NEW GENERAL INTEGRAL OPERATORS OF
p-VALENT FUNCTIONS
Integral Operators of p-valent
Functions
B.A. FRASIN
Department of Mathematics
Al al-Bayt University
P.O. Box: 130095
Mafraq, Jordan
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
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EMail: bafrasin@yahoo.com
Received:
10 May, 2009
Accepted:
14 October, 2009
Communicated by:
N.E. Cho
2000 AMS Sub. Class.:
30C45.
Key words:
Analytic functions, p-valent starlike, convex and close-to-convex functions, Uniformly p-valent close-to-convex functions, Strongly starlike, Integral operator.
Abstract:
In this paper, we introduce new general integral operators. New sufficient conditions for these operators to be p-valently starlike, p-valently close-to-convex, uniformly p-valent close-to-convex and strongly starlike of order γ (0 < γ ≤ 1) in
the open unit disk are obtained.
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Contents
1
Introduction and Definitions
3
2
Sufficient Conditions for the Operator Fp
7
3
Sufficient Conditions for the Operator Gp
12
4
Strong Starlikeness of the Operators Fp and Gp
17
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
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1.
Introduction and Definitions
Let Ap denote the class of functions of the form:
p
f (z) = z +
∞
X
an z n
(p ∈ N ∈ {1, 2, . . .}),
n=p+1
which are analytic in the open unit disk U = {z : |z| < 1}. We write A1 = A. A
function f ∈ Ap is said to be p-valently starlike of order β (0 ≤ β < p) if and only
if
0 zf (z)
Re
>β
(z ∈ U).
f (z)
We denote by Sp? (β), the class of all such functions. On the other hand, a function
f ∈ Ap is said to be p-valently convex of order β (0 ≤ β < p) if and only if
zf 00 (z)
Re 1 + 0
>β
(z ∈ U).
f (z)
Let Kp (β) denote the class of all those functions which are p-valently convex of
order β in U. Furthermore, a function f (z) ∈ Ap is said to be in the subclass Cp (β)
of p-valently close-to-convex functions of order β(0 ≤ β < p) in U if and only if
0 f (z)
Re
>β
(z ∈ U).
z p−1
Note that Sp? (0) = Sp? , Kp (0) = Kp and Cp (0) = Cp are, respectively, the classes
of p-valently starlike, p-valently convex and p-valently close-to-convex functions in
U. Also, we note that S1? = S ? , K1 = K and C1 = C are, respectively, the usual
classes of starlike, convex and close-to-convex functions in U.
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
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A function f ∈ Ap is said to be in the class UC p (β) of uniformly p-valent closeto-convex functions of order β (0 ≤ β < p) in U if and only if
0
0
zf (z)
zf (z)
Re
−β ≥
− p
(z ∈ U),
g(z)
g(z)
for some g(z) ∈ US p (β), where US p (β) is the class of uniformly p-valent starlike
functions of order β (−1 ≤ β < p) in U and satisfies
0
0
zf (z)
zf (z)
(1.1)
Re
−β ≥
− p
(z ∈ U).
f (z)
f (z)
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
Uniformly p-valent starlike functions were first introduced in [10].
For αi > 0 and fi ∈ Ap , we define the following general integral operators
α
α
Z z
f1 (t) 1
fn (t) n
p−1
(1.2)
Fp (z) =
...
dt
pt
tp
tp
0
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and
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Z
(1.3)
Gp (z) =
z
pt
0
p−1
f10 (t)
ptp−1
α1
...
fn0 (t)
ptp−1
α n
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dt.
If we take p = 1, we obtain of the general integral operators F1 (z) = Fn (z) and
G1 (z) = Fα1 ,...,αn (z) introduced and studied by Breaz and Breaz [3] and Breaz et al.
[6] (see also [2, 4, 8,9]). Also for p = n = 1, α1 = α ∈ [0, 1] in (1.2), we obtain the
R z f (t) α
integral operator 0
dt studied in [12] and for p = n = 1, α1 = δ ∈ C, |δ| ≤
t
Rz
1/4 in (1.3), we obtain the integral operator 0 (f 0 (t))α dt studied in [11, 15].
