Internat. J. Math. & Math. Sci.
Vol. I0 No. 2 (1987) 241-258
241
ON QUASI-CONVEX FUNCTIONS AND RELATED TOPICS
KHALIDA INAYAT NOOR
Department of Mathematics
Girls College for Science Education
Sitteen Road, Ai-Malaz, Riyadh
Saudi Arabia
(Received January 28, 1985 and in revised form March 20, 1985)
&BSCT.
Let
be the class of functions
S
f
which are analytic and univalent in
the unit disc E with f(0)
I. Let C, S
and K be the classes of
0, f’(O)
of quasiconvex, starlike and close-to-convex functions respectively. The class C
convex functions is defined as follows:
Let
f
E
be analytic in
there exists a
gC
Re
(zf’ (z) )’
g’(z)
>
0, f’(0)
f(O)
and
such that, for
I.
Then
if and only if
fEC
E
O.
In this paper, an up-to-date complete study of the class
basic properties, its relationship with other subclasses of
,
C
Its
is given.
S, coefficient problems,
arc length problem and many other results are included in this study.
Some related
classes are also defined and studied in some detail.
WDDS AD PRABS.
Univalent convex, alpha-convex, quasi-convex, alpha quasiconvex, close-to-convex, arclength, coefficient, radius of convexity, order 8 type Y,
Livingston’ s and Libera’ s operators.
1980
1.
SJE’ SIFITION ODE.
30A32, 30A34.
IN1"IODOC’rION.
Denote by
E
unit disc
S
the class of functions
f(O)
and satisfy
f
which are analytic and univalent in the
O, f’(0)
I.
The subclasses
S
and
C
of
starlike and convex functions respectively are well-known and have been extensively
studied, see [I], [2] and [3].
for
A function
f
is said to be in
S
if and only if
zE
Re
and
C
zf’ (z)
f(z)
>
(1.1)
0.
are related by the Alexander relation [4], that is
,
fEC if, and only if
zf’ES
K.I. NOOR
242
Hence a function
f
is said to be in
(zf’(z))’
f’(z)
Re
K
The subclass
be
to
K
in
zEE
(1.3)
0
,
and many properties of
said
if and only if for
consisting of close-to-convex functions is also well known [5]
S
of
>
C,
S
if
can be extended to the wider class
K.
and only if there exists a convex function
A function
g
f
is
such that, for
zeE
Re
G=zg’
Since
f’(z)
g’(z)
>
zf’ (z)
>
G(z)
,
zeE
convex, (1.4) can be written as
is starlike for g
Re
for
(*.4)
0
GES
and
(,.5)
0
G(z)=f(z) in (1.5) one sees that
Taking
that
K,
S
which shows
,
CoS CK
2.
QUASI-CO FNTIONS.
We proceed now to define and discuss a subclass of
S
K
which is related to
by
an Alexander type relation [6].
)KFIIITION
Let
2.I.
f
with
g(0)--0,g’(0)--
E
>
f(0) -0, f’(0)
I.
Then
f
g
such that, for
(zf’(z))’
g’(z)
Re
E
if and only if there exists a convex function
be analytic in
is said to be quasi-convex in
with
zE
(2.1)
0
,
Denote the class of quasi-convex functions by
It is clear that, when
f(z)ffig(z)
,
show now that
C
C K,
TEOREM2.|.
geC,
then (2.1) holds.
Hence
CCC
,
We
so that every quasi-convex function is univalent.
,
Let
Re
and
C
Then, for
feC
zf’ (z)
g(z)
>
0,
zE,
geC
,
and
so
C
univalent in
K S,
E.
thus,
every quasi-convex function is close-to-convex and hence
QUASI-CONVEX FUNCTIONS AND RELATED TOPICS
A result of Libera [7] shows that, if
PROI.
in
E
s(0)ffit(0)ffi0
with
tS
and
t’(z’) >o
Re
and
t
are functions analytic
zCE,
then for
s’ (z)
Re
s
23
s(z)
-f(>O
An immediate application of this with s(z)-zf’(z) and
It follows at once from the definition (2.1) that
t(z)-g(z)
proves the theorem.
,
fgC
if and only if
(2.2)
zf’eK
,
We now extend some results to the class
C
which are known to be true for
Z
a z
n
C,
see [1], [2] and [3].
