Internat. J. Math. & Math. Sci.
VOL. 13 NO. 2 (1990) 411-414
411
RESEARCH NOTES
ON THE COEFFICIENT DOMAINS OF UNIVALENT FUNCTIONS
M.M. ELHOSH
Department of Mathematics
The University College of Wales
Abe rystwyth, Wales
United Kingdom
(Received March I0, 1987)
ABSTRACT.
Coefficient domains for functions whose derivative has positive real part
in the interior of an ellipse are given in this paper.
KEY WORDS AND PHRASES. Univalent functions.
1980 AMS CLASSIFICATION CODES. Primary 30C45, 30C50.
INTRODUCTION.
Let f(z) be regular and satisfy the condition
Re f’(z)
Eo
r
n
(I.I)
0
Then it is wellknown (see
in a domain D.
D.
)
[I, p. 582], [2]) that f(z)
Let D be the interior of a fixed ellipse
{z
T < 2, s
cosh(s + IT), 0
o
o
s
o
tanh-l(b/a), a > b >
+ b be the sum of the seml-axes of the ellipse
a
+ iT and
0
<
s
<
s
o
E
o
0} wlth fool +/-I.
and set z
Then we see from the operator
is univalent in
Let
cosh rl where
lz- ln
that (I.I)
becomes
Re
for z in
z/-I f’(z)
)
0
(1.2)
Int(E ).
o
In this note we shall study coefficient domains for functions which are regular
Int(E ).
in
In connection with this problem see [3, Problem
o
6.54], [4, Problem 7.2], [5], [6], and [7] and [8, p.141].
and satisfy (1.2)
M.M. ELHOSH
412
be regular and satisfy (1.2) in
Let f(z)
n=
we have the sharp inequalities
anTn(Z)
THEOREM.
n
for
,,lanl
2
a
n
2/n slnh
2
sinh ns
o
n
2
2
cosh ns
(1.4)
4/n
o
The inequality (1.4) shows that the coefficients
4/n(r2
at the origin and seml-axes
a
n
+ I n that
n
a
Re[l+-I
r:n)
+
f’(z)]
.
n(a
where
Re[l +
/f-1
f’(z)]
>
sinh ns
n
n=l
21[
.....
cosh(s
z
o
+ iT), so
cos uT- 8 cosh ns sin nT)
o
n
s
o
we then see that
Re[l+/z-1 f’(z)] dT
21[
fo Re[l+ z--1
f’(z)] cos nT
n1[lanSlnh nSo,
(1.6)
f21[Re[1+-I
o
f’(z)] sin nT
n1[i cosh ns
o
n
(1.7)
Using (1.5), (1.6), and (1.7) we obtain
[an[
Jan+
f(1[
sinh n(s
o
+ iT) Re[l+ z/-I f’(z)]
slnh ns
21[
nW sin’h
fo
’ns’ o
o
cosh ns
Re[l+z-I
Islnh n(s o
+ iT)
4
cosh ns
o
dT
o
f’ (z)
2/n slnh us o
since
log r
0 in Int(Z ).
o
Since this is a Fourier series for fixed
-[0
o
lle in ellipses with centre
a
where n-1,2,3
We see from (1.2), setting
PROOF OF THE THEOREH.
and
+/-
Then
(I.3)
nSo,
2
+
Int(E 0 ).
This is (I,3).
dT
o
COEFFICIENT DOMAIN OF UNIVALENT FUNCTIONS
413
We also see form (1.5), (1.6) and (1.7) that
lansinh nSo + Is. =oh nol li
S"
Re[ I+
z-I f’ (z)]eniTdT
2/n.
This gives
a
sinh2ns o + B2n cosh 2
2
n
ns
o
4/n
2
as required in (1.4) and the proof of the theorem is complete.
Finally we see from [9, Theorems 2 and 6] that
cosh(s
where
0
T
<
+ iT’), z
cosh(s + iT), 0
<
s
<
s
o
0
< T’
<
2 and
2, plays the role of the extermal function in this case.
REMARK.
in
o
Normalizing in the sense of
[3, Remark 2]
we obtain the analogous results
[2, Theorem I].
REFERENCES
5.
KAPLAN, W. Advanced Calculus, Addison-Wesley (1952).
Trans.
Functions whose derivative has a positive real part.
MacGREGOR, T.H.
Amer. Math. Soc. 104 (1962), 532-537.
Research problems in complex
ANDERSON, J.M., BARTH, K.F. and BRANNAN, D.A.
analysis, Bull. London Math. Soc. 9 (1977), 129-162.
HAYMAN, W.K. Research problems in function theory, Athlone Press, London (1967).
ROYSTER, W.C. Functions having positive real part in an ellipse, Proc.
6.
Amer. Math. Soc. I0 (1959), 266-269.
Coefficient problems for functions regular in an ellipse, Duke
ROYSTER, W.C.
I.
2.
3.
4.
(1959), 361-372.
Coefficient regions for Schllcht functions,
7. SCHAEFFER, A.C. and SPENCER, D.C.
Amer. Math, Soc.#...colloq. Pub. 35 (1950).
8. SHAFFER, D.B. The Biekerbach conjecture, Contemporary Math. 38 (1985), 139-141.
9. ROYSTER, W.C.
A Poisson integral formula for the ellipse and some
applications. Proc. Amer. Math. Soc. 15 (1964), 661-670.
I0. ELHOSH, M.M.
On mean p-valent functions in an ellipse, Proc:" Roy. Soc:
Edinburgh 92A (1982), I-II.
Math. J. 26