There are many papers in which various sufficient conditions for multivalent starlikeness have been obtained. In this paper, we derive new sufficient conditions for the
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operators Fp (z) and Gp (z) to be p-valently starlike, p-valently close-to-convex and
uniformly p-valent close-to-convex in U. Furthermore, we give new sufficient conditions for these two general operators to be strongly starlike of order γ (0 < γ ≤ 1) in
U.
In order to derive our main results, we have to recall here the following results:
Lemma 1.1 ([13]). If f ∈ Ap satisfies
zf 00 (z)
1
Re 1 + 0
<p+
f (z)
4
Integral Operators of p-valent
Functions
(z ∈ U),
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
then f is p-valently starlike in U.
Lemma 1.2 ([7]). If f ∈ Ap satisfies
00
zf (z)
<p+1
+
1
−
p
f 0 (z)
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then f is p-valently starlike in U.
Lemma 1.3 ([16]). If f ∈ Ap satisfies
a+b
zf 00 (z)
<p+
Re 1 + 0
f (z)
(1 + a)(1 − b)
then f is uniformly p-valent close-to-convex in U.
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(z ∈ U),
where a > 0, b ≥ 0 and a + 2b ≤ 1, then f is p-valently close-to-convex in U.
Lemma 1.4 ([1]). If f ∈ Ap satisfies
zf 00 (z)
1
Re 1 + 0
<p+
f (z)
3
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(z ∈ U),
(z ∈ U),
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Lemma 1.5 ([17]). If f ∈ Ap satisfies
zf 00 (z)
p
Re 1 + 0
> −1
f (z)
4
then
s
Re
√
p
zf 0 (z)
>
2
f (z)
Lemma 1.6 ([14]). If f ∈ Ap satisfies
zf 00 (z)
γ
Re 1 + 0
>p−
f (z)
2
then
0
arg zf (z) < π γ
f (z) 2
or f is strongly starlike of order γ in U.
(z ∈ U),
(z ∈ U).
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
( z ∈ U),
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(0 < γ ≤ 1; z ∈ U),
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2.
Sufficient Conditions for the Operator Fp
We begin by establishing sufficient conditions for the operator Fp to be in Sp? .
Theorem 2.1. Let αi > 0 be real numbers for all i = 1, 2, . . . , n. If fi ∈ Ap for all
i = 1, 2, . . . , n satisfies
0 zfi (z)
1
(2.1)
Re
< p + Pn
(z ∈ U),
fi (z)
4 i=1 αi
then Fp is p-valently starlike in U.
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
Proof. From the definition (1.2), we observe that Fp (z) ∈ Ap . On the other hand, it
is easy to see that
α
α
fn (z) n
f1 (z) 1
0
p−1
...
.
(2.2)
Fp (z) = pz
zp
zp
Differentiating (2.2) logarithmically and multiplying by z, we obtain
0
n
X
zFp00 (z)
zfi (z)
= (p − 1) +
αi
−p .
Fp0 (z)
f
(z)
i
i=1
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Thus we have
(2.3)
n
X
zFp00 (z)
=p 1−
1+ 0
αi
Fp (z)
i=1
!
+
n
X
i=1
αi
zfi0 (z)
fi (z)
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.
Taking the real part of both sides of (2.3), we have
!
0 n
n
X
X
zFp00 (z)
zfi (z)
(2.4)
Re 1 + 0
=p 1−
.
αi +
αi Re
Fp (z)
fi (z)
i=1
i=1
Close
From (2.4) and (2.1), we obtain
!
n
n
X
X
zFp00 (z)
1
(2.5)
<p 1−
αi +
αi p + Pn
Re 1 + 0
Fp (z)
4 i=1 αi
i=1
i=1
1
=p+ .
4
Hence by Lemma 1.1, we get Fp ∈
Sp? .
This completes the proof.
Integral Operators of p-valent
Functions
B.A. Frasin
Letting n = p = 1, α1 = α and f1 = f in Theorem 2.1, we have:
Corollary 2.2. If f ∈ A satisfies
0 zf (z)
1
Re
<1+
f (z)
4α
R z α
where α > 0, then 0 f (t)
dt is starlike in U.
t
vol. 10, iss. 4, art. 109, 2009
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(z ∈ U),
In the next theorem, we derive another sufficient condition for p-valently starlike
functions in U.