Let
THBOR 2.2.
fC*
f(z)
with
[an
(i)
+r
(iv)
[m[
n
Then, for
z
r
<
I,
n=2,3
(l+r)
(Ill)
z +
2
(l-r)
,
where
f(z)a
2
in
E.
All inequalities are sharp, equality being attained for
Z
fo(Z)
(2.3)
,
(i)
PROOF.
z +
Z
n2
Since
there
feC
(zf’(z))’
g’(z)
and
r. c z
n
n=O
h(z)
c
a
convex
g
function
with
g(z)
zcE
such that, for
b z
n
exists
h(z)
where
Re h(z)
>
O,
I.
o
So
g’(z)h(z)
(zf’(z))’
Equating the coefficient of
2
n a
n
n
bn
z
+ (n-l) c
n-1
on both sides, we have
bn-I
+ (n-2) c2
bn_2+
+
2Cn_ 2
b
2
+
Cn_
ice.
(2.4)
K.I. NOOR
244
Now, from the known results [7],
I, n--2,3,...,
bn
and
c
n
we have
n
and this implies
21
n +
a
n
la
2{ n(n-l)}
2
--n
2
I, n=2,3
Using known [1,2 and 3] distortion theorems for the functions
Integrating the right hand side of (2.5) from
r
l+r
i
0 (l-r)
o
o=
obtain
g
and
h,
3
0 to z,
we obtain
r
dr
(l-r) 2
[f’()[,
bound fo
we have
(2.5)
(l_r)3
(l+r)3
od=
(2.4)
n--2,3,...
foo.
pro=d
E
the radius of the open disc contained in the map of
by
zf’.
L
Let
b
dl
be the
z
0
point
increases with {r
[z
the image of
w=f’(z)
by
r
expands
and is less than
Hence the linear-segment connecting the origin with the point
zf’(z)
covered entirely by the values of
mapped by
w=zf’ (z)
in
onto this linear-segment.
l-r
)
(l+r)
3
E.
Let
d
_f’(z0)
will be
E
which is
be the arc in
Then
dr
r
(l+r)
Integrating (ii), we obtain (iii)
and by letting
r
in the
lefthand side of
(iii), we have (iv).
Waadeland [8] proved
g(z)
z +
Z
k=l
bmk+iZ
mk+l
that
every
starlike
m-fold
symmetric function
g,
with
satisfies
2
--+k-I
2
r()
(2.6)
QUASI-CONVEX FUNCTIONS AND RELATED TOPICS
,
In order to extend this result to
K
we need only to extend Waadeland’s result to
C
,
K.
and
C
and then use the relationship between
was done by Pommerenke [9] and so (2.6) is true for
C
, However
K
this extension to
fEC
,
follows exactly in the same way as for
C
The following result for the class
the class
245
[!0].
in
TKORKM 2.3.
Let g(z) f(z).
feC *
Let
z +
f(z)
with
z +
g(z)
and
n=2
Then, for all n,
s n([ z)
n
a z
l
l
k=2
bkZ
k
f(z),
where
n
Sn
(z)
+ l
z
k--2
bkZ
means "subordinate to")
[I0] showed that if
Clunie and Keogh
in
o(I) as n
n a
definite area then
has
fC
n
(R).
f(z)-- z + l
and
a z
n
n=2
This result has been extended to
with
f(E)
,
C
[6] as follows.
TJ14 2.4.
fEC*
Let
z +
f(z)
with
l
n=2
then
n a
n
o(I)
n
as
n
f(E)
If
has finite area,
n being best possible.
f(E
the closed curve which is the image of
C(r)
Denote by
the index of
",
a z
n
r
and by
L(r)
the
length of C(r). We prove:
TSOM 2.5 [6].
Let
2 (. A(r)
A(r)
Further, if
<
L(r)
0
for
r
lr)
<
I,
then
as
r+l
log-- shows
f(z)
0
2 (. A(-)
L(r)
o(1)(log
The convex function
Then, for
fC
r
<
(I og
I,
l-r I/2
that the factor
(log
(2.7)
--ll__r)/2
in (2.7) is the
best possible.
PROOF.
equality.
The
Since
left hand inequality follows at once from the Isoperimetrlc infC
zf’ (z)
F(z)
is close-to-convex.
2
2
0
0
r
2.
o
M(P,zf’)
--,
see [I, p.
