Internat. J. Math. & Math. Sci.
VOL. II NO. 3 (1988) 497-502
497
SOME CLASSES OF ALPHA-QUASI-CONVEX FUNCTIONS
KHALIDA INAYAT NOOR
Mathematics Department
College of Science Education for Girls, Sitteen Road,
Malaz, Riyadh, Saudi Arabia.
(Received April 13, 1985 and in revised form May 30, 1986)
Let C[C,D], -I<D<C<I denote the class of functions g,g(O)
0
g’ (O)=I, analytic in the unit disk E such that -(zg’ (z))
is subordinate
ABSTRACT.
to
I+CZ
g (z)
We investigate some classes of Alpha-Quasi-Convex Func-
zgE.
I+DZ’
tions f, with f(O)=f (O)-I=0 for which there exists a gEC[C,D] such that
f’ (z)
(zf’ (z))
I+AZ _I<B<A<I
+
is subordinate to
Integral rep-
(l-a)g-
g(z
resentation,
I+BZ’
coefficient bounds are obtained.
It is shown that some of
these classes are preserved under certain integral operators.
KEY WORDS AND PHRASES.
function,
AMS(MOS)
1.
starlike,
Convex,
Integral representation,
Subject classification
quasi-convex,
close-to-convex
Alpha-quasi-convex functions.
(1980)
Codes:
30C45,
30C55.
INTRODUCTION
Let S,K,S
f(z)=z
a z
n
n
and C denote the classes of analytic functions f:
which are respectively univalent,
2
(wit=respect
to the origin)
new subclass C
close-to-convex,
and convex in the unit disc E.
starlike
In [i],
of univalent functions was introduced and studied.
function f belongs to C
a
A
if there exists a convex function g such that,
for zgE,
(zf’ (z))
Re
It is also sknown that fgC
see Noor
if,
and only if,
zf’EK.
For complete study
[2].
A new class Q
of a-quasi-convex functions has been defined and dis-
[3].
cussed in some details in
real,
>0.
are called quasi-convex functions and C_C ._KCS.
The functions in C
of C
g’(z)
A function
f belongs to the class
if and only if there exists a convex function g such that,
Re
We note that
In
[4]
a function p,
Q0
K and
Q1
(l-a)
f’ (z)
(zf’ (z))
+ a
g’ (z)
g’ (z)
Janowski introduced the calss
I+AZ
I+BZ
(i.I)
C
P[A,B].
analytic in E with p(0)=l belongs
is subordinate to
> 0
Qe,a
for zeE
Also,
given C and D,
For A and B, I<E<A<I,
P[A,B],if p(z)
and S [C,D]
C[C,D]
-I<D<C<I,_
to
the class
K.I. NOOR
498
denote the classes of functions f analytic in E with f(z)=z +
n=2
(z)
(zf’ (z))
ePIC,D] and zf’
f(z)
f’(z)
and D=-I we note that C[I,-I]= C and S
P[C,D
such that
the classes
K[A,B;C,D]
ggC[C,D] such
z +
n
analytic in E,
n; <D<C<I
is said
if there exists a
g-g
that
It is clear that
a z
n
K[A,B; C,D], -I<B<A<I;
f (z)
P[A,B.
in the class
to be
n
[5] defines
Silvia
S
z
For C=I
respectively.
[i,-i]
n
as follows:
A function f:f (z)
Definition 1.1.
a
K and
K[I,-I;I,-I]
K[A,B;C,D]_K._S.
We now define the following:
in E.
Then
C,D], -I<B<A<I;
fgQ[A,B;
gC[C,D] such that,
exists a function
(i-)
It is clear that
2.
n
a z
be analytic
n
n=2
-I<D<C<I if and only if there
Let >0 be real and f:
Definition 1.2.
Q[I,-I;
f’ (z)
g’ (z)
i,-i]=
for zeE,
(zf’
g’
+
z +
f (z)
(z))
(z)
P[A B].
Q.
MAIN RESULTS
We shall now study some of the basic properties of the class
Qe[A,B;C,D].
From the definition
THEOREM 2.1.
Let F(z)
fEQ[A,B;C,D],
Then
we immediately have:
1.2,
(l-)f(z)+zf’(z),where 0<e<l is real and zeE.
