PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 47, Number 1, January
1975
SOMELINEAR SUBORDINATIONRESULTS FOR
CLASSES OF UNIVALENT FUNCTIONS
ROBERT B. BYERS
tions
ABSTRACT.
In this
on complex
numbers
to f(z) — z + a z
2
note
we determine
A and
necessary
p. such
that
...
+ ...
for all functions
and sufficient
Az/(1
— pa
/ in certain
condi-
z) is subordinate
classes
of univalent
functions.
1. Introduction.
analytic
Let
and univalent
is analytic
the class
in the unit disk
in F with
to be subordinate
S denote
g(0) = 0 and
of functions
fiz) - z + a.z
+...
F = \z : \z\ < 1 \. If / is in S and
g(F)
to /, and / is called
is a subset
of /(F),
a univalent
then
g
g is said
majorant
oí g. We express
subclasses,
respectively,
this by writing giz) -< f(z).
Let
K and
S
denote
of S. The well-known
the convex
result
and starlike
that for each
{z : \z\ < Y2\ is equivalent
to the statement
"]4 theorem"
< f(z)
implies
and F. R. Keogh
numbers
z/A
[1] examined
À and p such
in K. Keogh
[A] also
that
/ in K, f(E)
z/2
for all
examined
and sufficient
< Xz + ¡ia z
the question
the disk
Similarly,
/ in S. T. Basgoze,
necessary
z/2
contains
■< f(z).
the Koebe
J. L. Frank,
conditions
-< f(z)
on complex
for f(z) = z + a z +...
of z/A -< kz + pa z
-< f(z)
for
/ in S*.
In this paper
for functions
we answer
similar
of the form \z/(l
\z/(l
— paz)
map
tions
obtained
are equivalent
questions
- pa z). Since
F onto open disks
in the classes
the functions
(or half planes),
to covering
theorems
K, S , and
z/A,
z/2,
the various
for functions
S
and
subordina-
in the clas-
ses involved.
2. Convex
functions,
functions.
we present
Lemma 1. For all
Before
by the editors
Key words
and phrases.
the theorem
for the class
necessary
for its proof.
+ • • • in K, f(E)
contains
of convex
the disk
\a2\2).
July 14, 1973 and, in revised
AMS (MOS) subject classifications
starlike,
lemma
f(z) = z + a2z
\w-3ä2/2(A-\a2\2)\<3/(AReceived
stating
the following
form, November 19, 1973-
(1970). Primary 30A26; Secondary 30A32.
Subordination,
covering
theorem,
univalent
function,
convex.
Copyright © 1975. American Mathematical Society
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143
144
R. B. BYERS
Proof.
value
If the function
of /(F),
then
f(z) = z + a z
+ ...
[3] the inequality
(1)
|z7.2+ l/c|
of /(F),
c is an omitted
the function
is in S. This implies
If the function
is in S, and
f(z) = z + a z
< 2.
+ ...
is in K, and
c is an omitted
value
then the function
(2)
g(z) = f(z)_ll(l>
=z+(a
is in S [5], and
c/2
value
tain
< 2. The function
|íz
quality
+ 3/2c|
is sharp.
Theorem
is an omitted
The
1. Let
±)z2 + ...
of g(F).
f(z) = z/(l
Applying
— z) shows
(1) to (2), we obthat the above
ine-
lemma now follows.
A and
p be complex
numbers.
Then
z/2 < Xz/(l - pa2z) < f(z)
for all f(z) = z + a2z + ...
Proof.
function
A/(l
Without
z/(l
- z).
loss
in K if and only if À = lA+ Vip, 0 < \p\ < lA.
of generality,
The function
we can assume
|Az/(l
+ \p\) on |z| = 1. If the subordinations
- pz)\
\ > 0. Let
has a minimum
hold,
we then have
f(z)
value
be the
of
A/(l
+ \p\)
> Yt. Also for real x, —I < x < 0,
x/(l
Allowing x to approach
-1,
- x) < Ax/|l
- px\ < x/2.
we obtain A/|l + p\ = Vi. This implies
A = Vi+
Vip and p= \p\> 0.
If the function
.
p,
/ is in K, then
so is the function
fl(z + Q/il+c!z)]-f(t)
g(z,<f)=-.-=
/'(¿)U-|f|2)
z + Az
2
+...,
Ç <1,
2
where
The
the original
author
wishes
to thank
the
referree
for suggesting
proof.
