785
Internat. J. Math. & Math. Sci.
Vol. 8 No. 4 (1985) 785-793
GROWTH RESULTS FOR A SUBCLASS
OF BAZILEVI FUNCTIONS
RABHA MD. EL-ASHWAH and D. K. THOMAS
Department of Mathematics and Computer Science
University College of Swansea
Singleton Park
SWANSEA SA2 8PP
South Wales, U. K.
(Received March 29, 1985)
B()
0, let
For
ABSTRACT.
Choosing the associated starlike function
defined in the unit disc.
a proper subclass
BI()
be the class of regular normalized
B().
of
B(),
For
g(z)
z
functions
gives
correct growth estimates in terms of the
Several results in this direction are given for
area function are unknown.
KEY WORDS AND PHRASES. Bazilevic functions, subclasses
tive has positive real part, close-to-convex functions,
estimates.
1980 AMS SUBJECT CLASSIFICATION CODE. 30C45.
1.
Bazilevi
BI(1/2).
,
functions whose derivaof
coefficient and length-area
INTRODUCTION.
Let
S
be the class of regular, normalized, univalent functions with power series
expansion
f(z)
for
D
z
where
R,
Denote
S*,
D
z
zl
K
and
B()
vative has positive real part
z
+
B(a),
0
that, there exist
S*
such that for
B()
f(z),
D
which is the image of
and
A(r)
with
z
i0
0
and
B(1)
M(r)
max
best possible when
f
regular and normalized in
D, such
(1.2)
Let
C(r)
f(z),
L(r)
K
under the mapping
w
C(r)
For
f e
r
I,
denote the closed curve
be the length of
S*,
C(r)
it was shown [7] that,
I,
r
e(r)
where
p.221], closeFollowing [13] we define
O.
l-g (z)a
the area enclosed by the curve
re
which are functions whose deri-
D,
z e
zf’(z)
S*
S
[13] respectively.
to be the class of functions
g e
(i.I)
starlike with respect to the orgin [9
f(z)
g(z)
n
I}.
[8],
Rc
Then if
a z
n
the subclasses of
to-convex [6] and Bazilevic of type
f
nE__2
f(z)l,
f
is
0(I) (M(r) log
i_-)
as
(I.3)
and Hayman [4] gave an example to show that this estimate is
bounded.
In [14] this result was extended to starlike func-
786
R.M. EL-ASHWAH AND D.K. THOMAS
A(r)
tions with
S*,
f
A modification of this method also shows
A constant.
0(1) /A(r)(log _J__l
e(r)
as
l-r
r-
Thomas [14] also showed that (1.3) holds for the class
0
[13].
a _<
for
.that
I.
(I 4)
K
B(a),
and for the class
It is apparently an open question that (1.4) is valid for
K
f
or
B(a).
Pommerenke [II] showed that if
nla n
C
where
S*
f
then for
C/A(I-
-)’
C M(I-
In [12] the subclass
g(z) _= z
and a
a2, a3,
a
n
of B(a)
Bl(a
B(a)
by showing that
I).
n
The question as to whether (1.5) is valid for
which
(1.5)
is constant, and Noor [lO] extended this to
nla n
2
n
(I 6)
K
f
B(a)
or
is also apparently open.
consisting of those functions in
B(a)
for
was considered and sharp estimates for the modules of the coefficients
In [15] Thomas gave sharp estimates for the coefficients
were given.
4
I/N,
in (i.i) when a
N
a positive integer
In this paper we shall be concerned with the class
BI(1/2)
and will use the method
of Clunie and Keogh [I] to establish (1.5) and hence (1.6) and the method of Thomas [14]
(1.4) and hence (1.3).
to prove
The methods will in fact give results which are
stronger for this subclass.
2.
STATEMENTS OF MAIN RESULTS.
THEOREM 1.
Let
(i)
nla n
(ii)
L(r)
BI(1/2)
f
and be given by (i.i).
-I),
n
-< o(I) + 0(i) /A(I
0(i) (A(r) log
1r
Then
as
n
,
as
r
I.
