+l,Zl,f)
Aj_l(n,zl,f
(2.7)
Lemma 2.4
Let f be analytic in E and let the Hankel determinant of f be defined
by (26).
Then, writing
Aj=Aj(n,zl,f),
we have
A2q_2 (n)
A2q_3 (n+l)
A2q+3 (n+l)
A 2q-4 (n+2)
Aq_ 1 (n+q-l)
Aq_ 2 (n+q)
(2.8)
H (n)=
q
A
q-1
(n+q-l) A
q-2(n+q)
A (n+2q-3)
0
Lemma 2.5
With z
n
i
n---
y’
and
v>0_
Aj (n+v,zl,zf’)
any integer,
k=0l ()
k
y (v- (k-l) n)
--(n+l)- k
Lemmas 2.4 and 2.5 are due to Noonan and Thomas
Aj_k (n+v+k, y, f)
[i0].
82
K. I. NOOR AND S. A. AL-BANY
Lemma 2.6
[ii].
Let N and D be anlytic in E, N(0)--D(O) and D maps E onto many sheeted region which is starlike with respect to the origin.
N(z)
implies Re
DZ--
3.
Then Re
>0
> 0.
MAIN RESULTS.
Let feB
THEOREM 3.1:
k(8)
0<8<1.
k>2,
Then
Lr(f)<C(k_ ,8) MI-8(r) (I)_
C(k,8)
where
is a constant depending on k,8
the closed curve
f(Izl=r<l)
L (f) denotes the length of
r
only.
and M(r)
max
PROOF: We have
zf’ z
L (f)
r
z=re
de,
0
2
If
0
_<
1_ 8
(zig 8 (z)h(z)[de,
using (1.5), where geR
k
and
Re h(z)>0.
M1-8(r)
Ig 8 (z) h(z)lde
0
<
MI-8(r)
12 I
0
<
r
r2n
Mi-8(r)
18g’ (zlg 8-1 (z)h(z)+g 8(z)h’(z)
0
ir0
0
1
12 ir
where
H(z)
g(z)
is defined as
Using Lemma 2.1, Lemma 2.3 (li),
k
8(y
Lr(f)<C(k,8)M l-8(r) (r)
(z)IdrdO
]2 I
0
r
8 (z)zh’ (z)
i
ldrde}
1g
0
in Lemma 2 2
Schwarz inequality and then Lemma 2.2 for
both general and speclal cases (k>2, k=2)
depending on k,8
g 8 (z)zh
8(z)h(z) Idrde+
0
0
(z)h(z)Idrde
0
0
2rr
g
g(z)
Idrde.
we have
+ 1)
0<8<1, C(k,8)
is a constant
only.
Corollary 3.1
For k=2, feB(8) and
(-) 28
i
Lr(f)<C(8)Mi-8(r)
THEOREM 3.2.
Let feB
k(8),
0<8<1,
la n !_<
Cl(8,k)
and f(z)
k>2_
C
1
z
+
l a z
n=2
Then for n>2
1)-1
1
.n
(8,k)M1-8(1 )
is a constant depending only upon k and
n
8
BAZILEVIC FUNCTIONS
PROOF:Since, with z=re
83
Cauchy’s theorem gives
n a
zf’ (z)e -ine de,
1
n
2r
n
0
then
nla n-<
[2 Izf’ (z)Id@
f
i
2zr
n
i
70
2r
1
Using theorem 3.1 and putting r=l
e r (f)
n
we obtain
n
the required result.
Corollary 3.2
When B=I,fET
k
and from theorem 3.2 we have
la n
This
<A(k)n
result was proved in
k/.2
A(k) being a constant depending on k only.
-
[6].
Corollary 3.3
For k=2,
fgB(8) and
la n I<A 1 (8)
M
I-B (i-
2B-I
I).
n
n
n>2
THEOREM 3. 3.
Let f be as defined in theorem 3.2.
Then,
for n>2,_
k
Ilan+ l I-la II
0(z)M
n
where 0(i)
PROOF:For
n>2_
and z=re
i0
used
0
12 Z-zlll
i
2r
z z
n+l
+ i)-2
eE, we have
I(n+l)z I lan+ 1 I-nla n I1<
2r
8(
__I)
.n
n
B.
depends only on k and
ZlgE
(i-
k>
n+l
i
II
zf’(z)
Id@
0
f (z)
g (z)h(z)
Id@,
2,
where we have
(1.5).
TakiDg M(r)=max f(z)
(n+l) z
I
an+l l-n
an II
and using
<
M I-8 (r)
n+l
2r
(2.1)
(k )
4 8 4
(2.4) and (2.5), we have
i
2r
2
1-r
2
[
2
0
.k+2
Is z(z)
[---
i)
Ih(z) lae,
where S
is a starlike function.
1
Schwarz inequality, together with Lemma 2.2
starlike functions
(k=2) and subordination for
[12] yields
+ i)-2
27r8(
4(
r)
MI
2r2
de)1/2" (2 -I+3r2)1/2
an+l l-nlanll < 2rn’+ ()
i@ 8(k+l-r2
l-re
l-r2
0
( + 1)-i
<__C(k,B)M I- (r) (r)
k
(n+l) z
1
whre C(k,B) is a constant depending only on k and 8.
we obtain the required result.
