Internat. J. Math. & Math. Scl.
VOL. 15 NO. 2 (1992) 279-290
279
ON SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS OF HIGHER ORDER
KHALIDA INAYAT NOOR
Hathematlc. Department
P.O. Box 2455, King Saud University
Rtyadh 11451, Saudl Arabia
(Received October 30, 1990 and in revised form October 21, 1991)
ABSTRACT.
Tk(O),
The classes
using the class
Vk(O)
0
0
<
I, k > 2, of analytic functions,
of functions of bounded boundary rotation, are
defined and it is shown that the functions In these classes are close-to-
Covering theorem, arc-length result and some radii
convex of higher order.
We also discuss some properties of the class
problems are solved.
Vk(P)
including distortion and coefficient results.
’1980 AMS SUBJECT CLASSIFICATION. 30C45.
KEY WORDS AND PIIRASES: Analytic functions, close-to-convex, unlvalent,
bounded boundary rotation, coefficient, positive real part.
I.
THE CLASS
Let
{z:IzI<l}
Pk(o)
Pk(o)
be the class of functions p(z) analytic in the unit disc E
satisfying the properties p(0)
f 2Re_(z)
_,-
0
where z
re
i0
p
Id0
k2 and O<p<l.
case k
(1.1)
k.,
This class has been introduced in
note that, for offi0, we obtain the class
0, k
and
Pk
[I].
We
defined by Pinchuk [2] and for
2, we have the class P of functions with positive re’a1 part. The
2 gives us the class P(p) of functions with positive real part
greater than p.
Also we can write
2
p(z)
+ (I
0
2p) ze
ze
-it
-it
d(t)
where (t) is a function with bounded variation on [0,2] such that
J
0
2
dr(t)
2
and
(1.3)
2
0
From (I.I), we have the following.
280
K.I. NOOR
TIIEOREFI 1.1.
k
P(0),
pi
Pk(O).
p c
(Z+-)
pCz)
whee
Let
pl (z)
Then
(--)
p2 (z),
1,2.
We now prove.
THEOREH 1.2.
PROOF.
Let 1t 1,
--
Pk(p)
The class
i12r. Pk(O).
is a convex set.
We shall show that, for a, B
0
Pk()"
belongs to
From Theorem 1.1, we can write
tl(z)
[{(gk + ) PlCZ)
pie P(O),
where
{(_u4 ’+-)
+
k
( )
k
(" ’) P4 (z)
P3 (z)
1,2,3,4.
Now, writing
(1-O)
pi(z)
+ o,
hi(z)
i=1,2,3,4, see [3],
we have
tl(z)-o
o
k
(+7
(k
where
fl
I-g;
(z)
and
f2
.
(_k 7) l-(
(- (z) + fh (z))}
3
(_k
+
2
h (,-)
(z),
P, since P is a convex set, see [2] and this gives us the
required result.
THEOREH 1.3.
27
(i)
f
0
T;
Let p c
ip(re i0 )i 2
Pk(O)
and be given by
l+[k2(l-o) 2-t
dO
r
r
+
(z)
E
e z
Then
2
2
and
f Ip’( rei o) Ido
(ii)
k(l-o)
0
-r
PROOF. (t)
1___
2rr
21
f
0
ip(r e
2
Using Parseval’s identity, we have
)i2dO
n=O
+
I n 2 2"
k2(1-0)2
E
r
2n
I+[ k2 (l-o)
n=l
where we have used an easily establtsled .qharp
result
nl.
-1
r
(1 -r 2
Icn lk(1-o),
for all
281
SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS OF HIGHER ORDER
(ii) By using Theorem I.L, we can write
p(z)
where
(_4 + )(I-P) h l(z)
o
(l-o)
h2(z),
c P.
h2
hi,
( -)
Therefore,
(1.4)
Now, for all h.P, we have
2w’(z)
h’(z)
where
(l+w(z-)-) 2
w(z) is a Schwarz function [3], and
2w
o
iO)ldo
W 0f
2
l-r
21I
(1.5)
2
llence, from (1.4) and (1.5), we have
Ip
J
(re i0)
0
id0
l-r
2-’
which t8 the required result.
