425
Internat. J. Math. & Math. Sci.
VOL. 13 NO. 3 (1990) 425-430
A NOTE ON NEIGHBORHOODS OF ANALYTIC FUNCTIONS
HAVING POSITIVE REAL PART
JANICE B. WALKER
Department of Mathematics
Xavier University
Cincinnati, Ohio 45207
(Received July 7, 1989 and in revised form October 18, 1989)
ABSTRACT.
D
{z
lllzl
Let P denote the set of all functions analytic in the unit disk
< 1} having the form
+ s pk
k--1
p(z)
zk with
For
Re{p(z)} > O.
a > O,
let
qk zk analytic in D with s. Ip k -qk I.< 6. We
k=l
k=l
1 + pk
denote by P’ the class of functions analytic in D having the fore p(z)
k=l
with R [zp(z)]’} > O. We show that P’ is a subclass of P and detemine a so that
N(p) = P for p P’.
Na(p)
be those functions q(z)
i + S
zk
KEY WORDS AND PHRASES. Functions having positive real part (Carathodory class),
subordinate function, a-neighborhood, and convolution (Hadamard product).
1980 AMS SUBJECT CLASSIFICATION CODES. 30C60, 30C99.
I.
INTRODUCTION
Let denote the class of functions f analytic in the unit disk D
with f(O)
0 and f’(O)
i.
z +
For f(z)
T.
k=2
ak zk
in Z and
z + s.
bkzk
a > O,
{z
Ilzl <
1}
let the
bkl,<
6}.
kla
k-2 k
k=2
For h(z) z, Goodman [I] has shown ihat Nl(h) = S* where S* denotes the class of
univalent functions in /z which are starlike with respect to the origin. St. Ruscheweyh
a-neighborhood of f be given by
proved that if f(z)
univalent functions in
C were replaced by
,
=(g
{g(z)
Na(f)
_Z
akzk
lies in C, where C denotes the class of convex
z + E
k=n+l
then Na(f)= S* for a n 2 -2/n. Fournier [3] found that if
cllZg:’Iz)l
lg
z)
<
z
D}
and S* by
T
g(z)
Brown
D}
then N a (f)= T for a n e -1/n.
[4] extended the results of St. Ruscheweyh and
n
Fournier and provided simpler proofs. We shall focus on a class of functions directly
related to S* and to other classes of univalent functions. Let P denote the class of
426
J.B. WALKER
.
k
pkz with Re{p(z)} > 0 for
k=1
zl < I. This family is usually called the Carathodory class. For f in Y, recall
that f S* if and only if p(z) zf’(z)/f(z) lies in P.
Let P’ denote the class of functions analytic in zl < 1 having the form
functions analytic in
1 having the form p(z)
1 +
zk
+ Z pk with Re{[zp(z)]’} > 0 for Izl < I. In this paper we shall define
k=1
a neighborhood of p P’ and determine ; > 0 so that N(p)= p.
2. PRELIMINARY RESULTS.
We begin by defining P and P’ in terms of subordination. Recall that g is
subordinate to h, written gh, if g(z)
h(w(z)) where w is analytic in
<
w(O)
0
and
I,
<
i
for
lw(z)l
Izl
positive rea] part in
Izl < I. Since 1+z
zl < 1, is univalent, and is when z O, it is not difficult to show that
p(z)
has
p(z)
p
P if and only if
p
P’ if and only if
1+z
(2.1)
and that
One can also show that P’= P.
[zp(z)]’-
.
1+z
(2.2)
For according to (2.2), if p
P’ then
1+z
[zp(z)]
and thus we have
l+z
Since
obtain
is convex and univalent, we can apply a lemma
z(z)
(see Brown
[5],
p. 192) to
l+z
i
Z
from which it follows that
p(z)
"
l+z
Hence, by (2.1) p P and P’c P.
Now let us establish a criterion for a given function to belong to P. By (2.1)
1+z
l+z
q c P if and only if q(z)-.
Since is
univalent, then q c P if and only if
iO
+
e
< I. That is,
q(z)
iO for O< O< 2 and Izl
1 -e
q
for O< B< 2,
P if and only if
Izl
(1
eil))q(z)
(1 + e iE)
O,
(2.3)
< 1.
We can express (2.3) in terms of convolutions.
; a.z k and g(z)
unit disk D. Recall that if f(z)
Let f and g be analytic in the
;
b,z k
k=O K
k=O K
(or Hadamard product) of f and g, denoted by f-g, is
f,g
k=O
akbkzk.
then the convolution
NEIGHBORIOODS OF ANALYTIC FUNCTIONS HAVING POSITIVE REAL PART
(1
Thus,
eiO)q(z)
(1 + e iQ) can be written as
1_i-* q(z)]
[
( l-eiQ (i )
(I- e iO)
Let h O(z)
he(z) ,
-(i +
+ e iO
1
be defined by
Then it follows
ho(z)
that he(O)
[1
2e iB’
e iO
(1 + e iO
Z
1
L
and for O< 0 < 2,
1 + z pk
k=1
DEFINITION. For any p(z)
N(p),
ei@)
zk
* q(z)
* q(z).
q(z) O.
3. THE MAIN RESULT.
We define a 6-neighborhood of p for p
denoted by
427
Izl
1J"
P if and only if
< 1, q
(2.4)
P.
in P and 6 >, O, the S-neighborhood of p,
is
N(p)
q(z)
kE=llPk
k:lqk z
1
Our main result is the following theorem.
qk .<
zk
1 + E pk belongs to P’, then N6(p)=P, where
k=l
.3862944. This result is sharp.