451
Thus
K.I. NOOR
246
(2.9)
Z)
( A()F{)
log
We can show (2.8) easily by taking
(2.7) that for
the factor
fgC,
For
2.I.
L(r)
fC
(log
I-=r) I/2
< =.
A(r)
L(r) (2M(r).
it is well-known that
(log --’--Z_)/2
O(l) M(r)
r
as
1,
It follows from
The question of whether
can be removed is still open.
It is well-known [II] that
(zf’(z))’
f’ (z)
Re
>
+
0
zf’(z)
f(z)
Re
,
>
that is, every convex function is starlike of order
,
C
a relationship exists between
and
.
zeE,
It is natural to ask if such
The following example shows that this is
K.
not in fact the case.
z
l<a<__"
Then
Re
(zf’(z))’
g (z)
Re
zf’(z)
g(z)
Re(l-az)
2
>
zE,
0,
but
Re(l-az)
and
zf’ (z)
Inf Re
g(z)
<
for
<
< I__
a
I’
zSE.
Now, following the same method as in [12], we have
I/3
f’(z)
Then Re(g(z))
THEOR 2.6. Let fC* and g(z)
f’(-z)"
>
0,
for
zZ.
This result is sharp as can be seen from the function
zf
(cosY)e i
where
3.
RKIATIONSHII’ OF
(i)
The class
The class
write
[z(l-gz)]/[(l+z)2],
l(z)
0<<
,
C
and
*
flC
with respect to the convex function #(z)
WITH OTHER SOBCLASSES OF
[-le z]
(l+z)
S.
C.
C
of convex functions is a proper subclass of
C
In fact if we
247
0UASI-CONVEX FUNCTIONS AND RELATED TOPICS
f[(x+z)/(l+xz)
F(z)
-f(x)
xeE, zE,
z
where
f(z)
then the function
f,(z)
(l-z) z
E
defined in
by
Z
f.(z)
d
0
C*
belongs to
(II)
C,
but not to
The Class
S
The class
C*
-
while a proper subclass of the class
functions, is not contained in
l-i
f(z)
,
belong to
but
does
,
f
not
z
l-z
l+i
The class
R
studied by Pommerenke [9].
2
Re
0
01, 02
f
necessary condition for
It
the
such that
0
01 2’
<0
coefficient
result
Ra
f’(z)
as follows:
0
belongs to
)
-a,
’ 01 < 02 ’
2,, 0
dO
’ r ,<
C
Ra,
0’<I,
if and only if
I.
and
Re,
we need the following
C
fEC*
2
has
is starlike
and
with
0
been
re
z
and
--(zf’(z))’}
f’(z)
f
PROOF.
k(z)
-e<0<0.
of univalent functions was introduced by Reade [13]
from
clear
is
,
0
z
i0
the
{(zf’(z))’
f’(z)
such that
Let
e
C*
Before establishing a relationship between
THEOREM 3.1.
z
and
It
We define
An anlytic function
for all
when
0,
R
The Class
0
f()’ <
(l-z)
C
to
zf’(z)
e, Re
Also the Koebe function
,
distortion theorems for the class
(iii)
log (l-z)
2
is not starlike.
belong
of close-to-convex
K
For example, the function
S
but for sufficiently small
C
This means
see [6] for more details.
proved
d0
)
0r<l.
+
Then, for
02-01
(3.1)
2
in [14] that for
zf’
FeK,
and for all
2.
0
02-0 1,, f 2
2
O
Re
io
relO F’(re
,).}
10
F(re
dO, 2.
+
o2-o
2
01
0
2
K.I. NOOR
248
,
using this and the fact that
result.
We
that
note
0<e I.
where
in
if and only if
fEC
82-8
(3.1),
we obtain the required
zf’eK,
can be very small and we can take
2
82-8
2
Thus we conclude that
e, (0<a<l).
for some
RKMAK 3.1.
It is an open problem to find the exact value of
,
It should be some fixed
with
C
(iv)
The class of functions convex in one direction.
n,-,ber determined
Robertson [15] introduced the class
by
ee(0,1)
that goes
C
of convex functions in one direction.
C
These are the functions for which the intersection of the image region with each line
of certain fixed direction is either empty or one interval.
if
f has real coefficients, then feC
He has also shown that
zf’eT,
if and only if
T
where
is the class
of typically real functions, that is, the functions with real coefficients.