From Theorem I.I and tlle properties of the class P(O), we immediately
have the followlng.
THEOREM 1.4.
Let
-k(l-o)r + (l-2o)r 2
-r
-r
pPk(p).
Let
ro= 2/[k(l-o)
When
2.
Then
+ k(l-p)r + (l-2o)r 2
Re p(z)
2
THEOREM 1.5.
Pk(O).
p
+
[z] <
Then peP for
4k2(l-o) 2
2
4(l-2p)],
p
ro,
where r0
is given by
(1.6)
o=O, we obtain the results proved in [2].
’tl cAss
DEFINITION 2.1.
Let
Vk(0)
denote the class of analytic and locally
univalent functions f in E with normalization f(0) -0, f’(0)
and
satisfying the condition
(zf’(z))’
f’(z)
Pk(O),
0 < p
<
1,
k
2
When o-0, we obtain the class Vk of functions with bounded boundary rotation. The class Vk(p) also generalizes the class C(p) of convex functions
of order
.
It can easily be seen [l] that f e
Vk(o)
if and only if there
K.I. NOOR
282
exists F e V k such that
(F’(z))i-
f’(z)
(2.1)
In the followlng, we will study the distortion theorems for the class
Vk(o). We will use the hypergeometrlc functions
r(c)
r(a) r(b)
G(a,b; c,z)
r(a+n) r(b+n)
r(c+n)
z
n=O
r(c)
r(a)r(c-a)
u
a-I
z
n
n’.
(l_u)C-a-I (l_zu)-b
du,
0
where Re a>O and Re(c-a)>O. These functions are analytic for z.E [4].
In addition, we define the functions
b-I
2
Ml(a,b; c,r)----a [G(a,b;
c,-l)-
rla G(a,b;
c,-r
-I
)]
(2.2)
and
H2(a
where
2b-I [G(a
a
b; c r)
r
b; c,-l)
G(a b;
c-rl)
[z
r (0
r
rl
+ r
bt2(a,b;
c,r)
Vk(O).
Let f
THEOREH 2.1.
g
if(z)l
g
Then, for
Ml(a,b;
<
<
r
c,r),
I), we have
(2.3)
where
a
b
c
and
Hi,
M
(--
l) (I -o) + l,
(2.4)
=2o
(-
l) (l
o) + 2
are as defined in (2.2).
2
This result is sharp.
Using (2.1) and the well-known bounds for
PROOF.
[2],
IF’(z)l
with
FCVk,
we have
k
k
(+
k
-o
z_ )(
c
k
) (I-o)
c
I--I)
(2.5)
/
l)(l-o)
Let d r denote the radius of the largest schlicht disk centered at the
origin contained in the image of
r under f(z). Then there is a point
o
I’-o]
,
Izl <
such that
If(’- o )1
dr-
The ray from 0 to f(z
o
lies
entirely in the image of E and the inverse image of this ray is a curve in
Thus
d
r
If(z o )1
f If’(’-)l Idol
C
see
-
SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS OF HIGHER ORDER
k
k
c
(b
283
l) (l-o)
+ l) (i-o)
(’-"- t)
2
dt
k
+ l)
(l+t)(
k
l-t
Let
f
o
cT;7)
4-
Then
So
lC--o>l
2
r
1- z
Put
dt
(l+t)2CL-o)
-2
l+t
k
(1+6) -20
I
0
l) (L-o)
r
+r
d
)2- dt
and
(1+0 -20
d}
rlu
This gives
Iz
0
2b-I 1o a
a
)!
M2(a,b
b; c-1)
-r
G(a b;
c-rl)
c,r),
a,b,c and M2 are respectively defined by (2.4) and (2.2).
where
Similarly we can calculate the lower bound for
If(z)
and this
establishes our result.
