Internat. J. Math. & Math. Scl.
VOL. 14 NO. 2 (1991) 283-288
SOME
283
BAZILEVI FUNCTIONS OF ORDER BETA
S.A. HALIM
Department of Mathematics and Computer Sciences
University of Wales
University College of Swansea
Singleton Park Swansea SA2, 8PP, U.K.
(Paper Received April 4, 1988 and in Revised form September 13, 1988)
Distortion theorems and coefficient estimates are obtained for a new class
ABSTRACT.
of
Bazilevi
functions.
Bazilevi
KEYWORDS AND PHRASES.
Functions, Functions Whose Derivative has Positive
Real Part, Coefficient Estimates.
1980 AMS Classification Code.
1.
Primary 30C45.
INTRODUCTION.
Let S be the class of normalized functions regular and univalent in the unlt disc
D
<
z
z
and S
>
and Re h(z)
showed
that
the
P(B), h(O)
P(O).
We write P
D.
8 for z
Bazllevlc [I]
Denote by P(), the
the subclass of starlike functions.
class of functions which are regular in D and such that for h
class
of
normalized
regular
f
functions
with
representation
=f p(t)
f(z)
>
when
>
S
0.
and p
,
S
Re
[3],
(1.2).
(I.1)
it
follows
such that
z f’(z)
fCz)-gCz)
Singh considered
Thus f
D forms a subclass of S.
P for z
We denote this class of
See also [2].
Then
there exists g
In
dt) =
,
O, g
functions by B().
Let
-I
g(t) = t
BI()
if
the
easily from (I.I)
for z
>
that f
B() if, and only if,
D
(1.2)
0.
subclass
Bl(e)
and only if, for
of B(s) obtained
>
0 and z D
by
taking g(z)
z in
284
S.A. HALIM
z
Re
I-
of
(z)
>
O.
(z)
We extend this class of functions as follows:
DEFINITION.
Let f be regular in D with
f(z)
Then if
>
+
z
0 and 0
z
Re
n=E2 anzn
B
<
I, f
-uf ’(Z)_ >
(1.3)
BI(,)
e
If, and only if, for z
D
(,.4)
B.
(z)
R, the class of functions whose derivative has real part [4].
BI(I,) was considered in [5]. Zamorskl [6] and Thomas [7] solved the coefficient
problem for f
B(), in the case when N is a positive Integer. In [7], sharp
We note that
BI(I,O)
distortion theorems were obtained for f
i5 to extend these results to the class
BI()
BI(,8).
>
for
0.
The object of this paper
The class
BI(,B)
has also recently
been considered in [8].
2. RESULTS.
Distortion Theorems
THEOREM I.
BI(,).
Let f
(f)
If(z)
Q2(r)
(r)
re
D, O
r
<
I,
Ql(r)
<
(ii) if 0
ra-1 Q2
Then for z
a
(l-r)(l-B)(1+r)
+
B)
(]f’(z) ’
r
If’(z)l
r
Q1
(r)
q2
(r)
=
(+r)(-B)
(l-r)
+ B)
and if
r
where
and
Q1
(r)
(l-r) (l-B)
(l+r)
+
B]
(-1
(x
(l+r) (l-B)
(-r)
+
)
SOME BAZILEVIC FUNCTIONS OF ORDER BETA
f
Equality holds in all cases for the function
f
af
(z)
t
defined by
((l+tei)(l-B)
(l-te 10)
’I
285
+
B)dt)
(2.1)
0 or
where
PROOF.
and it follows from (1.4) that
BI(,8)
(i) Since f
z
(l-[)p(z)
l-f ’(z)
f(z)
P.
D and p
for z
l-a
Thus
f(z)a
and since
l+r
l-r
p(z)l
$
a
t
a-I
for z
(p(t)(1-8) + 8)dt
(2.2)
D, (see eg. [9])
Q1 (r)
To obtain the left-hand inequality in (i), write
h(z)
’(z)
z
f(z)
Then (1.4) shows that h
(2.3)
I-4
p().
Thus from,
[5] (Theorem
with c=1-28 and
n=l), we
obtain
(l-r) (1-3)
+ I
(1+r)
(i+r)(1-1)
lh(z)l
(l-r)
+ S
(2.4)
Hence from (2.3) and (2.4) we have
d."
Writing
to
f(zl )a,
f(z)
(2.5)
(1+r)
it follows that since f is univalent, the llne segment
lles entirely in the image of D.
