Irnat. J. Math. & Math. Si.
Vol. 6 No. 2 (1983) 335-340
335
ON THE RADIUS OF UNIVALENCE OF
CONVEX COMBINATIONS OF ANALYTIC FUNCTIONS
KHALIDA I. NOOR, FATIMA M. ALOBOUDI and NAEELA ALDIHAN
Mathematics Department
Science College of Education for Girls
Malaz, Sitteen Road
Riyadh, SAUDI ARABIA
(Received July i0, 1982 and in revised form February 21, 1983)
ABSTRACT.
We consider for a > 0, the convex combinations f(z)
(l-a)F(z) + azF’(z),
where F belongs to different subclasses of univalent functions and find the radius for
which f is in the same class.
KEY WORDS AND PHRASES. Univalent funtns, alpha-quasi-convex, stake, close-toconvex functio, nvex combinons.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
Primary 30A32, Secondary 30A34.
INTRODUCTION.
1.
Let S, K, S* and C denote the classes of analytic functions in the unit disc
{z:
E
convex.
zl
< i} which are respectively univalent, close-to-convex, starlike, and
In [1,2], a new subclass C* of univalent functions was introduced and studied.
A function f, analytic in E, belongs
tion g such that for z g
to C* if and only if there exists a convex func-
E,
Re
(zf’(z))’ > O.
g’(z)
Tile functions in C* are called quasi-convex and C
f
g
C* if and only if zf’
g
K.
(i.i)
C*
K
S.
It is shown [2] that
Recently the functions called a-quasi-convex have been
defined and their properties studied in [3].
A function f, analytic in E, is said to
be s-quasi-convex if and only if there exists a convex function g such that, for a
real and positive
Re{(l
a)
f’(z)
’-(z) +
a
(zf’ (Z))’
g’(z)
> O,
(1.2)
336
K.I. NOOR, C. M. ALOBOUDI and N. ALDIHAN
It has been shown [3] that F is s-quasl-convex if and only if f with
f(z)
(i- e)F(z) + zF’(z) is close-to-convex.
(1.3)
All s-quasl-convex functions are close-to-convex.
MAIN RESULTS.
2.
We shall now study the mapping properties of f:
(i -)F(z) + zF’(z),
f(z)
e > 0, when F belongs to different subclasses of univalent functions.
Let F
THEOREM 2.1.
> 0.
S* and
(i- )F(z) + zF’(z)
F(z)
is starlike in
zl
<
ro,
The function
(2.1)
where
r
o
+
2
2+
(2.2)
2
i
This result is sharp.
PROOF.
We can write (2.1) as
1
2
f(z)
z
1
i
F(z))’,
(Z
and from this it follows that
1
z
F(z)
z
-I
i
-2
z
f(z)dz
(2.3)
0
Then
1
z.F’(z)
F(z)
i
i--i
[z z---2
{((i- 1 )z c
f(z)dz + f(z))/(z
i
{(i-
0
1
z--2e f(z)dz +
{z
f(z)dz)}
z
0
i
1
---2
z
1
--1
i
f(z)}/{
z
f(z)dz}
z
h(z),
(2.4)
f(z)dz.
(2.5)
0
- -
where Re h(z) > 0, since F e S*.
From (2.4), we have
I
f(z)
Z
i
i)
z
C
i
f(z)dz
h(z)
0
z
C
0
Differentiating both sides of (2.5), we obtain
1
1
---2
I) z(
1
---i
f(z) + z
f’ (z)
1
1
---2
i)
zC
f(z) =h’(z)
i
1
z ---2
---2
e
z
f(Z)o
f(z)dz+h(z)z
0
Thus
(z)
f(z)
h(z),+ {h’(z)
z
0
1
i
O
O
f(z)dz}/{z
f(z)}.
UNIVALENCE OF CONVEX COMBINATIONS OF ANALYTIC FUNCTIONS
lh’(z)
Now, using the well-known result [4],
z
zf’(z)>
h(z){l2f(z)
Re
r
}"
1
I
z-2
i
-f(z)
zF’ (z) +
Fz)
Iz
1
i
>- Re{h(z) +
c
-i)}
0
F(z)
1
from which it follows that
1
F(z)}
i) z
i
(z
0
l{z-if(z)/ z-2 f(z)dz}
i
---2
1
F’(z) +
1F(z))
(z
f(z)dz
z{zc
F(z))’
z(z
f(z
z
i
---i
i
---i
(2.6)
2
z
1
r, we have
z-2f(z)dz
2
From (2.1) and (2.3), we have
Izl
i
’0
Re
1
---i
r 2),
-< {2Re h(z)}/(l
337
i)
>-
1
h(z) +
i
(-i) +
(-
1
1
+
i),
r
r
(2.7)
Using (2.7), we have from (2.6)
Re
zf’(z)
f(z)
> Re
2
h(z){l1
){(_.z._.