We shall need the following:
LEgIA
F(z) 2
I.
Let
BI(1/2)
f
2
f(z ).
and be given by (I.i).
For
pe
z
i0
0
-<
_<
2(-2) 2
f2
0
fr iF,(z)12
0
f2
fr
0
0
<_
f(z
4___ [15]
A(r 2 f)
I,
r
A(r,F)
Now
F
Then
A(r,F)
PROOF.
Define the function
pd0d0
Izf’(z2)i2 0d0d0.
f (z 2
h
and so using (I.I) we have
(-2)
A(r,F)
_<
4
(_2)2
2 (-2) 2
f2
fr
0
0
n =[
(z2)]2
nla n ]2r 4
A(r2,f).
2 (r-2)
zf,
od0dO
(where
lal
)
in
D
by
GROWTH RESULTS FOR A SUBCLASS OF BAZILEVIC FUNCTIONS
LEM 2.
(i)
S
the map
r
A(r)/(l+r)2r 2
(l-r) b
5__i_2_ A(r)
A(/r)
(ii)
PROOF.
For
for
r
0
is decreasing on the
787
(0, I),
interva|
I.
r
Since
f2
0
fr f’(z)[ 2
f2
0
]zf,(z)[2d 0
A(r)
0dod0,
0
we have
rA’(r)
f2rr
fr[f,(z)[[zf"(z
1
0
0
<- 2
The classical distortion theorem for
4r(r+2)l_r f02 irlf,
120
rA’ (r) -<
(z)
2
r2+4r+1
2A(r)
f2
0
+ 2
r
f’ (z)
12
ododO.
[3,p.5] gives
S
f
odpdO
0dod0 + 2
le0dod00
f2n0 /rlf’
(z)
}.
Thus
d
d- (logA(r)) -<
d
rr
(log
r2 (l+r) 2
(l-r) b
and part (i) of Lemma 2 is now obvious.
Part (ii) follows immediately.
PROOF OF THEOREM I.
(i)
BI(1/2)
f
Since
we can write from
f(z2) 1/2 h(z 2)
zf’(z 2)
where
Re h(z)
O, for
(1.2)
(2.1)
D.
z
l+w(z2),
h(z 2)
Set
l-w(z 2
w(z 2)
where
Then with
is
lw(z2)I
regular,
f(z2) 1/2
F(z)
D, w(O)
in
0
and
w(z)
nZ=l
w z
n
n
and
F(z)
z
+
nl=2b2n_l z
2n-I
(zf’(z 2) + F(z)) w(z 2)
(2.1) gives
zf’(z 2)
F(z).
Thus
{2z +
kl=2
(ka
k
Equating coefficients of
na
n
b
2 n-I
2
Wn_
+
b2k_l)Z
z
2n-I
+ (2a
2
2k-1
in
(2.2), we
+
b
Wn_ 2
3
This means that the coefficient combination
depends only on the coefficient combinations
the right-hand side.
{2z
+2
Ilence, for
(kak+b 2k-1 )z2k-1}w(z 2)
2k-1
kZ=2 (kak b2k_l) z
w(z
2
n
find that for
+
+ [(n-l)a
nan b2n-1
[2a
2
n
n-I
(2.2)
-> 2
+
b2n-3 ]w
on the left hand side of (2.2)
+ b 3],...,[(n-1)an_l +
b2n_3]
on
we can write
n
k---g2 (kak-b2k-1 )z 2k-l+
k=n+l ckz
2k-1
(2.3)
R.M. EL-ASHWAH AND D.K. THOMAS
788
Izl
Squaring the moduli of both sides of (2.3) and integrating round
say.
lw(z2)l
obtain, using the fact that
n
k $2
for
kak b2k_112 r4k-2
D,
z
k=Zn+! Ck 12
+
we
r
4k-2
+nk=2 kak + b2k-112
4
letting
<
r,
1, we have
r
n
kZ=2 Ika k b2k_112
n-1
2
Z__ Ika k
4 +
+
b2k_iI2.