Theorem 2.3. Let αi > 0 be real numbers for all i = 1, 2, . . . , n. If fi ∈ Ap for all
i = 1, 2, . . . , n satisfies
0
zfi (z)
p+1
(2.6)
(z ∈ U),
fi (z) − p < Pn αi
i=1
then Fp is p-valently starlike in U.
Proof. From (2.3) and the hypotheses (2.6), we have
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n
0
X
zFp00 (z)
zf
(z)
i
1 +
− p = −p αi
Fp0 (z)
f
(z)
i
i=1
0
n
X
zfi (z)
<
− p
αi fi (z)
i=1
n
X
p+1
<
αi Pn
= p + 1.
i=1 αi
i=1
Now using Lemma 1.2, we immediately get Fp ∈
Sp? .
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
Letting n = p = 1, α1 = α and f1 = f in Theorem 2.3, we have:
Corollary 2.4. If f ∈ A satisfies
0
zf (z)
2
<
−
1
(z ∈ U),
f (z)
α
R z f (t) α
dt is starlike in U.
where α > 0, then 0
t
Applying Lemmas 1.3 and 1.4, we obtain the following sufficient conditions for
Fp to be p-valently close-to-convex and uniformly p-valent close-to-convex in U.
Theorem 2.5. Let αi > 0 be real numbers for all i = 1, 2, . . . , n. If fi ∈ Ap for all
i = 1, 2, . . . , n satisfies
0 (a + b)
zfi (z)
P
<p+
(z ∈ U),
(2.7)
Re
fi (z)
(1 + a)(1 − b) ni=1 αi
where a > 0, b ≥ 0 and a + 2b ≤ 1, then Fp is p-valently close-to-convex in U.
Proof. From (2.4) and the hypotheses (2.7) and applying Lemma 1.3, we have Fp ∈
Cp (β).
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Letting n = p = 1, α1 = α and f1 = f in Theorem 2.5, we have:
Corollary 2.6. If f ∈ A satisfies
0 zf (z)
(a + b)
Re
<1+
(z ∈ U),
f (z)
(1 + a)(1 − b)α
R z α
where α > 0, a > 0, b ≥ 0 and a + 2b ≤ 1, then 0 f (t)
dt is close-to-convex in
t
U.
Theorem 2.7. Let αi > 0 be real numbers for all i = 1, 2, . . . , n. If fi ∈ Ap for all
i = 1, 2, . . . , n satisfies
0 zfi (z)
1
(2.8)
Re
< p + Pn
(z ∈ U),
fi (z)
3 i=1 αi
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
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then Fp is uniformly p-valent close-to-convex in U.
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Proof. The proof of the theorem follows by applying Lemma 1.4 and using (2.4),
(2.8) to get Fp ∈ UC p (β).
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Letting n = p = 1, α1 = α and f1 = f in Theorem 2.7, we have:
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Corollary 2.8. If f ∈ A satisfies
0 zf (z)
1
Re
<1+
(z ∈ U),
f (z)
3α
R z α
where α > 0, then 0 f (t)
dt is uniformly close-to-convex in U.
t
Using Lemma 1.5, we obtain the next result
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Theorem 2.9. Let αi > 0 be real numbers for all i = 1, 2, . . . , n. If fi ∈ Ap for all
i = 1, 2, . . . , n satisfies
0 zfi (z)
3p + 4
(2.9)
Re
> p − Pn
(z ∈ U),
fi (z)
4 i=1 αi
then
s
√
zFp0 (z)
p
Re
>
(z ∈ U).
Fp (z)
2
Proof. It follows from (2.4) and (2.9) that
!
n
n
X
X
zFp00 (z)
3p + 4
p
Re 1 + 0
>p 1−
αi +
αi p − Pn
= − 1.
Fp (z)
4 i=1 αi
4
i=1
i=1
By Lemma 1.5, we conclude that
s
√
zFp0 (z)
p
Re
>
Fp (z)
2
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
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(z ∈ U).
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Letting n = p = 1, α1 = 1 and f1 = f in Theorem 2.9, we have:
Corollary 2.10. If f ∈ A satisfies
0 zfi (z)
3
(2.10)
Re
>−
fi (z)
4
then
v
u
f (z)
1
u
(2.11)
Re t R z >
f (t)
2
dt
t
0
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(z ∈ U),
(z ∈ U).