We prove the follwing:
,
THKORM 3.2.
If
one direction.
,
C
Let
K
C
zf’K(R).
E
in
,
PROOF.
and
fC
C (R), K(R)
and has real coefficients, then it is convex in
CI(R)
and
classes
the
be
respectively and have real coefficients.
K(R)
But
Hence
T.
zf’eT
feC
and so
I(R).
of functions which are in
Let
fC (R).
,
Hence
C (R)
This implies
I(R)
and
f(E ).
The
C
this proves the theorem.
From Theorem 3.2 and the results for the class
,
THKORM 3.3.
Let
fEC (R).
(I)
Re
CI(R)
in
[15], we have:
Then
f(z)
>
and
f(z)
-----z
is subordinate to
(l+z)
I.
i8
(ii)
I+21 a21 r+r
z+ r.
f(z)
where
n=2
(iii)
2w r
,
L(r)
(l-r
2)
where
equality is obtained for
(iv)
larg
f(z)
z
arcsin
f(z)
Iz
and
arg f’(z)
(v)
2 arc sin
,
FC (R),
where
Re
2
re
n
a z.
n
L(r)
z
-z"
is the length of the closed curve
r
249
QUASI-COhWEX FUNCTIONS AND RELATED TOPICS
I
F(z)
z +
f(tz)d#(t)
(t)
n anZ
l
n=2
0
(0,I)
any real function monotonic increasing in the interval
is given by
moments sequence {
n
is
tn
n-- 0
d(t),
and the
I
Thus we have seen that
c=c*
()
,
(2)
cRacKcS
C (R) cO(R) cT
,
We now discuss the relationship of
way.
We have the following:
Let
TKORKM 3.4.
,
_
fC
E.
in
_
with other subclasses in a different
C
f
Then
Izl<r-42-5
maps
0.6568
onto a
convex domain, and this result is sharp.
This follows at once from the result of Lewandowskl
such that the image of
r
exact radius
(with respect to the origin) is
4-
r
We
see
that, from
5
fEK
is a starshaped domain
0.6568
result
this
Izl<r
[16] where he proved that the
by
zf’S
zf’K
fC
for
z
<4/-5
fC
for
Lewandowski’s method yields the existence of an extremal function which maps
E
onto the w-plane cut along a half-llne not passing through the origin consequently we
have the extremal function for theorem 3.4.
THEOREM 3.5.
(zf’(z))
Re
> 0
g’(z)
4.
(a)
Let
fC*
for
Izl
<
APPLICATIONS OF THE CLASS
The Class
K
gC
and
in
K.
For the proof
zf’(z)
g(z)
see [17].
If
Re
>
0, zE,
then
C
I.
We now introduce a new class
quasl-convex function.
K
by replacing convex function
g
in (1.4) with
This generalizes the concept of qasi-convexlty and close-to-
convexity both.
I}FIIITIO 4.1.
Let
f
be analytic in E and f(0)
,
gEC
if and only if, there exists a
f’(z)
g,z) >
,
Clearly
C
such that for
0, f’(0)
I.
Then
feK I,
zEE,
0.
KcKI.
We state some basic properties of the class K I.
We refer to [18] for the proofs.
250
K.I. NOOR
Let
THEOREM 4.1.
and be given by
feK
r.
z +
f(z)
n2
(i)
a
n
In,
for all n.
(-r)
(l+r) 3
(+/-+/-1
Then
a z
n
if,(z)l
(,+1
(i-r)
3
I
(iii)
(i+r)
2
(i-r)
2
(iv)
All inequalities are "sharp, equality being attained for
fo (z)
z
e K
(l-z) 2
(v)
where 0(I) denotes a constant.
Or<l.
The question whether the factor
(vii)
(b)
(l_--Ir)I/2
For
zf’
implies that
feK
Izl
is univalent in
< o-g"
Alpha-quasi-convex functions.
Mocanu [19] introduced the class
Let
can be improved is unsettled and remains open.
a
be
real
and
suppose
of alpha-convex functions as follows:
M
f:f(z)-z + Z
that
n2
f(z).f’(z)O.
feM
Then
zf’(z)
f(’Z) +
It has been shown [20] that all
they unify the classes of starlike
n
analytic
is
in
E
with
zeE,
if, for
Re {(i-a)
a z
n
a
(zf’(z))
f’(z)
a-convex
(affi0)
>
0.
functions are univalent and starlike and
and convex
(affil)
functions.
