I ntern. J. Math. & Math. Sci.
Vol. 5 No. 2 (1982)411-415
411
BAZILEVIC FUNCTIONS OF TYPE
K. INAYAT NOOR
Mathematics Department
Islamia University
Bahawalpur, Pakistan
(Received March 12, 1980 and in revised form December 7, 1980)
ABSTRACT.
type
B
In this paper, a new coefficient result for the Bazilevi functions of
is obtained.
KEY WORDS AND PHRASES. Bazrlevi functions, Stake, Coefficients, CZose-to-convex, Univalent functions.
980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
30C45, 30C50.
INTRODUCTION.
Let S denote the class of functions f which are analytic and univalent in
E
{z:
Izl
< I} and which satisfy f(O)
consisting of starlike functions.
0 and f’(0)
For z
-Sai
={
where a is real,
valent [i].
B
With a
8
l+a
2
I
Let S* be the class
Bazilevi@ [I] introduced a class of analytic
functions f defined by the following relation.
f(z)
i.
z
(H()
ai)
i+a
2
E, let
E
B/l +
-i
g
l+ai
a
()d
(i.i)
0
> 0, ReH(z) > 0 and g
E
S*.
Such functions, he showed, are uni-
0 in (I.i), we have [2] for z e E,
Re
zf’ (z)
fl-8(z) g8 (z)
(1.2)
> 0
We shall denote this class of functions by B(8).
We notice that, if 8
1 in
(1.2), we have the class of close-to-convex functions first introduced by Kaplan
[3].
K. INAYAT NOOR
412
2.
MAIN RESULTS.
Denote M(r,f)
max
If(z) l,
0
-<
nlan
<
K(e 8)n
Thomas [2] has proved that
e.
-< (i
r)
-e,
-< 2.
0 -<
We improve his result as follows:
< 8 < i, and be given by
B(8), for 0
Let f
THEOREM i.
r < i and M(r,f)
f(z)
z
+ [.
n=2
a z
n
n
Then, for n > 2,
nlan
2n.-.l
A(B)M
<
2n
f)
where A(8) is a constant depending only upon 8.
PROOF.
From (1.2), we can write
fl-8(z) gB(z)h(z),
zf’(z)
Re h<z)> 0; g e S*
(2.3)
Thus,
(zf’(z))’
Since, with z
re
i0
fl-8(z)f’z)gS(z)h(z) + 8fl-8(z)gS-l(z)g ’(z)h<z)
+ fl-8(z)gS(z)h’(z)
6)
(i
Cauchy’s theorem gives
2
i
2
n a
n
2rn
J
-in
z(zf’(z))’ e
8
0
we have from (2.4),
n
a
n
-<
2r
(i- 8)
n
2Izf’(z)
f-B(z)gB(z)h(z)Id8
0
2
Izg ’(z)fl-B(z)gB-l(z)h( z)
+8
0
fl-B(z) g[3(z) zh’ (z)
+
0
l__nr Ill + 12 + 131’
Now,
(1
)
r|
21]"
2
say.
[zf’ (z)f-B(z)gB(z)h(z)]d 6)
2r
8)
(i
using (2.3).
2z
0
(2.4)
BAZlLEVI
FUNCTIONS OF TYPE 8
413
In order to estimate this integral, we use the following well-known result
[4, p. 46].
1 _<
r < l, 0 < X _< 2.