Equality is attained in (2.3) for the function f o c
k
(t +
f’(z)
6tz)(62z)(k
o
(1
Vk(O)
defined by
t)
+ 1) (1
O)
now study the behaviour of the integral transform
z
f (z)
a
f (f,())c
0
dd
(2.7)
for fV (0)
k
This problem has been studied for the class of univalent normalized
functions in E and for the close-to-convex functions, see [3].
/e have
K.I. NOOR
284
V
Then f
.
Let
TIIEOREH 2.2.
given.
V
k(o),
0
<
0
l,
2
k
and let a, O<a<l be
mla(l-p)(k-2)+2].
for
From (2.[), we have
PROOF.
f’(z)
(F’(z)) I-0
f’(z)
(f’(z))
F e V
k
Now
a
(F’(z))
-log (l-e -it) a(l-o) din(t)
--exp
-log (l-e
exp
where
(-o)
do(t)
a(l-o) din(t) +
d0(t)
a(1-o)
-It)
d0(t),
a(l-O)]
[I
d_.t
Also
J
l-a(l-o)
din(t) +--
--W
2
dt
I
--1I
and
i,
Ido(t)l
+ l-a(l-0)
Idm(t)
a(l-o)
dt
--7
--f[
a(l-o)k
/
--1
211- a(l-o)]
Hence tile result.
We note that
fa
is univalent for
of univalent functions for 2
not unlvalent provided
a
<
a
Hence f
4.
m
2
< (l-o)(k-2)
since
Vm
consists
is unlvalent even if f is
2
(l-o)(k-2)"
Using tile standard technique, we can easily prove the
following.
THEOREM 2.3.
g,heVk(O)
Let
Z
and let
a
> O, 8 > 0 and
a
+ 841
(g’(t)) a (h’(t)) 8 dt
H(z)
0
iS convex of order
01
(I
[k
lk
I_--Z
for
[z <
r
where
2
4]
The result is sharp when
g’(z)
h’(z)
[(
l_z.(-)_
t( I+z)(-
I)(l-O)
+ I)(I-o)
(2.8)
o. Then
SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS OF HIGHER ORDER
285
We now prove the following.
THEOREM 2.4.
Let f:f(z)
z +
E
anz
E
Vk(O).
Then, for all n
>
3,
<=.
2 <k
(1-O)(-- +
I%1 < [k2(l-p) 2 + k(l-o)] 2-20 (._)2n
The function f
k
[(I-o)( +
2]
I)
PROOF.
2
(2.6) shows that the exponent
defined by
O
1)
is best possible.
By definition, we have
(zf’(z))’
f’(z) p(z),
Pk(p)
p
Set
F(z)
(z(zf’(z))’)’
[p
f’(z)
For
z
re
i0
(z) +
we have
21
n
zp’(z)]
lnl’ 27 r n-3
f
0
if’(z)l iP 2 (z), zp’(z)ldO
Using (2.5) and theorem 1.3, we obtain
(t+r)
n-3
n
(l-r)
(L+r)
r
Let
r
n-3
(t-)(2-)
(t-o) "-k-.
--)
{l+JkZ(l-P)-- 2_1 2}r
(i-o)()-t
{l+k(I-o)
k+2.
>
n
3.
+ k(l-0)
r
[k2(1-0) 2-1 Jr 2}
+
(l-o)(T+t
(l-r)
3
2
Then
k+2.
31ani
[k2(l-O) 2
+
k(l-o)]e 3.
3)
(2
.k+2.
[k2(l-p)2+ k(l-p)]e3.() l-o)(T-2] n:zThus, for n)3,
lan
< [kZ(1-p)2
THEOREM 2.5.
k
+
k(l_o)](2)-2o.
(.__)2n
Let f c Vk(p),
o
is given by (I.6).