Close
3.
Sufficient Conditions for the Operator Gp
The first two theorems in this section give a sufficient condition for the integral
operator Gp to be in the class Sp? .
Theorem 3.1. Let αi > 0 be real numbers for all i = 1, 2, . . . , n. If fi ∈ Ap for all
i = 1, 2, . . . , n satisfies
1
zfi00 (z)
(3.1)
Re 1 + 0
< p + Pn
(z ∈ U),
fi (z)
4 i=1 αi
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
then Gp is p-valently starlike in U.
Proof. From the definition (1.3), we observe that Gp (z) ∈ Ap and
00
n
X
zG00p (z)
zfi (z)
= (p − 1) +
αi
− (p − 1)
G0p (z)
fi0 (z)
i=1
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or
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(3.2)
n
X
zG00p (z)
1+ 0
=p 1−
αi
Gp (z)
i=1
!
+
n
X
i=1
αi
zf 00 (z)
1 + 0i
fi (z)
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.
Taking the real part of both sides of (3.2), we have
!
n
n
X
X
zG00p (z)
zfi00 (z)
(3.3)
Re 1 + 0
=p 1−
αi +
αi Re 1 + 0
.
Gp (z)
fi (z)
i=1
i=1
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From (3.3) and the hypotheses (3.1), we obtain
!
n
n
X
X
zG00p (z)
1
(3.4)
Re 1 + 0
<p 1−
αi +
αi p + Pn
Gp (z)
4 i=1 αi
i=1
i=1
1
=p+ .
4
Therefore, using Lemma 1.1, it follows that the integral operator Gp belongs to
the class Sp? .
Letting n = p = 1, α1 = α and f1 = f in Theorem 3.1, we obtain
Corollary 3.2. If f ∈ A satisfies
1
zf 00 (z)
Re 1 + 0
<1+
f (z)
4α
Rz 0
where α > 0, then 0 (f (t))α dt is starlike in U.
(z ∈ U),
Theorem 3.3. Let αi > 0 be real numbers for all i = 1, 2, . . . , n. If fi ∈ Ap for all
i = 1, 2, . . . , n satisfies
00 zfi (z) p+1
(3.5)
(z ∈ U),
f 0 (z) < Pn αi − p + 1
i
i=1
P
where ni=1 αi > 1, then Gp is p-valently starlike in U.
Proof. It follows from (3.2) and (3.5) that
n
00 n
X
X
zG00p (z)
zf
(z)
i
1 +
=
−
p
α
−
(p
−
1)
α
i
i
0
0
Gp (z)
fi (z)
i=1
i=1
n
n
X
X
p+1
− p + 1 < p + 1.
< (p − 1)
αi +
αi Pn
i=1 αi
i=1
i=1
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
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Therefore, it follows from Lemma 1.2 that Gp is in the class Sp? .
Letting n = p = 1, α1 = α and f1 = f in Theorem 3.3, we obtain:
Corollary 3.4. If f ∈ A satisfies
00 zf (z) 2
(z ∈ U),
f 0 (z) < α
Rz
where α > 0, then 0 (f 0 (t))α dt is starlike in U.
Integral Operators of p-valent
Functions
B.A. Frasin
Applying Lemmas 1.3 and 1.4, we obtain the following sufficient conditions for
Gp to be p-valently close-to-convex and uniformly p-valent close-to-convex in U.
vol. 10, iss. 4, art. 109, 2009
Theorem 3.5. Let αi > 0 be real numbers for all i = 1, 2, . . . , n. If fi ∈ Ap for all
i = 1, 2, . . . , n satisfies
a+b
zfi00 (z)
P
(3.6)
Re 1 + 0
<p+
(z ∈ U),
fi (z)
(1 + a)(1 − b) ni=1 αi
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where a > 0, b ≥ 0 and a + 2b ≤ 1, then Gp is p-valently close-to-convex in U.
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Proof. In view of (3.3) and (3.6) and by using Lemma 1.3, we have Gp ∈ Cp (β).
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Letting n = p = 1, α1 = α and f1 = f in Theorem 3.5, we obtain
Corollary 3.6. If f ∈ A satisfies
zf 00 (z)
a+b
Re 1 + 0
<1+
(z ∈ U),
f (z)
(1 + a)(1 − b)α
Rz
where α > 0, a > 0, b ≥ 0 and a + 2b ≤ 1, then 0 (f 0 (t))α dt is close-to-convex
in U.