Suppose that f is a really mean p-valent in E and that
1 -< 0 -< r and
Then there exists 0 such that 2r
If’(0 e i8)
I and
With 0
Since r <
1
+
4PM(r f)
d8 <
(2.5)
r)
(I
l-ld 8 <_ 4M(r,f)
(i
r)
If’ (0e i8) 12 If(0 e i %
0
I +
M(r f) is an increasing function, M(r,f) < M(
2
0 and
2
X-2
f(0e i8)
(2.5), we have
i in
2
2
0
f)
Also,
2
l-p
i
l-r
since 2r
Thus
I
i
Choosing 0
I
<_ 8(1- 8)
M
+
p
f)
2
)
O)
(i
we obtain for n > 2, see
n
I
re
8(1
r
0
i
M(
<
11
For z
i
[5, p. 238, 240],
2n
i
f)"
2n
n
i8
2
8___
12
zg, (z) fl-8 (z) gB-l(z)h(z)
2
d8
0
2- 8r
I
2
If’(z)(z)[d0,
where (z)g(z)
zg’(z); Nee(z)
0
Applying the Schwarz inequality, we have
12
<
8r
i
f’(z)
z)
0
2d
0
Now
i
If’(z)
Since the function log
log nr
2
dS=
0
n
n <
o
log n r
o
}-. n 2
n=l
lanl 2 r 2n-2
<
n=l
n} has a maximum at a point n
--(nr
log
i
elog i
r
nlanI2
r
I
o
log
i
r
n
max n r
we have
n-2
> 0.
K. INAYAT NOOR
414
n-2 <
i.e., nr
i
2
i
er logr
<
i
2
er (i
(2.6)
r)
4 M 2
(r ,f)
Also, it is well-known [6, p. 42] that M(r,f) <
(2.7)
and that the area principal for univalent functions gives
2
M (r,f), (see [7, p. 215]).
A(r,f) <
(2.8)
Using (2.6), (2.7) and (2.8), we obtain
If’(z) 12d8
Since Re
(z)
2
1__6
(2.9)
r3 (l_r)
e
> 0,
2
(z)
<
l+ze
i
I
0
ze
-it
d(t)
-it
+ ’.
i
n=l
c z
n
Thus
1
o
ICn 12 r 2n
i(.) 12de
< i
Icn
with
+ 4
.
<
r
see [4].
2;
2n
n=l
+
3r
2
i- r
i
2
(2.10)
From (2.9) and (2.10), we have
i2
Since r
<
.I + 3r_
M(rf)
48
and M(r,f)
-< l+r
2
1/2
4V28
M(r,f)
(i
is an increasing function, we have for r
n> 2
12
Finally, since 0 < 8 < i, z
i3
2
<
4V28
e---r
<_
88__ M( 2n2n
re
2n- i
2n
M(
e
ie
f). n
i
f). n
and
lfl-S(z)gS(z)
i__
zh’
(z)Ide
0
< H 1-8
(r,f)
< M l-B
(r f)
1
[gB(z)
I__
2r
1
Since it is known [8] that
r)
[zh’ (z)
r
zh’ (z)Ide
lg 8(z) lReh(z)dO
2
2
<
2rReh(z)
2
then by (3)
i
D1
n
and
BAZILEVI
13
< M
l-8(r,f)
FUNCTIONS OF TYPE
2r
r
i
2
Re
415
f8-1 [z)f’ (z)e -iarg g (z) d
0
Integrating by parts and using the fact that g is starlike, we obtain
13 -< Mi-8(r,f).
M(
-<8
=SM(
2n- i
2n
8
l+r
2
1
r
MB(r,f)
I
r
f)
f). n, on choosing r
i
I
n
n
>- 2
Thus, for n -> 2
nlanl
-< e
i
8((i
8) + 88
+ 8
M(
2n
i
2n
f)
(see [4, p. 45])
This completes the proof.
REFERENCES
i.
Bazilevi, I. E.
2.
Thomas, D. K.
353-361.
3.
Kaplan, W.
On a case of integrability in quadratures of the LoewnerVufarev equation, Mat. Sb. 37 (1955), 471-476, Russia.
On Bazilevi functions, Trans Amer. Math. Soc. 132 (1968),
Close-to-convex schlicht functions, Michigan Math. J. (1952),
169-185.
4.
Hayman, W. K.
5.
Hayman, W. K. On successive coefficients of univalent functions, J. London
Math. Soc. 38 (1963), 228-243.
6.
Thomas, D. K. Starlike and convex functions, Ph.D. Thesis, University of
London, 1967.
7.
Pommeranke, Ch. on starlike and convex functions, J. London Math Soc. 37
(1962), 209-224.
8.
Caratheodory, C.
Multivalent functions, Cambridge, (1967).
Conformal representation, Cambridge, 1932.