Using the concept of quasl-convexlty, we now define the following:
DEFINITION 4.2.
Let
a
be
real
and
f:f(z)=z + Z
n=2
a z
n
n
be
analytic
in
E.
Then f is said to be alpha-quasi-convex, if and only if there exists a convex function
g such that, for
zEE
251
QUASI-CONVEX FUNCTIONS AND RELATED TOPICS
f’(z)
+
[(l-a) g,(i)
Re
We
QI=C
,denote
the
class
of
(zf’(z))}
>
g’(z)
a
O.
a-quasl-convex functions
We
Qa"
as
Thus alpha-quasl-convex functions connect the classes
,
S
way as alpha-convex functions do
note that
K
and
C
Qo-K
and
in the same
C.
and
In [21], we proved:
(l-a) f(z) + azf’(z),
if and only If, FeK.
(1)
Let
(ii)
feQa’
F(z)
if and only if, for
such that, for
a
>
a
and
O,
be real, a
)
O, zeE.
Then
feQa
there exists a close-to-convex function
F
zeE
-2+--
---z
f(-)
F(t) dt
z
0
(iii) Every
(iv)
a-quasi-convex function, for
Let FeK in E.
a
0
is close-to-convex.
Then F will be a-quasi-convex in
Izl<r 0
I/(2ad4(4a 2-
2a + I)).
This result is sharp.
(v)
Let
fQa
z +
and be given by f(z)
r.
a z
n
n
2,
Then, for n
n
This result is sharp as can be seen from the function
f0(z)
(c)
z
1---z
a
---1
ta
(l-t) -2 dt
0
,
The class
C (8,Y)
A function
fgS
Re
We note this class by
Also
feS
of quasl-convex functions of order
is called a convex function of order
(zf’ (z))’
f’(z)
>
if, for zE,
8,
C(8).
is a starlike function of order
Re
8 type
8
8, 0
zf’
(.z)
f(z)
>
8, 0
8
I, for
zcE
8,
and we call this class as S (8).
These two classes were introduced by Robertson [22].
In [23], Libera introduced the close-to-convex functions of order
8
type
K.I. NOOR
252
DEFINITION 4.3.
f(0)=0, f’(0)=l,
0 4 Y 4 I,
and
A function
f
E, normalized by the conditions
analytic in
said to be close-to-convex of order
is
if and only if there exists a function
Re
zf’(z)
g(z)
>
8
such that, for
K(0,0)=K.
It is clear that
K(8,Y).
We now introduce terminology of order and type together in the class
,
0
8
geC(Y)
(zf’(z))’
g’(z)
0
and
>
8
,
I.
Y
We
such
call
class
a
We shall now state some results on the class
C (8,).
as
Clearly
,
For the proofs, we refer
C (8,Y).
[24].
llOll]i
4.3.
Every quasi-convex function of order
of the same order and hence univalent.
RKM 4.1.
,
C(8,Y)
C (8,Y),
THEOREM 4.4.
Let
if and only if
fC (8,Y)
that is
(4.1)
zf’gK(B,Y).
and be given by
f(z)
z +
7.
n=2
(i)
(ii)
I]an
(2(3-Y)
Then we have
a z
n
(n-2Y)[n(l-8)+(8-Y)]
n.n!
1-r)d_r
,F0 (l+r)2-2Y[l+(l-28)r]
r(l-Y)(1-28) + (8-Y)[l-(l-r)
Y#2, Yl
(I-28)
(
T
Izl=r,
,
0<r<l.
THEOREM 4.5.
Izl<ro=
2(I-8)
+-------r
Y
The first result and rlght-hand side of the second are sharp.
Let feS
and
log(l-r)
2-2Y]
log(l-r) + (28-I), Y=l,
2(8-1)
where
is close-to-convex
type Y
we can see that an Alexander-type
K(8,Y),
and
,
feC (B,Y)
8
,
From the definition of
relation holds between the classes
for
as:
zcE
such that for
c (0,0)=c
to
,
f, analytic in E, normalized by the conditions
if and only if
to be quasi-convex of order 8 type y,
is said
there exists a function
where
C
A function
DEFINITION 4.4.