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Theorem 3.7. Let αi > 0 be real numbers for all i = 1, 2, . . . , n. If fi ∈ Ap for all
i = 1, 2, . . . , n satisfies
zfi00 (z)
1
(3.7)
Re 1 + 0
< p + Pn
(z ∈ U),
fi (z)
3 i=1 αi
then Gp is uniformly p-valent close-to-convex in U.
Proof. In view of (3.3) and (3.7) and by using Lemma 1.4, we have Gp ∈ UC p (β).
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
Letting n = p = α = 1 and f1 = f in Theorem 3.7, we have:
Corollary 3.8. If f ∈ A satisfies
1
zf 00 (z)
<1+
(z ∈ U),
Re 1 + 0
f (z)
3α
Rz
where α > 0, then 0 (f 0 (t))α dt is uniformly close-to-convex in U.
Using Lemma 1.5, we obtain the next result.
then
(3.9)
Re
√
zG0p (z)
p
>
Gp (z)
2
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Theorem 3.9. Let αi > 0 be real numbers for all i = 1, 2, . . . , n. If fi ∈ Ap for all
i = 1, 2, . . . , n satisfies
zfi00 (z)
3p + 4
(3.8)
Re 1 + 0
> p − Pn
(z ∈ U),
fi (z)
4 i=1 αi
s
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(z ∈ U).
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Proof. It follows from (3.3) and (3.8) that
!
n
n
X
X
zG00p (z)
3p + 4
p
Re 1 + 0
>p 1−
αi +
αi p − Pn
= − 1.
Gp (z)
4 i=1 αi
4
i=1
i=1
By Lemma 1.5, we get the result (3.9).
Letting n = p = 1, α1 = 1 and f1 = f in Theorem 3.9, we have
Corollary 3.10. If f ∈ A satisfies
zfi00 (z)
3
(3.10)
Re 1 + 0
>−
fi (z)
4
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
(z ∈ U),
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then
Contents
s
(3.11)
Re
1
zf 0 (z)
Rz
>
0
2
f (t)dt
0
(z ∈ U).
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4.
Strong Starlikeness of the Operators Fp and Gp
Applying Lemma 1.6 and using (2.4), we obtain the following sufficient condition
for the operator Fp to be strongly starlike of order γ in U.
Theorem 4.1. Let αi > 0 be real numbers for all i = 1, 2, . . . , n. If fi ∈ Ap for all
i = 1, 2, . . . , n satisfies
0 zfi (z)
γ
Re
> p − Pn
(z ∈ U),
fi (z)
2 i=1 αi
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
then Fp is strongly starlike of order γ (0 < γ ≤ 1) in U.
Letting n = p = 1, α1 = α and f1 = f in Theorem 4.1, we have
Corollary 4.2. If f ∈ A satisfies
0 zf (z)
γ
Re
>1−
(z ∈ U),
f (z)
2α
R z f (t) α
where α > 0, then 0
dt is strongly starlike of order γ (0 < γ ≤ 1) in U.
t
Applying once again Lemma 1.6 and using (3.3), we obtain the following sufficient condition for the operator Gp to be strongly starlike of order γ in U.
Theorem 4.3. Let αi > 0 be real numbers for all i = 1, 2, . . . , n. If fi ∈ Ap for all
i = 1, 2, . . . , n satisfies
zfi00 (z)
γ
Re 1 + 0
> p − Pn
(z ∈ U),
fi (z)
2 i=1 αi
then Gp is strongly starlike of order γ (0 < γ ≤ 1) in U.
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Letting n = p = α1 = 1 and f1 = f in Theorem 4.3, we have
Corollary 4.4. If f ∈ A satisfies
zf 00 (z)
γ
Re 1 + 0
>1−
(z ∈ U),
f (z)
2α
Rz
where α > 0, then 0 (f 0 (t))α dt is strongly starlike of order γ (0 < γ ≤ 1) in U.
Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
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References
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Integral Operators of p-valent
Functions
B.A. Frasin
vol. 10, iss. 4, art. 109, 2009
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[11] Y.J. KIM AND E.P. MERKES, On an integral of powers of a spirallike function,
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