Volume 10 (2009), Issue 4, Article 109, 9 pp.
NEW GENERAL INTEGRAL OPERATORS OF p-VALENT FUNCTIONS
B.A. FRASIN
D EPARTMENT OF M ATHEMATICS
A L AL -BAYT U NIVERSITY
P.O. B OX : 130095 M AFRAQ , J ORDAN
bafrasin@yahoo.com
Received 10 May, 2009; accepted 14 October, 2009
Communicated by N.E. Cho
A BSTRACT. 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.
Key words and phrases: Analytic functions, p-valent starlike, convex and close-to-convex functions, Uniformly p-valent
close-to-convex functions, Strongly starlike, Integral operator.
2000 Mathematics Subject Classification. 30C45.
1. I NTRODUCTION AND D EFINITIONS
Let Ap denote the class of functions of the form:
∞
X
p
f (z) = z +
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-toconvex functions of order β(0 ≤ β < p) in U if and only if
0 f (z)
Re
>β
(z ∈ U).
z p−1
129-09
2
B.A. F RASIN
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.
A function f ∈ Ap is said to be in the class UC p (β) of uniformly p-valent close-to-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)
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
fn (t) n
f1 (t) 1
p−1
(1.2)
Fp (z) =
...
dt
pt
tp
tp
0
and
0 α n
0 α 1
Z z
fn (t)
f1 (t)
p−1
(1.3)
Gp (z) =
pt
.
.
.
dt.
ptp−1
ptp−1
0
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,
R z f (t) α
9]). Also for p = n = 1, α1 = α ∈ [0, 1] in (1.2), we obtain the integral operator 0
dt
t
studied inR [12] and for p = n = 1, α1 = δ ∈ C, |δ| ≤ 1/4 in (1.3), we obtain the integral
z
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 operators Fp (z)
and Gp (z) to be p-valently starlike, p-valently close-to-convex and uniformly p-valent close-toconvex 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
then f is p-valently starlike in U.
Lemma 1.2 ([7]). If f ∈ Ap satisfies
00
zf (z)
f 0 (z) + 1 − p < p + 1
(z ∈ U),
(z ∈ U),
then f is p-valently starlike in U.
Lemma 1.3 ([16]). If f ∈ Ap satisfies
zf 00 (z)
a+b
Re 1 + 0
<p+
(z ∈ U),
f (z)
(1 + a)(1 − b)
where a > 0, b ≥ 0 and a + 2b ≤ 1, then f is p-valently close-to-convex in U.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 109, 9 pp.
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I NTEGRAL O PERATORS OF p- VALENT F UNCTIONS
Lemma 1.4 ([1]). If f ∈ Ap satisfies
zf 00 (z)
1
Re 1 + 0
<p+
f (z)
3
3
(z ∈ U),
then f is uniformly p-valent close-to-convex in U.
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)
>
f (z)
2
(z ∈ U),
(z ∈ U).
Lemma 1.6 ([14]). If f ∈ Ap satisfies
zf 00 (z)
γ
>p−
Re 1 + 0
f (z)
2
then
0
arg zf (z) < π γ
f (z) 2
( z ∈ U),
(0 < γ ≤ 1; z ∈ U),
or f is strongly starlike of order γ in U.
2. S UFFICIENT C ONDITIONS FOR THE O PERATOR 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.
Proof. From the definition (1.2), we observe that Fp (z) ∈ Ap . On the other hand, it is easy to
see that
α
α
f1 (z) 1
fn (z) n
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
Thus we have
(2.3)
n
X
zFp00 (z)
1+ 0
=p 1−
αi
Fp (z)
i=1
!
+
n
X
i=1
αi
zfi0 (z)
fi (z)
.
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
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 109, 9 pp.
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4
B.A. F RASIN
From (2.4) and (2.1), we obtain
!
n
n
X
X
zFp00 (z)
1
1
=p+ .
(2.5)
Re 1 + 0
<p 1−
αi +
αi p + Pn
Fp (z)
4 i=1 αi
4
i=1
i=1
Hence by Lemma 1.1, we get Fp ∈ Sp? . This completes the proof.
Letting n = p = 1, α1 = α and f1 = f in Theorem 2.1, we have:
Corollary 2.2. If f ∈ A satisfies
0 1
zf (z)
<1+
Re
f (z)
4α
R z f (t) α
where α > 0, then 0
dt is starlike in U.
t
(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)
< Ppn+ 1
(2.6)
−
p
(z ∈ U),
fi (z)
i=1 αi
then Fp is p-valently starlike in U.