Thus
Ina n
(where
Ibl I=
fall
b
-< 4 + n-I
k_Z_2
2n_112
Ib2k_l I,
klakl
(2.4)
I).
Hence
Inan -b2n_112
-4n+1
_<4r
0
for
r < I, and so
Ina n b2n-I
Lemma
=Ii klakl2)(k--El
(k klak 12 r4k)(k= (2k-l)Ib2k_l12 r4k-2)
4
2
<4 r
/(A(r 2 f))
-4n+l
/(A(rF)).
now gives
4
Inan b2n-112
and choosing
r2
72(-2)
r
-4n+
IA(r 2 ,f).
we obtain
n
Inan b2n-I 12
C A(I
),
where
C
BI(),
Finally, it is easy to see that from the definition of
n
2,
2
Ib2n_ll
(2.5)
is constant.
and so [8] for
F e R
Thus (2.5) gives
-Z"
nlanl
o(I) + 0(I)
/(A(I
))
=.
n
as
This proves part (i) of Theorem I.
(ii)
Since
L(r)
12
Izf’(z)Id8
0
and
e(r2
f2
iz2f
0
< r
/27
[r
0
0
ll(r)
+
<
(zm)id 8
IF ,(z)
12(r)
h(z 2)
F(z) 2
f(z 2)
y2 IF(z)
h(z 2)
0
Idod@
+ 2r
(2.1) gives
Id8
/2z ir iF(z
0
0
h’(z 2)
lOdod@
(z=oe
say.
Again using (2.1) we have
Ii(r)
where
h(z)
+
r
/r /2 lh(z 2) 12d8do
0
L
n=l
0
hn zn
for
2r
I r (1 +
0
z e D, and since
lhnl
n_E_ lhnl204n)dO
_<
2
for
n
_>
[2 p. I0],
GROWTH RESULTS FOR A SUBCLASS OF BAZILEVIC FUNCTIONS
_<
I l(r)
fr(l
2nr
nl 04n)d0
+ 4
0
0(I)
789
log(z
as
I.
r
Also
12(r)
2r
0
2r
where
(t)
z
-
h(z2
increases and
Therefore
h’ (z 2)
and so
[h’(z2)[
f
say.
D
we may write
2
+
0
f2
0
1
i
<
f2w
[h’ (z 2) [od0d0)
0
0
(Jl (r)) (J2(r))
0, for
[0d0d0) (fr
h’(z 2)
iF(z)2
0
Re h(z 2)
Since
27
(/r
z2 e
-it
z2
-it
e
(2.6)
-it
z2 e
(I
f2
0
d(t),
I. [5 p. 68].
d(t)
f2
0
e
-it) 2
d(t),
d(t)
[I-
e-itl2
z2
Thus
fr f2r
Jl (r)
Since
F
0
0
2
F(z)
f
odOdla (t)do
-it
z2 e
0
is an odd function we may write
m(z)
z2 e
Z
-it
S2n-1 (t)z
n=
2n-I
(2.7)
and so
2".
F(z)
2
0
0
z2 e
We now show that for
where
F(z)
2
04n-2 f2
IS 2n-I (t) 12
0
n=l
d(t)
I,
n
02
d0d(t)
-it
S2n_l (t) 12
n__El b2n_iZ
2n-I
for
d(t)
_<
n
jlI= lajl Ib2j_l
27
D.
z
From (2.7) we have
S
l
n=l
and so for
2 n-1
(t) z
2n-i
(nZ=l
b
2n-I
z
2n-I
(nZ=0
e
-int
k_Zl b2k_l
e
-i(n-k) t
Now (2.1) gives
(nZ_l
h
0
2n)
I,
n
S2n_l(t)
(where
z
1)
na z
n
2n-1
and so for
(n__Z b2n_l z
2n-1
n
n
Z
b
h
2v-I n-v
It is easy to see from (2.6) that, for
k
l,
na
n
v=l
)(
n=ZO hnZ 2n)
(2.8)
R.M. EL-ASHWAH AND D.K. THOMAS
790
and so
na
f27
0
,
hk
n
e
-kit
d(t).
