# BAZILEVI( ```Internat. J. Math. &amp; Math. Sci.
Vol. 8 No. 4 (1985) 779-783
779
ON A SUBCLASS OF
BAZILEVI( FUNCTIONS
D. K. THOMAS
Department of Mathematics and Computer Science
University College of Swansea
Singleton Park
Swansea SA2 8PP
Wales, U. K.
B()
Let
ABSTRACT.
v
be the class of normalised Bazilevic functions of type
with respect to the starlike function
g(z)
BI()
Let
g.
be the subclass of
0
B()
when
Distortion theorems and coefficient estimates are obtained for functions
z.
BI().
belonging to
KEY WORDS AND PHRASES. Bazievic functions, subclasses
tive has positive heal part, cose-to-convex functions,
of S, functions whose derivacoefficient and length-area
estimates.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
30C45.
INTRODUCTION.
Let
be the class of normalised functions
S
IzI&lt;1}.
{z
D
in the unit disc
S*
Let
D
f, regular in
tions
f(z)
for
g
S*
contains both
P
p
S*
principal values).
a
S
be the subclass of
and satisfy there the conditions
v
Bazilevic [i] showed that if
which are regular and univalent
consisting of
P, the class of functions which are reguP.
0 for p
p(O)
I, Re p(z)
functions which are starlike, and denote by
lar in
f
are real numbers, with
and
rz
[(a+i)
and
0
p(t)g(t)
tiB-1 dt] I/a+i
D, also form a subclass of
z
(I.I)
S
(Powers in (I.I) are
and the class of close-to-convex functions.
When
O,
B
we write
B(a,B)
B(a).
B(a,), which
denoted by
Zamorski  and the author
f e B(I/N),
N
a positive integer,
and more recently Leach  has shown that the conjecture is true for
f
B(a),
I.
a
Si,gh  considered the subclass
like function
a 2, a
O, then func-
D, and having the representation
 gave proofs of the Bieberbach conjecture for
0
a &gt;
3,
and
g(z)
z
BI()
of
B(), obtained by taking the
star-
and gave sharp estimates for the modules of the coefficients
a4, where for
z
D,
D.K. THOMAS
780
z
[z)
a z
n
n=2
We note that
BI(1)
Re f’(z)
for
0
n
S
is the subclass of
[1.z)
which consists of functions
f
for which
D .
z
sharp estimates for the coefficients
a
in (2)
n
BI(I/N)
f e
when
Bl(a)
f
In this paper, we shall obtain some distortion theorems for
N
and give
is a positive
integer.
2.
DISTORTION THEOREMS.
THEOREM I.
f e
(i)
Q2(r)
(ii)
and if
1/a
a-I
If(z)I
_&lt;
Then with
z
re
i0
0 N r
I,
Ql(r) I/,
_&lt;
a-&lt; I,
I-
0
If
r
and be given by (1.2).
Bl(a)
Let
l-r &lt;
a
Q2 (r)
ra-IQ
if ,(z)l
I+--
l+r
1-r
a
(r)
a e
1-a
1-e
r
a-I
l-r &lt;
a
Q1 (r)
re-IQ 2(r)
f’(z)l
i+---{
a
l+r
I---
where
Ql(r)
a
Q2(r)
a
I
0
rOa-I (t_0)v- l+ do,
and
(o)do.
froa-1
0
Equality holds in all cases for the function
f(z)
0
where
PROOF.
(i)
or
z
Taking
D
and
Ip(z)I
and since
defined by
ta_l.l+tei)dt)i/a
(l_tei
fz
0
(2 I)
n.
8
equation
for
(a
f
_
0
P.
p
&lt;_
l+r
z
1-a
-=
g(z)
and
f’(z)
in (I.I), it follows that
z
f(z)
l-a
f
satisfies the
(2.2)
p(z)
Thus
f(z)a
a f
z
ta-lp (t)dt,
0
for
z e D
, we have
at once
If(z)I
&lt;
Ql(r) I/a.
Re p(z) &gt; 0
To obtain the left-hand inequality in (i), we observe that, since
for
z
D,
Re p(z) -&gt;
1-r
r
dz
Now let
Zl,
[Zll
r
,
and so from (2.2)
]a
&gt;_ ar
a-
be chosen so that
l-r
(2.3)
(]-\$r)
[f(zl)a[
-&lt;
[f(z)a[
for all
z
with
[z[
r.
v
781
SUBCLASS OF BAZILEVIC FUNCTIONS
w
f(zl)a
fl(z )(
w
Then, writing
it follows that the line segment
D.
L
Let
from
0
w
to
be the pre-image of
%
then
The proof follows at once from (2.2) and (i) on noting that for
p
P,
lies entirely in the image of
by (2.3) we have
dw
[dzl
If(zl)la fxldwl fL ]zl
(1.1.1)d
fr pa-1
a
0
Q2(r),
p
which is the left-hand inequality in (i).
(ii)
1-r
1+r .
1+-’-’- lp(z)l
Equality is attained in (i) for
and in (ii) for
f0
f
when
o
0
and for
a e 1.
when
We remark that as
0, the results of Theorem
should in some way correspond
(univalent) functions .
to the classical distortion theorems for regular starlike
The following shows that the bounds in Theorem
0
tortion theorems as
THEOREM 2.
are asymptotic to the classical dis-
Let
Q1(r)
(i)
Ql(r)
Q2(r)
and
I.
be defined as in Theorem
Then for
0
r @
I,
as
(ii)
(iii)
’2 r
I/a
r
(l_r)
i/a
r
(l-r)
2
I.