[A,B;C,D].
-I<D<C<I if and only if FEE
-I<B<A<I;
We now give the integral representation for the functions in the
class
,B;C,D].
THEOREM 2.2.
A function
fEQ[A,B;C,D],_
>0,
for
-I<B<A<I;
-I<D<C<I,
and only if there exists a function FgK[A,B;C,D] such that,
z
f(z)=
PROOF.
From
(2.1),
if
for zEE,
(2.1)
F()d
it follows that
1
1
(
1
2
1
1
f (z)+z
i) z
-2
z
f’ (z)
F (z)
so
(l-)f(z)+zf’ (z)
F(z)
and the result follows immediately from theorem 2.1.
THEOREM 2.3.
fK[A,B;C,D]
PROOF.
Let
fEQ[A,B;C,D]
0< < 1 and -I<B<A<I;
-I<D<C<I.
Then
and hence is univalent.
Silvia
[5]
has proved that if
z
l+l
Fl(Z)
l
t
i_i
fleK[A,B;C,D],
fl (t) dt
Re
then so is
(2
l >0
z
bsing this result and the integral representation
for
f.
[A,B;C,D,
we
obtain
the required result.
(2
2)
with
1
i
a
2)
SOME CLASSES OF ALPHA-QUASI-CONVEX FUNCTIONS
499
For our next theorem,
we need the following result due to Silvia
7, b z n
Let FeK[A,B;C,D] and P(z)
z +
Then
n
n=2
LEMMA 2.1.
-
[5].
ib21 < (C-D) +2 (A-B)
and
(A-B) (C-D+1)
+
3
(C-D) (C-2D)
THEOREM 2.4.
Let
(A-B) (C-D+l)
3
+
6
FQ[A,B;C,D],
C-2DI >I
0<<i and f(z)
z +
7,
a z
n=2
n
n
Then
+ (A-B)
(C-D)
i
la21<--
2
and
D___Z) +
(A-B) (C-D+1)
3
l
a3 1<(1+2,
(C-D) (C-2D)
2
PROOF.
Since
fEQs[A,B;
C,D]
KEA,B;C,D].
(A-B) (C-D+1)
(C-2D)
3
by theorem 2.1,
F (z)
belongs to
+
the function
+ szf’ (z)
(l-s) f (z)
7. b z n
z +
Let F(z)
I> 1
n
n=2
Thus
(l-)[z +
7.
a z
n
3+
n
n=2
sz
[i +
n=2
n
na z
n
z +
Z
n=2
b z
n
n
or
Z a z n + s 7, na z n
(l-s)
n
n=2
n=2
Equating coefficients of z
n
7,
n
n=2
using Lemma 2.1 and the relation
REMARK 2.1.
Let FeK
n
on both sides, we have
[(l-s) + sn]a n
Now,
b z
n
[A,B;1,-1]
b
(2.3)
n
(2.3), we obtain the required result.
and be given by F(z)
z +
7,
n=2
b z
n
n
Then
I1
1
-+).
This result is sharp for the function
F
3.
THE CLASS
0
I
(z)
z
0
F0eK[A,B,I,-I]
(l+Aw)
2
(I+Bw)
and defined by
dw.
(l-w)
Q[I-26,-I; i-2,-13
In definition 1.2,
if we put A=I-28
B= -i;
C=l-2y, D
-i,
then we
have the following:
Definition 3.1.
A function f,
convex of order 6 type y,
if,
analytic in E,
and only if,
is said to be alpha-quasi-
there exists a function
K.I. NOOR
500
geC[l-2y,-l]
such that
f’ (z)
(1-)g-?
H(,f)
REMARK 3. i.
Let g be analytic in E.
Then
f’ (z)
(z)
It follows
and only if
if,
if and only if
that
Re[(l-) g’
REMARK 3.2.
geC[l-2y,-l]
i]
>y, zeE.
g’ (z)
eP[l-28,-l] implies
P[I 28
g’(z)
(zg’ (z))
Re
Thus H(,f)
(zf’ (z))
+
+ a
(z))
(z)
(zf
g
from the definition 3
(l-a) f+azf’} e
>8,
that
i,
zEE.
feQa
K[I-28,-I; l-2y,-l].
We now have the following:
THEOREM 3.1.