Choosing
z
I
r
n
__n_
n+l’
84
.
K.
NOOR and S. A. AL-BANY
Corollary 3.4
If 8=1,
fT k
and we obtain a known
k
llan+ll-lanll
O(1).n
[6] result, for k>3,
2
We now proceed to study the Hankel determinant
problem for the class
Bk(8)
THEOREM 3.4.
Let
8(+
feBk(8),
q
(n) of f be
Then
k
8( 7
+ 1)-1,
H (n)ffiO (1)M 1-8 (r)
q
PROOF: Since
the Hankel determinant H
0<8<1, k>2 and let
feBk(8)
defined as in definition 2.
q=l
i) q-q
q>2,
k>_l
2
we can write
1-8 (z) g 8 (z)h(z), Re h(z)>0, gR
k
any non-zero complex number and
Then for j>l,_ z
Let F(z)fzf’(z).
I
zf(z)--f
consider
Aj(n,Zl,F)
(2.7).
as defined by
2rn+J
2r
i0
Then
(Z-Zl)JF(z)e -i(n+j)@ d 01
1_.,
IA j (n,Zl,F)
zfre
0
1-8(z)gS(z)
I
n+j
h(z)IdO
0
<
Using (2.1),
MI-8 (r)
iZ_Zll ig
(2.4) and (2.5), we have
(2r22)
8(
Ml-B(r)--------(--4r)
(n,Zl,F)I<
IAj
--2rn+J
j
1-r
12n
k
S
lcz)l
(
1
+ )-J
IhCz)
Id0
0
Schwarz inequality together with subordination for starlike functions
k
+ i
[12] and (2.2) gives us, for
J>0,
8(
)k
IA j (n
where
+ l)-J
)8(
I<C(k
j)MI-8(r)(Tr
.....
’1
F)
z
C(k,8,J)
is a constant which depends
Applying lemma 2.5 and putting
for k > 2
2
Zl=
,
uponok
)e
P
io
j(n,e
n
k
f);O(1)Ml-8(r) n (
+ 1)-j-1
k, and J only,
We now estimate the rate of growth of H (n)
q
A
a’
For q=l, H (n)
(n) and
0
where 0(1) depends on
q
n
8 and J only.
(n+)
r-i
1
---’n
we have
BAZILEVIC FUNCTIONS
8(k
O(1)M I-8 n
H (n)
q
85
+ i)-i
q=l
For q>2, we use the Remark
due to Noonan and Thomas
in [i0] and we have
k
2
+ 1)q-q
H (n)
0(1)M i-8 (r)n
q
q_>2,
0<B<I,
and k >
2
This
(
completes the proof.
Corollary 3.5
B--l,
When
feT
and
k
q=l
H (n)=O (i)
q
k
+ 1)q-q 2
(-
q>2,
This result is known
[13].
Definition 3.1
A function f is called
{(l-s)
Re
_
-convex
zf’ (z)
+
if, for >0,
(zf’ (Z))
’We now prove the following
THEOREM 3.5.
Let feB k(B), k>2 and
k>Sq- i0.
f"(’z’)’} >
Then f is 1
B>I.
0
zeE.
convex for
[z[<ro,
where
r0
PROOF:Since
fKBk(B),
we have
fl-B(z)gB(z)
zf’(z)
_i (zf’ (z))’
h(z); Reh(z)>O,
1
zf’(z)
+ (i- )
’f’(z)-
--f(’z’)--
geRk,
from which it follows that
+ 1 zh’ (z)
g(z)
(8>I)
h----
Thus
Re
Since
1
zf’(z) ] >Re zg’(z)
(zf’(z))’ + (i- 1
[ F
zG’=geR k
implies
Re
GeVk,
Re
g(z)
h(-z)
(zG’ (z))
G’(z)
[14],
2
>l-kr+__r
l-r
(3.2)
2
[15] that
it is known
(z)l<__2,r_
(3.3)
(3.3), we have
Using (3.2) and
Re
81
we have from a known result
and for functions h of positive real part,
zh’
1 zh’ (z)
g
f
[i
(zf’(z))’ + (If (z)
-
zf (z)
f
>. 8(l-kr+r
8(l-r
2
)-2r
2
and this gives us the requaired result.
For k=2,
8>1, feB(8) is
i
convex for
Izl<r I
(8+1)
/-/’f
.
K.
86
I implies
8
(ii)
This result
fgT
NOOR AND S. A. AL-BANY
zI<r 2
and it is convex for
k
(iii) When k=2, and 6=1,
[(k+2)-
+4k
[6].
see
is known,
fgK and
it
IzI<2-.
is convex for
THEOREM 3.6.