We need several leas.
LEMMA I. If p c P’, then z(p*h is univalent for each 0 < 0 < 2.
PROOF. Fix O< O< 2. Then
If p(z)
THEOREM.
2 In 2
O)
-
[z(p,ho)]’
el
I
(1
zp(z)
e
I + e iO
(zp(z)),
1
for
iQ
1
eig
e
iO]Izl
i + e
e iO
i
By definition of P’, the range of (zp(z))’ for
i8
of 1 +e iO lies on the imaginary axis.
-e
(1 + e iO
iO) p(z)
1
=- [
i
Q
e
’i(}
(I
ei@
< i lies in
Thus, we can choose
e -iO
Re(z)
(3.1)
> 0 and that
so that
Re{eiEz(p*hB)(Z)] ’} > 0
Izl
(l+e
1
arg{-(1
eiB)-leiO}.
By the Noshiro-Warschawski Theorem
O, 0 < 0 < 2.
each
for
is
univalent
z(p*ho)
r for
>
P’, then
]
Izl r< I, O< @< 2.
+
e
we define F(w)
PROOF. Using expression (3.1) for
iO
it
1+re
w
where w
0,< t,< 2. Now F(w) may be rewritten as
i(
e
1 re 1"t
0 < @ < 2.
F(w) e’iO{ (1 + e
(I
< 1, namely
(Duren [6], p. 47),
LEMMA 2. If p
[z(p*he)
’I
]-
Iz(P*h()l ’r,
)
iB)
Thus,
F(w)
1 -w + e iO
l+w
ei@)w},
e-i@(1
i@)
1
428
J. B. WALKER
11 + wile iO reitl
11 +wlll- rei(t-o)I
r)I1
> (.i
Ii
Since
+
reitl I]2Ireit
+
+2 r
+
w’
E
r
2 1l+r"
P’ and (3.1)holds, by letting w
[zp(z)]’ we get the desired inequality.
+
re T
>
it is clear that
IF(w)I >
p
Since
That is
[z(p*h 8
The lena is proved.
LEMMA 3 If p
PROOF.
choose
Zo
P’
p,hoI
then
n1
t dt
6, where 6
2 In 2
1.
P’. Then by Lemma i, z(p,h 8) is univalent. For fixed 0 < r < I,
Let p
with
1 $ r
IZo
r such that
mini z(
Izi
*h 8 )1
IZo(P*h )(zo)l.
Since z(p*hs)is univalent, the preimage L of the line segment from 0 to
is an arc inside ]zl # r. Hence, for lz]
r we have
Zo[(P*hs)(Zo)]L.
Iz(p*ho)l > IZo(P*ho)l
[z(p*%)] Idzl
>
fO
r
[z(p*hla)] ’lldzl-
Accordingly, we apply Lemma 2 to get
[P*hla] (z)l
>E
[z(p*hla)] ’1 dzl
>-1/or
l’tdt
l+t
21n
r
The function g(r)
-2 In (I +
r)+
,< 0 for r
> O,
2
-In
(1 + r)
r’(’l’2r} < O.
(I + r)
1.
I is decreasing for r > 0 if g’(r)
from which it follows that g’(r)< 0 for r > O.
p*hol >
This completes the proof of Lemma 3.
PROOF (OF THEOREM). Let p(z)
N(p)
want to show that every q
but fixed function in
Na(p).
(I + r) In (i + r)
It is not difficult to show that r
2 In 2
i.
Now we may prove the theorem.
i +
pk E P’ and let a be as in Lema 3.
zk
k=1
belongs to P, where q(z)
Hence, Ip k
k=l
h O*ql
Hence
qkl
,<
(ho*P) + ho* (q
a.
P)I
1 +
qk
k=l
Observe that
e
is an arbitrary
NEIGHBORHOODS OF ANALYTIC FUNCTIONS HAVING POSITIVE REAL PART
>
Ihg*pl
Ihg*(q P)I
-e
a- z
k=l
Therefore,
N(p)
ho*q
-
0 for
Izl
< 1.
ig
(qk
2
k=l
>
429
qk
pk
>
Pk )z
k
O.
By (2.4), it follows that q
P.
Consequently,
l+z
Now we prove that the result is sharp. Let p(z) be defined by (zp(z))’
1
2
2
In (i z) + 6z
Then p(z) =-I
In (I z). Now let q(z) p(z) + z:-I
q(-1).
Clearly, q N(p). However, as z/ -1, then q(z)-I + 2 In 2
I, then q(-1)< 0 and consequently Re q(z)< 0 for z near
Therefore, if > 2 In 2
-1. This contradicts Re q(z) > 0 for Izl < i. This completes the proof of the
theorem.
1.
2.
3.
4.
5.
6.
7.
REFERENCES
GOODMAN, A. W., Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc.
.8_8 (I 957 ), 598-601.
ST. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 8_1
(1981), 521-527.
FOURNIER, R., A note on neighborhoods of univalent functions, Proc. Amer. Math.
Soc. 87 (1983), 117-120.
BROWN, J. E., Some sharp neighborhoods of univalent functions, Trans. Amer. Math.
Soc. 287 (1985), 475-482.
BROWN, J. E., Quasiconfomal extensions for some geometric subclasses of univalent
functions, International Journal of Math. and Math. Sciences (1984), 187-195.
DUREN, P. L., Univalent functions, Springer-Verlag, New York, 1983.
HALLENBECK, D. J. and MacGREGOR, T. H., Linear problems and convexity techniques
in geometric function theory, Pitman Publishing Limited, 1984.