0
shows that this result is sharp.
domain where r
$
(l-o)(+
1/2.
t)
k
(
-)
(1-o)(-
2
Then f maps
The function f
3
O’
[z] <
r
0
onto a convex
defined by (2.6)
K.I. NOOR
286
The proof is straightforward and follows immediately from the definition and Theorem 1.5.
Furthermore it can easily be shown that if f c Vk(O)
( r where r is given by (2.8).
3.
then f is con-
Izl
vex of order 0 for
VIlE CLASS
A class Tk of analytic functions related with the class V
k
introduced and studied in [5]. We now define the following.
k
)
f’(z)
P
g’(z) e
2,
be analytic in E.
O, f’(0)
with f(O)
DEFINITION 3.1. Let
fcTk(o),
p<l, if there exists a function
0
has been
Then
gcVk(p) such that
for z e E.
Note that
Tk(O)
T
and
k
T2(O)
is the class of close-to-convex
functions.
THEOREM 3.1.
Tk(O).
Let
If(z)l )biz(a+l
where
H2(a,b;
c,r)
Then
b; c+l,r),
is defined by (2.2) and a,b,c are given by
(2.4).
This result is sharp.
PROOF.
Tk(o),
Since f c
g’(z) h(z),
f’(z)
It Is well-known that for
g c
we can write
Vk(0),
h c P.
h e P
(3.1)
Thus, using (3.1) and (2.5), we have
k
]-’(z)
.(t
zl)
(
*)(*-) +
k
(1 +
Proceeding in the same way as in Theorem 2.1, we obtain the required
result.
REMARK 3.1. When 0=0, fcTk and since in this case b
0<1, c
l+a-b,
1. Letting r
G(a,b; c, -1)
1, with 0- O, in Theorem 3.1, we
we have
see that the image of E under functions f in Tk constalns the schllcht
disk
Iz[ < k+’-"
We now give a necessary condition for a function f to belong to the
class
Tk(O).
THEOREM 3.2.
Let
f c
Tk().
Then, with
z
re
10 and
01<
02;
O(o<l,
-
SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS OF HIGHER ORDER
fo
02
(zf’(z))’
dO
,(..)
Re
(gl’(z))
f’(z)
I-p
So
(gl(z)
f’(z)
fi e
-k(l-o)
We can write
PROOF.
for some
>
T
(h (z))
I-p
hL(z))l-
for some
(f(z))
gl e
Vk, hle
-p
287
P.
(3.3)
k.
Hence
(zf’(z))’
f’(z)
(zf(z))’
(x-o)
ft’(z)
+ o
The required result follows on noting that, for
f0
02
REMARK 3.2.
(zfl(z))’
f[(z)
Re
dO
k
>
01< 02, fl e Tk
see [5].
In [t], Goodman introduced the class K(8) of normalized
analytlc functions which are close-to-convex of order 8
0 and showed that
if f is analytic in E and f’(z)
re
O, then for 8 > O, feK(B) If for z
and
01 <
0
2
fz
Re
(zf’(z))’
dO
f’(z)
>-
8n
B < I, K(B) consists of univalent functions, whilst if
I, f need not even be finitely-valent.
When 0
B
>
We note that Theorem 3.2 shows
tl, at
z(k(1-O))
Tk(p)6__._
--’-
Hence Tk(p) consists entirely of univalent functions if 2
also follows easily from the definition that the class
set of a llnear-invariant family of order
(k
2
(I--"
Tk(p) forms
It
a sub-
[--(l-p)+l].
Using the method of Clunle and Pommerenke as modified by Thomas
[7],
we can easily prove the followlng:
of the image of
THEOREM 3.3. Denote by L(r,f) the length
I,
max If(re i0). Then, for 0 < r <
u[ffir under f and by M(r)
L(r) < A(k,p)M(r)
log-i-i{
only on k and p.
where A(k,p) is a constant depending
Let P
in E given by
denote the class of functions p(z)
the circle
K.I. NOOR
288
+ e
p(z)
+
l.