Let
Let, for
GERk, k>2._
any positve integer,
1
2,3,...,n be
defined as
2
z
h(z)
G (t) dt
t
0
1
I +8)
(
Then h is
r
(zh’ (z))
h’(z)
PROOF:
Since GeR
+
1)
(3.4)
zG’ (z)
G(z)
[16] that
is known
zG’
IzI<r 0.
Let GeR k,
k>_2
I
i-
g(z)
and
II
z
(
--1
t
PROOF
[z
I
(
-E
0
0
given by (3.4)
Now
N’ (z)
8
valency
--
and
where
IzI<
r
r
0, where
i
2
G (t)dt
t
are defined as in
+ z
i
is given by
G (z)
(3.4).
N(z)
E
_l
which is (+ a -1)
valently
starlike
for
from theorem 3.6.
zG’ (z)
N’ (z)
for
1
G3(t)dt,
where D(z)
Thus,
4).
t
to
1
r
G(t)dt,
_i_ 2
G 8(t)dt
c
z
Izl<r O,
_
-2
Then g is starlike for
theorem 3.6.
Re
i
Izl<r 0 (kI
The (- i+)
from the argument principle.
THEOREM 3.7.
and
for
Re---->0
Hence h is convex and thus starlike in
foliows
where
/k 2 -4)
i
(k-
0
(I
it
k,
_lzl<ro,
valently starlike for
Izl<r O,
8Re
using
r
0
given by
zG’ (z)
eg(----
Re
have
>0, since GR k"
lemma 2.6,
z’(z)
(3.4), we
N(z)
for
D--- > 0,
Izl <
r O,
it follows that
proof.
and this completes the
functions.
result [17] for starlike
obtain Bernardi’s
(i) For k=2, 8 =1, we
=I/2, we obtain a result
(ii)Also, for k=2,
proved
in
[18].
BAZILEVIC FUNCTIONS
87
THEOREM 3.8.
Let
FeBk(B)
fB(z)
3.6.
1
z
zeE and let
1-1/
t
Then fEB(B) for
(i/)-2 8
F
IzI<ro,r 0
(t) dt, e and 8 defined (3.5) as in theorem
is given by (3.4).
PROOF:Let GER and let g be defined as in theorem 3.7.
k
for
where
0’
r0 is given by (3.4)
JzJ<r
- -
Then 8 is starlike
Now, from (3.5), we obtain
i
(i-)
B zf’ (z_)
fZ-8(z)gS(z
I
z
1
t
0
z
1
2
B
F (t)dt + z
1
2
t
rz
where D(z)
1
i
(8+-i)
is
F (z)
N(z)
D(z------
(t)]dt
C
GS(t)dt
t
l
valently starlike for
0
Also
D-’-(z)
Thus
Izl<r O,
s
FI-8
(B).
ce
using lemma 2.6, we obtain the desired result that fgB(B)
where r
is given by (3.4).
0
for
Corollar[ 3-8"
(i)
For k=.2, FEB(B)
zeE and it follows from theorem 3.8 that fB(B) for
(li) For k=2, 8=1 implies FeK and from theorem 3.8 it follows that f also
belongs to K for
zl <i.
(ill) Let B=I, then
to-convex for
Izl<
FeTk,
r0,
sult proved in [13]
for
and it follows from theorem 3.8 that f is close-
given by
@=
(3.4).
This is a generalization of a re-
1
_THEOREM. 3.9
Let
feBk(S).
O<el<e2_<2,
Then for z-re
o, f’(z)#O
f(z)
in E and
0<r<l, we have
2
(zf’ (z))
Re {
f (z)
PROOF: Sine
feBk(8)
zf’(z)
zi
de>-
Re h(z)>O,
geR k.
+ (B-l)
f(z)
l
8k.
we can write
fl-B(z) gS(z)h(z),
fl-B(z)gB(Z)hl(Z)
where
fl-S(z) (zT’ (z))8
where TeT
h(z)ffihSl(Z)
Re h (z)> 0
1
k
Therefore,
_(zf’ (z))
f’ (z)
(z)
(_zT’ (z))
+ (S-l) zf’
f(z) S --T"()
(3.5)
K.I. NOOR and S. A. AL-BANY
88
[6] for the class T k, we have by integrating both sides
(3.5) between O I, O 2,
Using a known result
of
0<01<02<2
Re{ (zf’ (z))
f’ (z)
01
+ (8-i) zf’_(z)
fz-
d0>-
kn
The following theorem shows the relationship between the classes
B(B).
More precisely it gives the necessary condition for
Bk(8)
fgBk(8)
and
to be
univalent.
THEOREM 3.10
Let feB (8).
Then f is univalent if
k
k_<
where
0<8<1._
PROOF:The proof follows immediately from Theorem 3.9 and the result of
Shell-Small [19]
REFERENCES
i.
BAZILEVIC, I.E.
equation, Math’.
2.
’THOMAS,
On a case of integrability of the Lowner-Kufarev
sb. 37(1955), 471-476 (Russian).
D.K. On Bazilevic functions, Trans. Amer. Math. Soc. 132(1968),
353-361.
3.
BRANNAN, D.A. On functions of bounded boundary rotation, Pr.oc. Edin.
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