I nternat. J. Math. Math. Sci.
(1980) 477-481
No.
Vol.
477
ON THE HANKEL DETERMINANTS OF
CLOSE-TO-CONVEX UNIVALENT FUNCTIONS
K. INAYAT NOOR
Department of Mathematics
Kerman University
Kerman, Iran
(Received July 9, 1979 and in revised form August 29, 1979)
ABSTRACT.
The rate of growth of Hankel determinant for close-to-convex functions
is determined.
The results in this paper are best possible.
KEY WORDS AND PHRASES. Starlike and close-to-nvex Functns, Hankl Dtina
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
30A32.
INTRODUCTION.
Let K and S* be the classes of close-to-convex and starlike functions in
y
{z:
z
< i}.
Let f be analytic in y and f(z)
z
+
n--2
Hankel determinant of f is defined for q > i, n > i by
an
H (n)=
q
an+l
an+q-1
an/1
an+q_ I
an+2q_ 2
a z
n
n
The qth
478
K. INAYAT NOOR
For f S*, Pommerenke
2] has solved the Hankel determinant problem completely.
Following essentially the same method, we extend his results in this paper to
the class K.
2.
MAIN RESULTS.
THEOREM i.
are numbers
Let fK and f(z)
Ym and
c
(
mu
+
a z
Then, for m
n
n= 2
m) that satisfy
z
0
0,i,..., there
i and
Cmm
Cmo
k--0
such that
m
c
=0
an+
my
0
(1)n-l’l"Ym
The bounds (2.1) are best possible.
PROOF.
Since f % K, there exists g % S* such that, for z E y
zf’(z)
(z)h(z), Reh(z)
Now g can be represented as in [I], g(z)
> 0
(2.2)
12
z exp
0
where 9(t) is an increasing function and (2)
be the jumps of (t), and
We may assume that
81
t=Ol, 02,..
0. Then
ul +
(0)
1
log.
-It d(t)]
l-ze
2.
Let
l
>
u2
a
be the values at which these Jumps occur.
e
< 2 and
I+2
+
+ sq
2
for some q if and oly if g is of the form
q
g(z)
We define
z
Jl
(l-e
-iOj
-2
q
z)
Cm by
m
i0
4)m(Z) VTI1
(1-e
and
m" Sm+l
We consider the :hree cases i.e.
(m= 0,i,...)
(2.3)
HANKEL DETERMINANTS OF UNIVALENT FUNCTIONS
0 <
(i)
-< i,
I
as in [ 2]
1 <
(ii)
<
=i
3
(iii)
3 <
i
479
< 2
and the first part, that is the bounds (2.1), follows similarly.
For
the rest, we need the following which is well-known [ 2].
LEMMA.
Let
81 < 82
...,
j (j’l,
> 0,
<
<
q).
8q
then
81
-i 0
1I (1-e
j =i
let
lq
i
Jz) -Xj
y.
n=l
0(i) n
b
+ 2,
be real, and
If
q
(z)
<
as
b z
n
n
(2.4)
rt->
We write
m
bm(Z)
c
.
and
=0
m
bm(Z)zf’(z)
b
n= 1
ms
m
zn+m+ n=l
z m-
(n+m)a
z
n+m
(2.5)
where
n
b
[
mn
u=0
mn
=0
(n+u) c m- an-v
m
an+
i.
There are two cases.
(a)
Let g in (2.2) have the form (3); that is, a + a +
I
2
v
With_
m
8m,
m
q-l, a
I
2
it follows that 7 <
m+l
m
a
.
q
Y0 + 71 +
< 3 and
+ aq
m
2.
2
implies
m+l
2
q
=--.
Now from (2.2), (2.5) and the Cauchy Integral formula, we have, with
m+nl
B (r)=
m
r
m
k=l
(n+m)
[bmk irk+m,
lamn
<
1
n+m
2r
0
[,m(Z)g(z)h(z ) ld0+Bm(r
(2.6)
K. INAYAT NOOR
480
Applying the Schwarz inequality, we have
amn
(n+m)
When we write
lj
<
2Ym
<
2r
n+m
[#m(Z)g(z)]
2
in the form (2.4)
q: m > 0).
(j=l,
I#m(Z)g(z)12d0)*’(
0
dO < A n
.
A8o
12 {h(z){2dO
1
2
{dn
But
<
2, n
n=0
0
e
the exponents -l
< i+4
2Ym-I
(n+).
{dn[2r2n (do--l)
.
1, and so
{h(z) 12d0
lh(z) 12d0)1/2 + Bin(r)
j
satisfy
Hence, using the Lemma, we have
), {,2
.{m(Z)g(z
0
0
2
l+3r
r n-2
n--i
l-r
(2.7)
Reh(z) > 0
2
<An, n>l
(2.8)
From (2.7) and (2.8), we have
(n+m)
i.e.
a
lamn
<
An7m
(n-)
0(1) nYm-i
(n=).
This proves the theorem in this case.
(b)
in
Let g in (2.2) be not of the form (2.3).
[2],
it follows that, for z
0
re
Then using arguments llke those
i0
{#m(Z)g(z)h(z)ld0
0(1)(l-r)
Hence from (2.6), we have
a
where a
is defined by (5).
O(l)n
(n-),
HANKEL DETERMINANTS OF UNIVALENT FUNCTIONS
The function f
o
f (z)
o
z(l-z q)
-2/q
(2/q9-1)z
[
=0
bounds (i) are best possible.
481
9q+l
shows that the
We also note that except in the case where
m=(q-l) and g in (2.2) is not of the form (2.3), one can choose 0 -<
.
Ym
>
2
m+l
from theorem (i) and Pommerenke’s method [ 2], we can now easily prove the
following
THEOREM 2.
Let f
K and f(z)
z
+
n=2
a z
n
n
Then for q > i, n > i,
H (n)
q
This estimate is best possible.
(2.3), there
such that
exists a 6
H (n)
q
0(1)n
0(1)n
2-q
(n-)
In particular, if g in (2.2) is not of the form
6(q,g) > 0
2-q-6
(n-=).
REFERENCES
[i]
Pommerenke, Ch. On Starlike and Convex Functions, J. London Math. Soc. 37
(1962) 209-224.
[2]
Pommerenke, Ch.
On the Coefficients and Hankel Determinants of Univalent
Functions, J. London Math. Soc. 41 (1966) 111-122.