Proof. From (2.3) and the hypotheses (2.6), we have
X
0
n
zFp00 (z)
zfi (z)
1 +
− p = αi
−p 0
Fp (z)
fi (z)
i=1
n
X
zf 0 (z)
<
αi i
− p
fi (z)
i=1
n
X
p+1
<
αi Pn
= p + 1.
α
i
i=1
i=1
Now using Lemma 1.2, we immediately get Fp ∈ Sp? .
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
f (z)
α
α
Rz
where α > 0, then 0 f (t)
dt is starlike in U.
t
(z ∈ U),
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 zfi (z)
(a + b)
P
(2.7)
Re
<p+
(z ∈ U),
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).
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 109, 9 pp.
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I NTEGRAL O PERATORS OF p- VALENT F UNCTIONS
5
Letting n = p = 1, α1 = α and f1 = f in Theorem 2.5, we have:
Corollary 2.6. If f ∈ A satisfies
0 (a + b)
zf (z)
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 U.
t
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
then Fp is uniformly p-valent close-to-convex in U.
Proof. The proof of the theorem follows by applying Lemma 1.4 and using (2.4), (2.8) to get
Fp ∈ U C p (β).
Letting n = p = 1, α1 = α and f1 = f in Theorem 2.7, we have:
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
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
Re
√
zFp0 (z)
p
>
Fp (z)
2
(z ∈ U).
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
0
(z ∈ U),
(z ∈ U).
t
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 109, 9 pp.
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6
B.A. F RASIN
3. S UFFICIENT C ONDITIONS FOR THE O PERATOR 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
zfi00 (z)
1
(3.1)
Re 1 + 0
< p + Pn
(z ∈ U),
fi (z)
4 i=1 αi
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
or
(3.2)
n
X
zG00p (z)
=p 1−
αi
1+ 0
Gp (z)
i=1
!
+
n
X
i=1
αi
zf 00 (z)
1 + 0i
fi (z)
.
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
From (3.3) and the hypotheses (3.1), we obtain
!
n
n
X
X
zG00p (z)
1
1
(3.4)
Re 1 + 0
<p 1−
αi +
αi p + Pn
=p+ .
Gp (z)
4 i=1 αi
4
i=1
i=1
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
zf 00 (z)
1
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
X
00 n
X
n
zG00p (z)
zf
(z)
i
1 +
=
−
p
α
−
(p
−
1)
α
i
i
0
G0p (z)
f
(z)
i
i=1
i=1
n
n
X
X
p+1
< (p − 1)
αi +
αi Pn
− p + 1 < p + 1.
i=1 αi
i=1
i=1
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I NTEGRAL O PERATORS OF p- VALENT F UNCTIONS
7
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
f 0 (z) < α
where α > 0, then
Rz
0
(z ∈ U),
(f 0 (t))α dt is starlike in U.
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.
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
<p+
(z ∈ U),
(3.6)
Re 1 + 0
fi (z)
(1 + a)(1 − b) ni=1 αi
where a > 0, b ≥ 0 and a + 2b ≤ 1, then Gp is p-valently close-to-convex in U.
Proof. In view of (3.3) and (3.6) and by using Lemma 1.3, we have Gp ∈ Cp (β).
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.
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 ∈ U C p (β).
Letting n = p = α = 1 and f1 = f in Theorem 3.7, we have:
Corollary 3.8. If f ∈ A satisfies
zf 00 (z)
1
Re 1 + 0
<1+
(z ∈ U),
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.
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
then
s
√
zG0p (z)
p
>
(z ∈ U).
(3.9)
Re
Gp (z)
2
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 109, 9 pp.
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8
B.A. F RASIN
Proof. It follows from (3.3) and (3.8) that
!
n
n
X
X
zG00p (z)
p
3p + 4
= − 1.
Re 1 + 0
>p 1−
αi +
αi p − Pn
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
(z ∈ U),
then
s
(3.11)
Re
zf 0 (z)
1
Rz
>
0
2
f (t)dt
0
(z ∈ U).
4. S TRONG S TARLIKENESS OF THE O PERATORS 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
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 γ
zf (z)
>1−
(z ∈ U),
Re
f (z)
2α
R z α
where α > 0, then 0 f (t)
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.
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.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 109, 9 pp.
http://jipam.vu.edu.au/
I NTEGRAL O PERATORS OF p- VALENT F UNCTIONS
9
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