02n
b2 ,-I
e
-i(n-’)t
(2.9)
Now
S2n_l(t) S2n_l(t)
IS2n_l(t)l
k
n
2 Re
iS 2n-I (t)l 2
b
2j-I 2k-I
e
n
-i(k-j)t
Z__ [b2_i[2
Therefore, using (2.9)
where we have used (2.8).
f2
0
jZ__
n
f2e-i(k-j)t d(t)
jl kE__l b2j ib2k_l 0
2 Re
d(t)
=I Ib 2-I
Z
[2
b2-I
RejZ__ b2j_l
2
n
2
n
jE=I jaj b2j_l
<- 2 Re
n
jE__Ij lajl Ib2j_l I.
2
Thus
Jl(r)
<_
4
r n=l
<_
47
fr
0
<_ 4n
fr
4n-I
O
4
is
6
r
increasing on
A(0, f) do, by lemma I.
-$
(0, i)
4
(-2)7 A(r,f)
Now
fr
0
J2(r
ih ,(z2)
do
r_
(-2) A(r,f) f0 I-0
Jl (r)
and since
(2j-l)
O
A( 0 F) 1/2
1-0
(-2)/2
A(0)
n
n
j=l
f)1/2
A(O
0
Since
(j=l jlaj] ]b 2 j-I [)do
n=l 0
n=|
0
< 4
4n-I
0
f2
0
).
Ih’ (z2)lododO
Re h(z 2)
_<
log(
for
0 _< r < 1,
1-r
J2 (r)
_<
r
2
0
f
-< 4 f r
0
since
h
is harmonic in
27 Re h(z 2)
0
1_
do
l_pU
D
Thus
J2(r)
_<
47 log
l-r"
pdOdp
Ib2_l
o
4]-2
1/2
do
CROWTH RESUI.TS FOR A SUBCLA3S OF BAZILEVIC FUNCTIONS
Combining the estimates for
Jl(r)
12(r)
J2(r)
and
/(A(r))
0(i)
COROLLARY i.
Let
f
nlanl..
(i)
(ii)
nla n
(iii)
L(r)
then as
(r)
,
i, P(r)
lanlr
n=Ig
A(r)k(Zog(1--r))-
PROOF
From (2 4) and the fact that
0
i,
r
n
i)
(log
n
-< o(1) + 0(i) A(I
0(I)
n
Ina n
b
_I
r
n)
r
as
Ib2k_l
2n-I
12
<-
2
2k--I
for
k
I,
it follows that, for
klakl
n-Iz
k=l 2k-I
_< 8
4r-np(r).
<_
Choosing
r
as
!)1/2
n
o(I) + 0(i) e(1
0 <_ r
where for
BI(1/2)
shows that
log
and the result is proved.
791
(i) follows.
(ii) follows since
P(r)
(n=l nlan]2rn)1/2(n=l rn)1/2"n
_<
It follows trivially from(2.1) that
e(r)
0(I) M(r)
log
-r
as
r
and so (iii) follows at once on noting that
M(r) 2
_<
A(/r)_
log.l-r
and on using lemma 2.
REMARK.
In view of Theorem
and Corollary i, it is possible that for
following conjectures are valid
2 2
0(i) M(I
(i) n a
n
(ii)
(iii)
n
4 4
an
e(r)
0(I) A(I
0(1) (A(r))
I)
as
n
,
)
as
n
=o,
as
r
I.
log(r)
f
BI(1/2)
the
We note that (ii) is stronger that (i) and that we have proved (ii) and (iii) in the
A(r)
case when
is finite.
The following extensions to Theorem
3.
support the above conjectures.
INTEGRAL MEANS.
For
f
regular in
D
define for
ll(r,f
real,
if(rei0)lid8"
f2
0
792
R.M. EL-ASHWAH AND D.K. THOMAS
For
THEOREM 2.
