Journal of Applied Mathematics & Bioinformatics, vol.4, no.3, 2014, 87-93
ISSN: 1792-6602 (print), 1792-6939 (online)
Scienpress Ltd, 2014
On a certain generalized condition
for starlikeness and convexity
Deborah Olufunmilayo Makinde1
Abstract
In this paper, we applied Salagean differential operator to the integral operator of the form
Z
F (z) =
k µ
zY
0 i=1
fi (s)
s
¶1
α
ds.
and obtained the starlikeness, convexity and convolution properties.
Mathematics Subject Classification: 30C45
Keywords: Salagean differential operator; integral operator; Starlikeness;
Convexity; Convolution
1
Department of Mathematics, Obafemi Awolowo University, Ile-Ife 220005, Nigeria.
E-mail: domakinde.comp@gmail.com; dmakinde@oauife.edu.ng
Article Info: Received : August 3, 2014. Revised : September 29, 2014.
Published online : October 15, 2014.
88
1
On a certain generalized condition for starlikeness and convexity
Introduction
Let A be the class of all analytic functions f (z) defined in the open unit
disk U = {z ∈ C : |z| < 1} and S the subclass of A condisting of univalent
functions of the form:
∞
X
f (z) = z +
an z n
(1)
n=2
While
½
¾
¡ zf 0 (z)
¢
S = f ∈ S : Re
> 0, z ∈ U
(2)
f (z)
¶
¾
½
µ
zf 00 (z)
c
S = f ∈ S : Re 1 + 0
> 0, z ∈ U
(3)
f (z)
are called respectively univalent starlike and convex functions with respect to
the origin.
∗
Lemma 1.[1] Let M and N be analytic in U with M (0) = N (0) = 0. If N (z)
maps onto a many sheeted region which is starlike with respect to the origin
© 0 (z) ª
© (z) ª
and Re M
> in U , then Re M
> in U .
N 0 (z)
N (z)
Seenivasagan [4] obtained a sufficient condition for the univalence of the
integral operator of the form:
( Z
¶ α1 ) β1
k µ
z
Y
fi (t)
tβ−1
Fα,β (z) = β
dt
t
0
i=1
Where
fi (z) = z +
∞
X
ain z n
(4)
n=2
While Makinde and Opoola [2] obtained a condition for the starlikeness of the
integral operator of the form:
¶1
Z zY
k µ
fi (s) α
ds, α ∈ C
Fα (z) =
s
0 i=1
The differential operator Dn (n ∈ N0 ) was introduced by Salagean [5]. Where
Dn f (z) is defined by
Dn f (z) = D(Dn−1 f (z)) = z(Dn−1 f (z))0 with D0 f (z) = f (z)
(5)
89
Deborah Olufunmilayo Makinde
Makinde [3] investigated some properties for Γnα (ζ1 , ζ2 ; γ) of Dn fi (z) in the
integral operator of the form:
Z
Fα (z) =
0
z
¶1/α
k µ n
Y
D fi (s)
s
i=1
ds, α ∈ C |α| ≤ 1
Where she gave the Salagean Differential operator for the function fi (z) in (4)
to be of the form:
∞
X
n
(6)
D fi (z) = z +
nk ain z n
n=2
P∞
Let Dn fi1 (z) = z + n=2 nk ain1 z n an Dn fi2 (z) = z +
the convolution of Dn fi1 (z) and Dn fi2 (z) by
n
n
n
n
P∞
n=2
D fi1 (z) ∗ D fi2 (z) = (D fi1 ∗ D fi2 )(z) = z +
∞
X
nk ain2 z n we define
nk ain1 ain2 z n
n=2
In the present paper, we investigated certain conditions for starlikeness and
convexity of the integral operator of the form:
Z
z
FDα (z) =
0
¶
k µ n
Y
D fi (s) α
1
i=1
s
ds,
α∈C
and its convolution properties.
2
Main Results
Theorem 2.1. Let FDα (z) be the function in U defined by
Z
z
FDα (z) =
0
¶
k µ n
Y
D fi (s) α
1
i=1
s
ds,
α∈C
ds,
α∈C
If Dn fi (s) is starlike, then FDα (z) is convex.
Proof. Let
Z
z
FDα (z) =
0
¶
k µ n
Y
D fi (s) α
1
i=1
s
90
On a certain generalized condition for starlikeness and convexity
Then
Qk
µ
Dn fi (s)
s
¶ α1
i=1
0
zFDα
(z)
=
µ
¶ α1
FDα (z)
R z Qk
Dn fi (s)
i=1
0
s
µ
Dn f1 (z)
z
z
=
¶ α1 µ
Dn f2 (z)
z
¶ α1 µ
¶ α1
Dn fk (z)
...