Re
zeE
8.
We denote such a class of functions as
f(0)=0, f’(0)=l,
0 4 8 4
where
Y
, type
gS (Y)
feC *
and gEC.
for
Let
Izl<rl
Re
zf’ (z)
3-48"
g(z)’ >
8,
for
zE.
Then
fEC* (8,0)
253
QUASI-CONVEX FUNCTIONS AND RELATED TOPICS
Q(e,8,7)
We can also define the class
8
type Y
of alpha-quasi-convex
functions of order
as:
DKFIITIOM 4.6.
quasi-convex
Let
8
of order
A function
’0.
if
Y,
type
and
f, analytic in
E, is said to be alphagC(Y)
only if there exists a function
such that
f’(z)
g, (z) +
Re[(l-)
for
zeE,
clear that
(zf’ (z))’
g’(z)
e
>
8,
8,7e[0,I]. We denote this class of functions as Q(e,8,7).
Q(s,0,0)
8, and 7, we have
Qe. For different values of
and
,
Q(0,8,)
K(8,)
Q(0,0,0)
K
Q(I,8,’)
C (8,’),
Q(I,O,O)
C
,
We notice that this class unifies the two
K(8,Y)
classes
and
C (8,Y),
It is
and it
follows from the definition that
feQ(e,8 ,)
(l-)f + s zf’} K(8,7
if and only if
Integral representation and coefficient problem can be solved in the same way as we
did for the class
(d)
oPgaTos
Let
os
mmssgs
f=T(F), where
properties of
f
when
T
c (8 ,v ).
is an operator.
FC (8,Y), 8,
Now we shall be dealing with the mapping
[0,I]
and
T
is a differential or integral
operator.
Here we shall discuss the case when
u-H^(E)
FHo,
I(F)
In [7], Libera
I
considered the operator
where for
T is an integral operator I.
Iflf(z)
f
f(z)
z +
l a z
n
n=2
n
and analytic in E
and
2
"
z
f
F(t) dt.
0
He proved that
I(C)=
c
I(S) s
I(m)
m
(4.2)
K.I. NOOR
254
A
of (4.2) has
generalization
defined as
In H0 H0’ In(F)
f(z)
nz
f
-n+l
been
considered in
[25] by taking operator
I
n
and
i tn-2 F(t) dt,
(4.3)
nffil,2,3
0
A simple proof of (4.2) is given by Mocanu, see [19], where it is also shown that
s (T)
i(s
(4.4)
(-3 +
where
/4
and is the same in both expressions.
Pascu [26] considered the operator
f(z)
11,
04141, I
HO, 11(F)ffif,
H
0
(4.5)
F(z) dz
z
0
which generalizes the results in
(4.2) and (4.4).
In [27], Salagean studied the operators (4.5) for the classes
S ($)
and
By using the same techniques used in [26] we obtain the similar results
classes K(B ,$) and thus consequently we have:
Then
(i)
feC (,)
If
Let f be defined by (4.5) where FeC (B,$),B,$[O.I].
Let 0<141.
THEOREM 4.6.
B4<I
where
0<14
is defined as follows:
and
I
2(I-I) 4y<l,
and
then
---[21"f+k-2+44k2f2-1212’+81T+9k2-41+4]/41>
<ffio
C(Y).
for the
0
and
(ii)
<141
If
and
offio
2
I-I
31-8
21
4
4
y,
then
[21T+l-,/412TZ-1212T+912-81]/41 >
0
and
(ill)
<141
If
<
and
<Y<I,
then
Special Cases:
1/2, =0,
(i)
For
1
(ii)
For
8ffi0, Yffi0,
FE C
I:0,
we obtain a known result for the class
we see that
fE C
C
see
[16].
QUASI-CONVFX FUNCTIONS AND RELATED TOPICS
where
0<II
Let
4.7.
TF)JM
0Y I, >0.
Then
and 08<1.
Let
f
255
given by (4.5) and
be
FeQ(a,8,y)
feQ(e,8 ,).
For the proofs of the above results, we refer to [28].
Special Cases:
(I)
For
(ll)
For
,
see [21].
Qa,
we obtain this result for the class
=0, 8=0,
=0, 8=0, we see that
FeQe
feQe
Using the integral representation of
4.2.
fgQ(s,8,)
and theorem 4.7,
0<
we notice that, for
K(8,).