Let feQ
Then we have,
for n>2
[1-28, -l;l-2y,-l]
and be given by
f(z)=z + Y. a z
n
n=2
n
lanl<2__(3-2y) (4-2T)..L...+/--(n-2x)[n(l-8)+8-y_].
n! [l+a(n-l)
This result is
sharp and the equality holds for the function
f0
defined as
1
f0 (z)
z
z
PROOF.
Since
feQ[l-28,-l;l-2y,-l]
the function
F(z)=(l-) f(z)+zf’ (z)
belong to
K[l-28,-l;l-2y,-l.
Let F(z)= z +
7.
n=2
[6]
Libera
for
has proved that
n
n>2
(n-2y) [n(l-8)+8-y]
2(3-2y) (4-2y)
(3.1)
n!
n
from relation
Now,
b z
n
(2. 3)
we have
b
a
Using this and
THEOREM 3.2.
and
-
(3.1), we obtain the required result
Let f be defined as
Let 0<I<i and 0<8<i.
1
1
1
1
f(z)
z
FeQ[l-28,-1;l-2y,-l]
Iz
--2
where 0<<i,
Let
PROOF.
n
i+ (n-l)
n
F
1
FleK
Fea[-2g,-1,1-2-l]
l-2g,-1; l-2y,-i]
(z)
feQ[l-28,-l;l-2T,-l]
0
groin
remark
ant to sho that
(I-a) f(z)+azf’ (z)
Now
(3.2)
F1 (c)d"
z
t ollos
e
Then
1
l-
fl(z)
here
>0.
(l-e)F(z)+ezF’ (z)
(z)
and le
Since
1
-->l.
F()d
(3.2)
(3.?)
3.2 that
fea[l-2g,-1;
can be written as
1-2,-1
SOME CLASSES OF ALPHA-QUASI-CONVEX FUNCTIONS
F
1
(z)
(l-e)F(z)+ezF’ (z)
1
2
fl
we obtain from
I
-z
(z)
1
I-
So,
F (z))
(3.3)
1
1
2
1
2
(
-i
(
F()
’d
0
z 1
)
z
1
(z
[z
1
i
1
1
- ez
and using this,
501
1
1
-I
(
F ()) ’d
0
integrating by parts,
T
-z
fl(z)
-
F(z)+
i
1
1
i-
=[-i-F(Z)
-
[z
]+
1
1
T- E
(z
E
-i
F(Z))-
I
=az[T
z
-i
i
(--
1
1
1
-(i- --)*
F(z)+
1
i
T(I- )z
--
[i
zf’ (z)+(l-e)f(z)
(3.3)
have used
FIEK[I-28,-I;I-2y,-I
(2.2)
with
1
Y1
i-
-
d
Iz
1
1
and this completes the proof.
d]
0
F()d
0
F()d]
-2
F()d].
(3.4)
fleK[l-28,-l;l-2,-l],
l,A=l-28,B=-l,C=l-2yand D=-I.
follows from remark 3.2 and the relation
ACKNOWLEDGEMENT.
i-
F()
iIzl
and so
F())
2
0
z
1
T-
0
I 1
-(-- -1)] z
c,
i
(-- )
)z
+ (i-o)
Now in
1
i
1
1
I
z
(3.4)
that feQ
where we
Thus it
[i-28,-i;I-2y,-i],
I am indebted to the referee for helpful comments which
improved the exposition of this work.
REFERENCES
1o
NOOR,K.I.
and THOMAS,D.K.
On quasi-convex univalent functions,
InternatoJ.Math. & Math. Sci. 3 (1980)
255-266.
2
NOOR,K.I.
Sci.
ON quasi-convex functions and related topics,
to appear.
Int. J.Math.
3o
and AL-OBOUDI, F.M.
NOOR,KoI.
Alpha quasi convex univalent functions,
Carr. Math. J. 3 (1984)
1-8.
4.
JANOWSKI,W., Some external problems for certain families of analytic
functions, Ann. Polon. Math.
28 (1973)
297-326.
5
SILVIA, E.M.
Math
Subclasses of close-to-convex functions,
(1983)
449-458.
Internat.
J.
& Math __S _i_.__
IBERA,R.J., Some radius of convexity problems,Duke Mat!_..(1964),143-1c.