Re hi
le
r
4r-
+ r2
- -r
2
1
c
+
f (z)
o
{C(z(
1
)}
2)r
2)r2)}/{(1
i
The right hand side of (2.8) is positive for r < r
result is sharp as can be seen by
1
(_
i
where r
o
2)z))}/(l
i
r)(
o
+
i
2r)}
is given by
(2.2).
F (z)
o
RE}L%RK 2.1.
is given by
(2.2).
z
(i- z)
2
This
z)
(i- o)F (z) + OzF ’(z),
o
o
where
(2.8)
(2.9)
S*
Let f e C, then f, given by (2.1), is convex for
zl
< r
o
The proof follows on the same lines as in Theorem 2.1.
where r
o
See also
[5] and [6].
REMARK 2.2.
In [6], Nikolaeva and Repnina treated the same problem, with a dif-
ferent notation, for the convex and starlike functions of order 8.
from their result when we take
’8
0 for 0 <
< i.
Theorem 2.1 follows
On the other hand, our proof of
Theorem 2.1 is much simpler and the result holds for all
> 0.
338
K.I. NOOR, F. M. ALOBOUDI and N. ALDIHAN
Let F e K and f(z)
THEOREM 2.2.
zl
close-to-convex in
< r
is given by (2 2).
r
0
e)F(z) + zF’(z),
(i
> O.
The function f
0
Then f is
in (2 9) shows
that this result is sharp.
Since F E
PROOF.
Now let g(z)
r
O
)G(z) + zG’(z).
(i
is defined by (2.2).
Then by Theorem 2.1
E
Using the same technique of Theorem 2 1
zf’(z
zl
> 0 for
g(z)
REMARK 2.3. For a
that Re
> 0.
E, Re zF’(z)
G(z)
g is starlike for Izl <r
O
K, there exists a G e S* such that, for z
< r
1/2
we can easily show
o
[7].
this result has been proved in
As an easy consequence of (1.3) and Theorem 2.2, we have the following.
Let F e K and f(z)
COROLLARY 2.1.
zl
quasi-convex in
< r
(i
e)F(z) + azF’(z), a > O.
Then F is e-
This means that the radius of a-quasi-convexity for close-
O
to-convex functions is given by (2.2).
Let F
THEOREM 2.3.
in
Izl
C*, for
PROOF.
<
ro,
r
g
C* and a > 0.
Let f(z)
a)F(z) + azF’(z).
(i
Then f is
(2.2).
is given by
C*, there exists a G e C such that for z e E, Re (zF’(z))’ > 0,
G’ (z)
)G(z) + zG’(z), then g is convex in zl < r
We can write
Since F
Now let g(z)
(i
O
1
2
f(z)
(i
a)F(z) + azF’(z)
z
g(z)
(i
e)G(z) + ezG’(z)
z
1
i
F(z))’
and
i
2
i
1
G(z))’o
Thus
i
(zf’ (z) )’
g’(z)
i
i
2-E zE-I F(z))’)’)’)/(z 2- ((
((z(z
i
z-
1
O(z))’)’)
(2.10)
Now
2
(z(z
1
(z
1
1
F(z))’)’)’
1
2
(Z
Let zF’(z)
1
i
o.
1
o
Z
1
+ zF’(z))’)’
(z((-I)F(z)
zF’(z) +
F’(z) +z
F"(z)))’
2F" (z))
i
i
i
i
z
zF’ (z)) ’) ’o
(z
(z
H(z), then from (2.10), we have
(zf’(z))’
g’ (z)
2
(z
1
e
1
z
2
i
H(z)) ’)’ / (z
1
a
z
1
a
i
G(z)) ’)’
2
Since from Theorem 2.2, the function (l-)tt(z)
longs to K with respect to a convex function g:
g(z)
+ ztt’(z)
1
1
1
It(z))’
(l-e)G(z) + ezG’(z) in
be--.