Q(s,8,Y)
In [29], Livingston has studied the converse question considered by Libera [7].
f defined by
In fact he studied the mapping properties of the function
where
D
is a differential operator and
then
FeS
example he has proved that if
(4.6)
(zF(z))’,
D(F(z))
f(z)
F is one of the subclasses of S. For
f, given by (4.6), is starlike for z "2
and, in general, in no larger disc centered at the origin.
[30]
Padmanabhan
,
izl<{_2 (y2
has
OY
for
+
of
f’1,
then
order
.
defined
4e}/2T.
+
Libera
and
of
by
Livingston
imposing
His main theorem shows that if FeS
F.
by
results
the
refined
restrictions on the character of
(4.6),
starlike
is
of
the
He obtains analogous result when
Livingston
[31]
F
order
same
further
(Y),
for
is a convex function
extended and generalized the results of
They extended to include the range of Y when
zf’ (z)
and generalized by finding, the sharp radius of the disc in which {Re f(z) >}
Padmanabhan
the
in
following
ways.
,
when
FgS
(), 0<I, 0<I
and
for the complimentary case when
).
<Y,
They were not able to obtain suitable results
but in [32] Bajpal and Slngh gave a method
which covers both of the cases and their result is the best possible.
D
We can generalize the Livingston differential operator
Dl(f)
where
l>O
f(z)
(4.7)
(l-l) F(z) + IzF’(z),
zE. The mapping properties of the function
and
S
the subclasses of
as following:
f, when
F
is in one of
have been studied in [33].
We generalize Libera and Livingston’s result by replacing Livingston’s operator
(4.6) by the operators (4.7) and have the following:
TKORKM 4.8.
given by (4.7).
Let
Then
081, 0’l, ’1
feC (U,)
for
Izl<r
and
,
8il.
where
r
Let
FeC (8,’)
is given as
and
f
be
K.I. NOOR
256
r
r
where
0
min (r
O,r2)
is the smallest positive root of
2
(I-o) + 2{(y-o) + X(l-Y) (o-2)}r + (2Y-o-l)(l-2k(l-Y))r
and
r
2
O,
is the smallest positive root of the equation
[l-(l-2X(l-Y))r][(l-) + 2{(B-p) + X(B+Y+(l-Y) -2)}r
+ (213-p-1)(1-2),(1-Y))r
2]
0
,
see [34].
B=O, this result reduces to one for the class C
YI, BI and yol. If
R 4.9. Let >0 and %>0. Let
where
FQ(,B.) and f is given by (4.7), then feQ(,p,o) for
For
defined as in theroem 4.8.
When
=Y=O,
we obtain this result for the class
,
Izl<r
see [34].
r
is
For the proofs
of the above theorems we refer to [35].
We can demonstrate the relationship between all the subclasses of
S
as follows:
Set inclusion, O<a;l, and B,yc[O,I].
257
QUASI-CO/VEX FUNCTIONS AND RELATED TOPICS
EFERENCES
Multivalent Functions, Cambridge University Press, U.K. 1967.
I.
HAYMAN, W.K.
2.
DUREN, P. L.
3.
GOOKL&N, A. W. Univalent Functions, Vol. I, II, Mariner, Tampa, FL, 1983.
ALEKANDER, J. W. Functions Which Map the Interior of the Unit Circle Upon Simple
Regions, Ann. Math. 17 1915-16, 12-22.
KAPLA, W. Close-to-convex Schlicht Functions, Mich. Math. J. __I 1952, 169-185.
NOOR, K. I. and THONAS, D.K. Quasi-convex Univalent Functions, Int. J. Math. &
4.
5.
6.
Univalent Functions, Springer-Verlag, Berlin, 1983.
Math. Sci. 3 1980, 255-266.
7.
LIB.RA, R.J.
8.
WAADELAND, H.
9.
I0.
I.
Some Classes
Soc. 16 1965, 755-758.
of
Regular
Univalent Functions, Proc. Amer. Math.
Uber k-fold Symmetrische Sternformige Schlichte Abbildungen des
Einheitskreises, Math. Scand. 3 1955, 150-154.
POMMRNK, Ch. On the Coefficients of Close-to-convex Functions, Mich. Math. J.