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this
simplification
of
SUBORDINATION RESULTS FOR UNIVALENT FUNCTIONS
145
a =A(i_i£i2)-ë.
2/' Ü)
fí fiz) = tan~lz
tions
and 0<£<1,
hold for all
then A 2 = -2<f/(l
+ £2). If the subordina-
/ in K, we then have
z
(H+ M<x)z
2
1 + 2&z/(l
+ Î2)
where
«(*,£) =-75
If z is real,
tan 1-—J-
g(z, £) is an increasing
function
-tan
of z.
»M.
Letting
z = 1, we obtain
the inequality
i<
*»*■ <iil7;-,.-fVy
2 -l*2f^(l+f2)-l-f2V
From this
we obtain
the inequality
ft|(l-e2)-2f(5
For
0 <£
-2tan-1e)]<(l+e2)(^
< 1, (1 — £ )-
2<f(77/2 - 2 tan-
- 2 tan"1 f) -(1-f2).
rf) is greater
than
zero.
Thus
we
have
< (l+^2)(77/2-2tan-1cf)-(l-c;2)
1 -¿f2-77<f+ 4^tan-1 Ç
Letting
ff approach
For the remainder
1, we obtain
p < Á-
of the proof we may assume
without
loss
of generality
that <?- > 0. For a < 1, 0< /i< 1, the images of E under (l+p)z/(2-2pn
ate nested.
Thus
completes
only show
that
3^/(4
— 2aJz)
-< f(z).
Lemma
1
the proof.
3. Starlike
Bernardi
we need
z)
and univalent
[2] is used
Lemma
Theorem
functions.
in the proof of the theorem
2. // f(z) - z + a z
2. Let
The following
A and
+ ...
for the univalent
is in S, then
p be complex
lemma due to S. D.
numbers.
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z/(2
Then
- a2z)
case.
-< f(z).
146
R. B. BYERS
z/A < Az/(l - pa2z) < f(z)
for all f(z) = z + a2z2 + ...
Proof.
all
Without
loss
f(z) = z + a z
be the Koebe
in S if and only if A = Va+ Vip, 0 < \p\ < Vi•
of generality
we can assume
+ . . . in S we have
function
z/(l
z/A
A > 0. Suppose
-< Az/(1
- z) . As in the proof
that
- pa z) -< f(z).
of Theorem
for
Let f(z)
1, we obtain
A = Va+ Vip and p. = \p\> 0.
Next
let
/(z)
be the function
z/(l
- z ) = z + z3 + • • . . Letting
z = ¿y,
0 < y < 1, we have
y/4<iy4+
Letting
y approach
lAp)y<y/(l
1, we obtain
For the remainder
+ y2).
p<Vi.
of the proof
we may assume
without
loss
of generality
that a > 0. For a < 2, 0 < ß < Vi, the images of E under (l + 2/¿)z/(4-4iia2z)
are nested.
Lemma
Since
cannot
2 completes
the functions
be improved
z/(l
the proof.
- z)2
for the class
and
z/(l
- z2)
are starlike,
Theorem
2
S .
REFERENCES
1. T. Basgoze,
J. L. Frank
and F. R. Keogh,
Cañad. J. Math. 22 (1970), 123-127.
2. S. D- Bernardi,
32 (1965), 23-36.
Circular
On convex
univalent
regions
covered
by schlicht
functions,
3. W. K. Hayman,
4
F. R. Keogh,
Math.
Multivalent
A strengthened
Essays
Dedicated
Ohio, 1970, pp. 201-211.
5. G. Sansone
plex variable.
II:
Duke
Math.
MR 30 #2135.
functions,
Cambridge
Tracts
Phys., no. 48, Cambridge Univ. Press, Cambridge, 1958, p. 3.
functions,
functions,
MR 41 #1983.
Ohio
and Math.
for starlike
Univ.
Press,
univalent
Athens,
MR 43 # 496.
and J. Gerretsen,
Geometric
form of the % theorem
to A. J. Maclntyre,
in Math,
MR 21 #7302.
theory,
Lectures
on the theory
Wolters-Noordhoff,
of functions
Groningen,
of a com-
1969, p. 198.
MR 41 #3714.
DEPARTMENT OF MATHEMATICS, OHIO STATE UNIVERSITY, MANSFIELD, OHIO
44906
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J.