UNIVALENCE OF CONVEX COMBINATIONS OF ANALYTIC FUNCTIONS
zl
< r
zl
so f is in C* for
0
REMARK 2.4.
< r
where r
0
is given by
0
1/2,
For F e C* and
339
(2.2).
Theorem 2.3 has been proved in
[1].
We now deal with a generalized form of (i.i) by taking g to be starlike and
prove the following.
Let F be analytic in E and let for z e E, Re (zF’ (z))’ >0, GeS*.
THEOREM 2.4.
Re
G’(z)
(l-)F(z) + zF’(z) and g(z)
Let f(z)
(.zf’ (z))’
g’(z)
> 0 for
Izl
<
rl,
I
PROOF.
492 + 1- 2
+
3
1/2, the problem has
Then
where
r
For e
(l-)G(z) + zG’ (z), with e > 0.
been solved in
[8].
From (2.3), we can write
I
iz
1
F(z)
1
---2
iz
C
f(z)dz
0
i
i--
1
zF (.)
i
i
[z
((1-) j
z
1
z
z
1
a
2
z
i
f(z)dz + z a
f(z))
0
f’ (z) dz)
0
Thus
1
I
(zef ’(z)
(-i)
(zF’ (z))’
G’(z)
fz
1
I
f’(z)dz)
z
0
z
z
1
a
h (z),
(2.ii)
i
g’ (z)dz
0
where Re h(z) > 0, z
E.
From (2.11), we write
1
zf, (z)
i
(-i)
1
--I
z
z
f’(z)dz
h(z)
0
I
1
--I
z
z
g’ (z)dz
0
Differentiating both sides, and simplifying, we obtain
i
h’ (z)(
z
lh’ (z)]
< 2Re
1
h(z)
r
2--
(2.12) gives
(2.12)
1
z
Using
g’ (z) dz)
J0
(zf’ (z))’ =h(z) +
g’(z)
g’(z)
340
K.I. NOOR, F. M. ALOBOUDI and N. ALDIHAN
i
Re (zf’ (z))_’
g’Cz)
> Re
h(z)[l
r
i
i
za
2
g’ (z) dz) / (z
g (z)dz) ]o
(2.13)
(z))’
+ (zG’
G’(z)
(2.14)
0
Now
i
(zg
(z))/(
I
z
i
E- g’ (z)dz)
z
(i/)G’(z) + zG"(z)
G’(z)
0
i
(-i)
Since G e S*, so
(z))’
G’(z)
l-4r+r
2
(zG’
2
(2.15)
From (2.13), (2.14) and (2.15), we obtain
Re
(zf’ (z))’ >- Re h(z)[l
g’(z)
4er
1
and this positive for
zl
<
rl,
2
1
1
4r
2)r
(i
2)r
+ 1
2a
where
r
2
2
2)
2)r
2
1
1
3c
ACKNOWLEDGEMENT.
r
6r- (I
1
Re h(z)
r(l- r
2
1
+
/9a
2
The authors are grateful for the referee’s helpful comments and
suggestions on the earlier version of this paper.
In particular, the reference to
Nikolaeva and Repnina was kindly supplied by him.
REFERENCES
I.
NOOR, K. INAYAT.
On a subclass of close-to-convex functions, Comm. Math. Univ.
St. Pauli 2__9 (1980), 25-28.
2.
NOOR, K. INAYAT and THOMAS, D.K. On quasi-convex univalent functions, Inter. J.
Math. and Math. Sci. 3 (1980), 255-266.
3.
NOOR, K. INAYAT and ALOBOUDI, F.M.
4.
LIBERA, R.J.
5.
CAMPBELL, D.M.
6.
NIKOLAEVA, R.V. and REPNINA, L.G.
7.
LIVINGSTON, A.E. On the radius of univalence of certain analytic functions, Proc.
Amer. Math. Soc. 17 (1966), 352-357.
8.
NOOR, K INAYAT and ALDIHAN, N.
Alpha-quasi-convex functions, to appear.
Some radius of convexity problems, Duke Math. J. 31 (1964), 143-158.
A survey of problems of the convex combination of univalent functions, Rockey Mount J. Math. 5 (1975), 475-492.
A certain generalization of theorems due to
Livingston, (Russian), Ukrain Mat. Z. 24 (1972), 268-273, MR45 # 5336.
appear.
A subclass of close-to-convex functions, to