Q2(r)
Ql(r)
2’
PROOF.
QI(r)I/
(
fr
0
p
-I
I/a
11__+_ )do
l_2ar-alog (l_r)) I/
r
r (l+2ar
-
re
a- O,
As
We prove (i), since (ii) and (iii) are similar.
fr
0
i_ do
-21og(l-r)
r
(l-r)
f(z)
Suppose that
COROLLARY.
lwl
Let
PROOF.
Let
L
lwl
w
Q2(1)
for
If(z)
L
as
for
at least once
a
then
as
0.
a
f(D)
be a point on the boundary of
e n D.
z e
D
z
I/a
denote the straight line from
sects
3.
O, and
a
w
0
(i) gives
Izl
lw]
r, for each
Q2(r)
I/a
and so
0 (from Theorem 2 (ii)).
A COEFFICIENT THEOREM.
NOTATION.
nZ=O anZ
n
nZ--O BnZ
n
means
l=.l -&lt; lnl
closest to the orgin.
its pre-mage in
L
and
w
to
Since the circle
Theorem
2
for
n
-&gt; O.
0
r
[w[
D
Then
inter-
Q2(1)
l/a
D.K. THOMAS
782
Let
THEOREM 3.
f
BI(I/N)
E
Suppose also that for
N
with
fo
fo(Z),
()N (N) (log
f(z)
(i)
(ii)
PROOF
(2.1).
is given by
Yn
z
+ n=2
as
n
Yn z
n
Then
n)
N-I
.
lenl -&lt; IBnl
We first note that if
(i)
(1.2).
D,
z
fo(Z)
where
a positive integer, and be given by
Z
(n=l
enZ
then for
1,2
m
8 n z n)m
(nZ-I
To see this, let
anzn) m nZ=O An(m) z n
(n=IZ
(nZ=O 8n zn)m
and
Bn(m)
nZ=O
z
so that
A(k)
n
A(k_l) a n-
=I
We now use induction on
IA(1)I
2
n
n
1,2
k
1,2
A(j+I)
n
and
,j.
(I)
’IA(k)
n
n -&gt;
Then [or
&lt;
1,2,...
n
n
Z__I
-&lt; B &quot;k’(
n
IA(k)
n
Suppose now that
n
IA (j) II an_
I
n
B
Bn
B(k_l)
=I
to show that for
k
lanl
D
n
B(k)
n
B
(j)
B n-
n
B
(j+l)
n
Thus (i) now follows at once, since from (22) we can write
f(z)
where
+
p(z)
pk z
kE__l
k
z[l+
When
2
Pk zk
kZ__l
z
N
k +I/N
[pk[
and since
f(z)
(ii)
I+
z
 we have
k
&lt; 2
]N
k
k+[/N
n
z[ i+ 2
f0 (z)
I/N, (2.1) gives
e
f0(z)
+
z
o(N) (.)’ (nl n+l/N
nE__2 Yn z
n
n
I n\$1/N ]N
z
z
n
Now trivially,
n
n
(nE--1
n
4-)
&lt;&lt;
n
E
n/l IN
&lt;&lt;
(nil
Write these three series as
Z
n=
Then
n
C(9)z
n
n=
n
D()z
n
E
E ())z
n=
n
and
n
l
n=9
E(9)z
n
n
z
)
(n 0 _.__z
n+l
respectively.
Clearly for
&lt; B (k)
n
for
v
SUBCLASS OF BAZILEVIC FUNCTIONS
Now a result of Littlewood [8, p. 193], states that if
783
is a flxed positive integer
and
(nE=O
n()
then
(log
n
n)-I
n
z
)
.
as n
n() z
n=0
n
Thus
E(V)
n
n-v
n=
C(V)zn
n
()
X(log
n
n)
Also
C
z
n=O
-I
D
()
(log
n
n
n)
=.
+
and so
-J
0 (j) (-I) n
(j)
()
n
as
n
and so
N
Yn
n
n
-I
v’--log
n
n
Thus
as
v\$O (N
()v D () ()()(log n)N-I
n
as n
REFERENCES
v
I.
BAZILEVIC, I. E., On a case of integrability in quadratures of the Loewner-Kurafew
equation, Mat. Sb. 37 (79), 471-476. (Russian) MRIT, #356.
2.
ZAMORSKI, J., On Bazilevic schlicht functions, Ann. Polon. Math. 12 (1962), 83-90.
3.
THOMAS, D. K., On Bazilevic functions, Math. Z. 109 (1969), 344-348.
LEACH, R. J., The coefficient problem for Bazilevic functions, Houston J. Math. 6
(1980), 543-547.
4.
5.
6.
7.
8.
v
SINGH, R., On Bazilevic functions, Proc. Amer. Math. Soc. 38 (1973), 261-271.
MacGREGOR, T. H., Functions whose derivative has positive real part, Trans. Amer.
Math. Soc. 104 (1962), 532-537.
POMMERENKE, Ch, Univalent Functions, Vandenhoeck and Ruprecht, Gttingen, 1975.
LITTLEWOOD, J. E., Theory of functions, Oxford, 1944.
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