Let
be the pre-lmage of
from 0
then by (2.5)
286
S.A. HALIM
which is the left-hand inequality in
From (2.1) we have for z
re
i8
(2.6)
if 0
<
a
If a
)
I, (1) gives
I, the
follow
inequalities
at
(2.6),
from
once
If(z)ll-e
Q2
(2.7)
(r)
Applying (2.4) and (2.7) to (2.6) gives the required result.
f0
and In (II) for
f0
when 0
The following shows that as a
<
and for
For 0
r
<
1, let
Ql(r)
f
and
Q2
>
Ql(r) a~
(Ii)
Q2(r)
(Ill)
r
(1+r)2(I-)
1.
PROOF.
We prove (i), since (l i) and (111) are similar.
As c*+O
re
-2(l-8)log(1-r)
r
(l-r)
COROLLARY.
Suppose that f(z)
for z e D, then
are asymptotic to the
(r) be defined as in Theorem I.
(l-r) 2(I-)
Ql(r)~ Qz(r)~
I.
0 (see eg. [9]).
as a+O
(I)
Equallty is attalned in
when a
0 the bounds In Theorem
distortion theorems for starlike functions of order 8
THEOREM 2.
and
-a
1-a
ql (r)
and (i) for
(2.4)
2(i-)
Then
SOME BAZILEVIC FUNCTIONS OF ORDER BETA
287
PROOF.
Let a
Let L
>
O, and
denote
a
be
the
0
from
L
and
to
closest
its
to
the
origin.
pre-lmage
Q2(r) =
II
(i) gives
once, Theorem
llne
straight
f(D)
of
boundary
the
on
point
Thus Theorem 2 (li) gives
as o.
O.
for n
O.
3. A COEFFICIENT THEOREM
Notation:
n=E0 n z
n
Let f e
THEOREM 3.
n
((
gnZ
0
z +
Then
(i) f(z) (<
Inl I%1
means
and be given by (1.3) where N is a positive integer.
BI(,
D,
Suppose also that for z
fo(Z)
n
n
0= YnZ
is given by (2.1).
fo(Z)
where
f0(z),
and
N
n
PROOF.
z
Write
p(z)=
+
k=Zl pkzk
f(z)N
=’"
n
%znl
jm
m
Then (2.2) gives
=: foz
t
[S(l.-Ig)zN
a-1
[[I
+
kl Pk t
+ (I-13) k=|
k
’:k z
(1+8) + ]dt
k4
+
k+N
k
z
1 +
(I-B)
N
kl
Pk z
)
(k+S)
Thus
f(z)
z(:+
(! -8)
N
gNzN
in
D.
S.A. HALIM
288
(1-)
z(l+
f(z)
Pkl
P, we have
and s l,ce p
f0(z)
k
PkZ
k=l
N
(li) Puttlng a
2 [6]. Hence
N
(k)
=
in
z +
nZ2 Tn zn
2z
(( z[l+
z[l
v
+-
2(I-8)N n=El(R)
n
(n
N
n
(n
+z )I
1"
n v
(v)
fo(Z).
v
2(I-8)
N
Dn (V)zn
(v
(log n) -I as n
IN
(N)
n
(log n)
N
+W)
n
Thomas [7] proved that D
n
N
(2.1), we have
n
Let
k
0, I, 2, 3
....
).
and so this gives
N-1
as n
-.
REFERENCES
I.
BAZILEVIC,
I.E.,
On a case of
the Loewner-
integrabillty in quadratures of
5.
Kufarev equation, Mat. Sb. 37(79)(1955), 471-476.
THOMAS, D.K., On Bazilevi functions, Ma.th Z. 109(1969), 344-348.
SINGH, R., On Bazilevi functions, Proc. Amer. Mat. Soc. 38(1973), 261-271.
Functions whose derivative has positive real part, Trans.
MA(GREGOR, T.H.,
Amer. Math. Soc. 104 (1962), 532-537.
TONTI, NORMAN E. & TRAHAN, DONALD H., Analytic fundtlons whose real parts are
6.
ZAMORSKI, J.,
2.
3.
4.
bounded below, Math. Z. 115(1970), 252-258.
Bazilevi
On
Schllcht functions, Ann. Polon. Math. 12 (1962), 83-
90.
7.
THOMAS, D.K.
Math.
On a subclass of
GOOEMAN,
functions,
Internat. J. Math. and
Scl.,, Vol.8, No. 4 (1985), 799-783.
HAL]M, ABDUL S.,
order 8.. To appear.
8.
9.
Bazilevi
A.W.,
On
Univalent
the
coefflclents
of
Vol.l
.,
functions,
some
Mariner
Bazilevl
functions
Publishing
Co.,
of
Tampa
Florida, 1983.
I0. POMMERENKE, Ch.,
Univalent functions, Vandenhoeck and Ruprecht,
Gttlngen, 1975.