z
µ
R z Qk
Dn fi (s)
s
i=1
0
ds
¶ α1
ds
Now, let
µ
Dn f1 (z)
0
M (z) = zFDα
(z) = z
z
and
¶ α1 µ
Z
z
N (z) = FDα (z) =
0
Dn f2 (z)
z
¶
k µ n
Y
D fi (s) α
1
s
i=1
Then
1
i=1 α
00
M 0 (z)
zFDα
(z)
=
1
+
=1+
0
0
N (z)
FDα (z)
Qk
i=1
But
Qk
i=1
Dn fi (z)
z
1 Dn+1 fi (z)
i=1 | α || Dn fi (z)
Dn+1 fi (z)
− 1| < 1
Dn fi (z)
Thus
|
M 0 (z)
− 1| < 1
N 0 (z)
¶
−1
¶ α1
¶
−1 |
¶ α1
|
− 1|
µ
¶ α1
Qk
Dn fi (z)
| i=1
|
z
Dn fi (z) is starlike by hypothesis implies that
|
Dn fi (z)
z
µ
Pk
≤
µ
Dn+1 fi (z)
Dn fi (z)
1
i=1 α
|
D n+1 fi (z)
D n fi (z)
µ
Pk
ds
µ
Pk
¯ 0
¯
|
¯ M (z)
¯
¯
¯
¯ N 0 (z) − 1¯ =
¶ α1 µ n
¶1
D fk (z) α
...
z
91
Deborah Olufunmilayo Makinde
Hence, Re
{
M 0 (z)
}>0
N 0 (z)
which shows that FDα (z) is convex and so by Lemma 1
Re {
M (z)
}>0
N (z)
which implies that FDα (z) is starlike.
This concludes the proof of Theorem 1.
Theorem 2.2. Let FDα (z) be the function in U defined by
Z
z
FDα (z) =
0
¶
k µ n
Y
D fi (s) α
1
i=1
s
ds,
α∈C
Then FDα (z) is starlike If Dn fi (s) is starlike.
Proof. From Theorem 1
¯ 0
¯
¯ M (z)
¯
¯
¯<1
−
1
¯ N 0 (z)
¯
This implies that
½
M 0 (z)
Re
N 0 (z)
and by Lemma 1 we have
½
M (z)
Re
N (z)
Which implies that
¾
>0
¾
½
0
zFDα
(z)
Re
FDα (z)
>0
¾
>0
Thus FDα (z) is starlike.
Theorem 2.3. Let
Z
z
FDα (z) =
0
¶
k µ
Y
(Dn fi1 ∗ Dn fi2 )(s) α
1
i=1
s
ds,
Then FDα (z) is convex if (Dn fi1 ∗ Dn fi2 )(z) is starlike.
α∈C
92
On a certain generalized condition for starlikeness and convexity
Proof. Following the procedure of the proof of the Theorem 1, we obtain
the result.
Theorem 2.4. Let FDα (z) be the function in U defined by
Z
z
FDα (z) =
0
¶
k µ
Y
(Dn fi1 ∗ Dn fi2 )(s) α
1
s
i=1
ds,
α∈C
Then FDα (z) is starlike If (Dn fi1 ∗ Dn fi2 )(z) is starlike.
Proof. Following the procedure of the proof of the Theorem 2, we obtain
the result.
Theorem 2.5. Let FDα (z) be the function in U defined by
Z
z
FDα (z) =
0
¶
k µ n
Y
D fi (s) α
1
s
i=1
ds,
α ∈ C, |α > 1
Then FDα (z) is convex if
k X
X
nk (n − 1)|ain | < 1
i=1
Proof. From Theorem 1, we have
¯ 0
¯ Pk 1 zDn+1 fi (z)
− 1|
| ||
¯ M (z)
¯
¯
¯ ≤ i=1 α Dn fi (z) 1
−
1
¶α
µ
¯ N 0 (z)
¯
Qk
Dn fi (z)
| i=1
|
z
Using (6) we obtain
¯
¯ 0
¯
¯ M (z)
¯≤ 1
¯
−
1
¯ α
¯ N 0 (z)
Pk P∞
nk (n − 1)|ain |
¯
µ
¶ α1 ¯
¯
Pk P∞ k i ¯ Qk
n f (z)
D
i
¯
1 + i=1 n=2 n |an |¯¯ i=1
z
¯
i=1
n=2
and by hypothesis, we have
¯ 0
¯
¯ M (z)
¯
¯
¯<1
−
1
¯ N 0 (z)
¯
z → 1−
93
Deborah Olufunmilayo Makinde
Thus FDα (z) is convex
Corollary Let FDα (z) be the function in U defined by
Z
z
FDα (z) =
¶
k µ
Y
(Dn fi1 ∗ Dn fi2 )(s) α
0
1
i=1
s
ds,
α∈C
Then FDα (z) is convex if
k X
X
nk (n − 1)|ain1 ||ain2 | < 1.
i=1
References
[1] S.A. Halim, On a class of analytic functions involving Salagean differential
operator, Tamkang Journal of Mathematics, 23(1), (Spring, 1992).
[2] D.O. Makinde and T.O. Opoola, On sufficient condition for sarlikeness,
General Mathematics, 8(3), (2010), 35-39.
[3] D.O. Makinde, Application of Salagean differntial operator to certain subclasses of harmonic univalent functions, International Journal of Scientific
and Research publications, 3(5), (2013).
[4] N. Seenivasagan, Sufficient conditions for univalence, Applied Mathemaitics E-notes, 8, (2008), 30-35.
[5] E. Yasar and S. Yalcm, Convolution properties of Salagean-type of harmonic univalent functions, Stud. Univ. Babes-Bolyai Math., 57(2), (2012),
283-289.