BI(1/2)
f e
I,
)‘
and
l)‘(r2,zf ’)
where
PROOF.
dp,
(t_p))’
o
)‘
is a constant depending only on
C()‘)
)‘12
M(0)
fr
<
(2.1) gives
f2
iz2f (z2)I)‘ d@
0
l)‘(r2,zf,
2n
izf,(z2)l)‘-11F#(z)
2
+
Jl’(r)
J2’(r)
(z2)Idedp
h,(z2)ipd6dp
say.
I,
)‘
Now for
jzf,(z2)j)‘-llF(z)
h
Jl(r)
-< Ir
fO
r
(M(p2,zf’)) -I
From the proof of Theorem
f2
0
02n IF’(z) h(z2)IdOp
(ii), we have with
iF,(z) h(z2)Id0
_<
0(i)
z
do.
pe
i.
as p
I__
I-;
Also (2.1) and the distortion theorem for functions of positive real part [4] gives
M(p 2 zf’) < 2p
M(o,F)
I_ 0
Thus
Jl’(r)
and since
F(z) 2
<
C()‘)
h
is
J2(r)
Combining the results for
where
PROOF.
Thus
Since
fr
0
M(p,f)
f
_
f e B
Jl (r)
BI(1/2),
l(r 2,
f)
f0(z 2)
f2n
2 0
then
C()
do
(l_p)
(l_p)
2
x
harmonic, we have
M(P-’-F)%(I-o) pldp
r
C(I)
0
C(%)
fr
0
J2’(r)
and
then for
1
I(1/2),
-I
%-i
Similarly, using the fact that
Let
M(p,F)
f(z2),
J1’(r)
THEOREM 3.
fr
0
0
_<
we obtain the result
I,
r
I
/2
M(p,f)
dp
(l-p)>,
l(r2,f 0)
z
dt)2
(f0 l+t----l-t
F e R.
if(z2)id0 =-#
,2n
0
r2 +
n
IF(z)l
2
dO
Ib2n_li2
r
4n-2
GROWTH RESULTS FOR A SUBCLASS OF BAZILEVIC FUNCTIONS
and since for
I,
n
ll(r2,f)
ib2n_l
r
+ 4
2
2n-I
nl=2
r
4n-2
(2n-i)
ad the theorem is proved.
REFEI{ENCES
-
793
2
/0 If0 (z -)Id0
|.
CLUNIE, J. and KEOGH, F. R., On starlike and schlicht functions, J. London Math.
Soc. 35 (1960), 229-236.
2
GOOD}LAN
].
HAY}LAN, W. K., #lultivalent functions, (Cambridge University Press 1958).
4.
HAYLAN, W. K., On functions
(1961), 5-48.
5.
HEINS, M., Selected topics in the classical theory of functions of a complex variable, (Holt, Reinhart, and Winston) (1962).
KAPLAN, W., C]ose-to-convex schlichtfunctions, Mic_____h.ian Math. J. (|95)(52). 169185.
6.
A
W.
Univalent functions, Volume 1
1983, Mariner Press, Florida
with positive real part, J. London Math.
Soc. 36
7.
KEOGH, F. R., Some theorems on conformal mapping of bounded star-shaped domains,
Proc. London Math. Soc. (3) 9 (1959).
8.
MacGREGOR, T. H., Functions whose derivative has a positive real part, Trans Amer.
Ith. Soc. 104 (1962), 532-537.
NEHARI, Z. Conformal Mapping, McGraw Hill, 1952.
9.
10.
NOOR, I. K., Ph.D. thesis (1972), University College of Swansea, Wales.
ii.
POMMERENKE, Ch., On Starlike and convex functions, J. London Math. Soc. 37 (1962),
209-224
12.
SINGH, R., On Bazilevic functions, Proc. Amer. Math. Soc. 3__8., No. 2 (1973).
THOS, D. K., On Bazilevic functions, Trans. Amer. Math. Soc. 132, No. 2 (1968)
353-361.
THOMAS, D.K., On Starlike and close-to-convex univalent functions, J. London Math.
Soc. 42(1967), 427-435.
13.
14.
15.
THOFS, D. K., On
a subclass of Bazilevic functions To appear.