9 1962, 259-269.
CHUNIE, J. G. and IGH, F. R.
On Starlike
London Math. Soc. 35 1960, 229-235.
STROAOER, F..
and
Convex
Beitrage zur Theorie der Schtichten
Schlicht Functions, J.
Funktionen,
Math. Zeit. 37
1933, 356-380.
13.
BURDIO, G. R., KEOGH, F. R. and MES, E.P. On a Ratio of Univalent Functions,
J. Math. Anal. Appl. 53 1974, 221-224.
READE, M. O. The Coefficients of Close-to-convex Functions, Duke Math. J. 23
14.
KIM,
15.
ROBERTBON, N. S. Analytic Functions Starlike in One Direction, Amer. J. Math. 58
1936, 465-47 2.
16.
NOOR, K. I.
17.
NOOR, K. I. Some Radius of Convexity Problems, C.R. Math. Rep. Acad. Sci. Canada
5 1982, 283-286.
NOOR, K. I. and I-OBODI, F.N. A Generalized Class of Close-to-convex Functions,
Pak. J. Sci. Res. 34 1982.
12.
.
1956, 459-46 2.
18.
J. and ES, E. P. On Certain Convex Sets in the Space of Locally
Schlicht Functions, Trans. Amer. Math. Soc. 196 1974, 217-224.
on a Subclass
Pauli 29 1980, 25-28.
of Close-to-convex Functions, Comm. Math. Univ. St.
Une Propriete de Convexite Generalizee Dams la Theorie de la
Representation Conforms, Math. (Cluj) (34), II 1969, 127-133.
19.
MO, P. T.
20.
MILLER, S. S., MOkNU, P. and READE, M.
21.
22.
All a-convex Functions are Univalent and
Starlike, Proc. Amer. Math. Soc. 2 1973, 553-554.
NOOR, K. I. and L-OBOUDI, F. M. Alpha-quasi-convex Functions, Car. J. Math.
3 (1984), I-8.
ROBERT$ON, M.
$.
On
the
Theory
of
Univalent
Functions,
Ann. Math. 37 1936,
374-408.
23.
LIBERA, R.J.
Some Radius of Convexity Problems, Duke Math. J. 1964, 143-158.
The Class C (B ,Y) of Quasi-convex Functions of Order Type, to appear.
24.
NOOR, K. I.
25.
BERNARDI, S.N.
26.
PAS(I, N. N.
27.
SALAGEAN, G.
28.
NOOR, K. I. and AI.-KRORSANI, H. A.
Convex and Starlike Univalent Functions, Trans. Amer. Math. Soc.
135 1969, 429-446.
Alpha-starlike Convex Functions, to appear.
Properties of Starlikeness and Convexity by Some Integral
Operators, Lect. Notes Math. No. 743, Springer-Verlag, Berlin, 367-372.
$.
Properties of Close-to-convexity Preserved
by Some Integral Operators, J. Math. Anal. Appl., 112 (1985), 509-516.
K.I. NOOR
258
29.
LIVINGSTON, A. K. On the Radius of Univalence of Certain
Proc. Amer. Math. Soc. 17 1966, 352-357.
30.
PADMANABHAN, K. S.
31.
LIBKRA, R. J. and LIVINGSTON, A. E.
32.
BJP&I, P. L. and SIGH, P.
33.
NOOR, K. I., AL-OBODI, F.M. and AL-DIHAH, N. A.
35.
Functions,
On the Radius of Unlvalence of Certain Classes of Analytic
Functions, J. London Math. Soc. (2)
1969, 225-231.
On the Unlvalence of Some Classes of Regular
Functions, Proc. Amer. Math. Soc. 30 1971, 327-336.
The Radius of Starlikeness
Functions, Proc. Amer. Math. Soc. 44 1974, 395-402.
Convex
34.
Analytic
Combinations
of
Analytic
of
Certain
Analytic
On the Radius of Unlvalence of
Functions, Int. J. Math. & Math. Scl. 6
1983, 335-340.
HOOR, K. I. and L-DIHAH, N. A. A Subclass of Close-to-convex Functions, Punjab
Univ. J. Math. 14-15 1981-82, 183-192.
NOOR, K. I. and AL-KHOSAHI, H. A. Generalization of Livingston’s Operator for
Certain Classes of Univalent Functions, to appear.