2
which satisfy the Inequality
has been introduced in [8] and it is shown there that, for
The class P
,
p’(z)
p(z)
where
(I + c)
(3.4)
(1 + cr)(l
r)
2a
c
We now prove the following.
Let g e
THEOREM 3.4.
Vk(o)
[z <
function of order p for
r
f’(z)
l" Then f
g’(z) c
(0,I) is the least
Pa,
and let
where r c
is a convex
positive root of the equation
[(p+c) + ck(l-o)]x 2 + [p(k-c)
(l-p)cx
(l+k)]x
+ (l-p)
0
We can write
PROOF.
(gl,(z))
f’(z)
l-p
p(z), gl V k,
Pa,
P
So
-O]
fi(z)
)(l-O) Re
1-
gi(z)
p(-)-
0 and (3.4), we have the required result.
Using Theorem 1.4 with p
Furthermore, if
T(r)
[(p+c)
(l-o)cr
+
ck(l-p)]r 2 + [0(k-c)
(l+k)]r
+
then we note that
T(O)
(l-p)
> o
T(1)
-20c
20
r e
(0,I).
Thus
COROLLARY 3.1.
Izl <
r
g
V
k.
o=O,
Then f is convex for
,
Izl <
r
unlvalent functions and in this case r
For a
O, k
4 and p
Ratti [9] and when k
0
0, f
Tk.
Thus f maps
onto a convex domain and this result is
sharp, see [5].
When
<
and o
0, c
When x
-1/2[(k+2) -Ck z + 4k]
COROLLARY 3.2.
k(1-p)
ck(1-p)
and then we have
=F.
=-
if’(z)
’g’(zO
<
for
For k-4, Vk consists of
This result is proved in [8].
0, we obtain the known result
r
3
22 of
2, we have the well-known result giving us the
radius of convexity for close-to-convex functions.
SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS OF HIGHER ORDER
289
Finally we have
Let
THEOREM 3.5.
Then F
r
E
2
F(z)
"r
Tk(0)
for all
z
r.
l-m
Vk(0)
z
m
E(z)
[z < r2,
2(l+m)/[(l-o)k
+
and let
I’,
m
1,2,3,.
where, for
,/(l-o)2k
4(l-20-m)(l+m)],
The proof Is straightforward when we note that
(zf’(z))’
F’(z)
and then use theorem 1.4.
The author is grateful to the referee for his valuable
and suggestions.
ACKNOWLEDGEMENT.
omments
REFERENCES
I.
PADMANABHAN, K.S. and PARVATHAM, R. ’Properties of a class of
functions with bounded boundary rotation’, Ann. Polon. Math.
31(1975), 311-323.
2.
PINCHUK, B. Functions with bounded boundary rotation,
I0(1971), 7-16.
3.
GOODMAN, A.W. Univalent Functions V.ol. I,!I
Company, Tempa, Florlda, U.S.A. (1982).
4.
WHITTAKER, E. and WATSON, G. A course of modern analysis, Cambridge
Univ. Press, New York, 1927.
5.
NOOR, K.I. On a generalization of close-to-convexlty, Int. J. Math.
& Math. Sci. 6(1983), 327-334.
6.
GOODMAN, A.W.
l.J. Math.
Mariner Publishing
On close-to-convex functions of higher order, Ann.
Etus Sect. Math. 25(1972), 17-30.
Unlv. Sci. Budapest
7.
THOMAS, D.K. On close-to-convex functions,
45(1989), 85-88.
8.
SHAFFER, D.B.
9.
PATTI, J.S.
Pub1. Instlt. Math.
Radii of starlikeness and convexity for special
classes of analytic functions, J. Math. AnaIysls and Appl.
45(1974), 73-80.
The radius of convexity of certain analytic functions,
Indian J. Pure Ap_pl. Math. 1(1970), 30-37.