Integral Operators ofp-valent Functions B.A. Frasin vol. 10, iss. 4, art. 109, 2009
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NEW GENERAL INTEGRAL OPERATORS OF p-VALENT FUNCTIONS
B.A. FRASIN
Department of Mathematics Al al-Bayt University P.O. Box: 130095 Mafraq, Jordan
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, Uni- formlyp-valent close-to-convex functions, Strongly starlike, Integral operator.
Abstract: In this paper, we introduce new general integral operators. New sufficient condi- tions for these operators to bep-valently starlike,p-valently close-to-convex, uni- formlyp-valent close-to-convex and strongly starlike of orderγ(0< γ≤1) in the open unit disk are obtained.
Integral Operators ofp-valent Functions B.A. Frasin vol. 10, iss. 4, art. 109, 2009
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Contents
1 Introduction and Definitions 3
2 Sufficient Conditions for the OperatorFp 7
3 Sufficient Conditions for the OperatorGp 12
4 Strong Starlikeness of the OperatorsFpandGp 17
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1. Introduction and Definitions
LetApdenote the class of functions of the form:
f(z) = zp+
∞
X
n=p+1
anzn (p∈N∈ {1,2, . . .}),
which are analytic in the open unit disk U = {z : |z| < 1}.We write A1 = A. A functionf ∈ Apis said to bep-valently starlike of orderβ (0≤ β < p)if and only if
Re
zf0(z) f(z)
> β (z ∈ U).
We denote bySp?(β),the class of all such functions. On the other hand, a function f ∈ Apis said to bep-valently convex of orderβ(0≤β < p)if and only if
Re
1 + zf00(z) f0(z)
> β (z ∈ U).
Let Kp(β)denote the class of all those functions which are p-valently convex of orderβ inU. Furthermore, a functionf(z)∈ Apis said to be in the subclass Cp(β) ofp-valently close-to-convex functions of orderβ(0≤β < p)inU if and only if
Re
f0(z) zp−1
> β (z ∈ U).
Note thatSp?(0) =Sp?,Kp(0) = Kp andCp(0) =Cp are, respectively, the classes ofp-valently starlike,p-valently convex andp-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 inU.
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A functionf ∈ Ap is said to be in the classU Cp(β)of uniformlyp-valent close- to-convex functions of orderβ(0≤β < p)inU if and only if
Re
zf0(z) g(z) −β
≥
zf0(z) g(z) −p
(z ∈ U),
for someg(z)∈ U Sp(β), whereU Sp(β)is the class of uniformly p-valent starlike functions of orderβ(−1≤β < p)inU and satisfies
(1.1) Re
zf0(z) f(z) −β
≥
zf0(z) f(z) −p
(z ∈ U).
Uniformlyp-valent starlike functions were first introduced in [10].
Forαi >0andfi ∈ Ap,we define the following general integral operators
(1.2) Fp(z) =
Z z
0
ptp−1
f1(t) tp
α1
. . .
fn(t) tp
αn
dt
and
(1.3) Gp(z) =
Z z
0
ptp−1
f10(t) ptp−1
α1
. . .
fn0(t) ptp−1
αn
dt.
If we takep = 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 forp=n = 1, α1 =α ∈[0,1]in (1.2),we obtain the integral operatorRz
0
f(t)
t
α
dtstudied in [12] and forp=n= 1, α1 =δ ∈C, |δ| ≤ 1/4in (1.3),we obtain the integral operatorRz
0(f0(t))αdtstudied in [11,15].
There are many papers in which various sufficient conditions for multivalent star- likeness have been obtained. In this paper, we derive new sufficient conditions for the
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operatorsFp(z)andGp(z)to bep-valently starlike,p-valently close-to-convex and uniformlyp-valent close-to-convex inU. Furthermore, we give new sufficient condi- tions for these two general operators to be strongly starlike of orderγ(0< γ≤1)in U.
In order to derive our main results, we have to recall here the following results:
Lemma 1.1 ([13]). Iff ∈ Ap satisfies
Re
1 + zf00(z) f0(z)
< p+ 1
4 (z ∈ U),
thenf isp-valently starlike inU. Lemma 1.2 ([7]). Iff ∈ Ap satisfies
zf00(z)
f0(z) + 1−p
< p+ 1 (z ∈ U),
thenf isp-valently starlike inU. Lemma 1.3 ([16]). Iff ∈ Ap satisfies
Re
1 + zf00(z) f0(z)
< p+ a+b
(1 +a)(1−b) (z ∈ U),
wherea >0, b≥0anda+ 2b≤1,thenf isp-valently close-to-convex inU. Lemma 1.4 ([1]). Iff ∈ Ap satisfies
Re
1 + zf00(z) f0(z)
< p+ 1
3 (z ∈ U), thenf is uniformlyp-valent close-to-convex inU.
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Lemma 1.5 ([17]). Iff ∈ Ap satisfies
Re
1 + zf00(z) f0(z)
> p
4 −1 (z ∈ U),
then
Re s
zf0(z) f(z) >
√p
2 (z ∈ U).
Lemma 1.6 ([14]). Iff ∈ Ap satisfies
Re
1 + zf00(z) f0(z)
> p− γ
2 (z ∈ U),
then
arg zf0(z) f(z)
< π
2γ (0< γ ≤1; z∈ U), orf is strongly starlike of orderγinU.
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2. Sufficient Conditions for the Operator F
pWe begin by establishing sufficient conditions for the operatorFpto be inSp?. Theorem 2.1. Letαi >0be real numbers for alli= 1,2, . . . , n.Iffi ∈ Ap for all i= 1,2, . . . , nsatisfies
(2.1) Re
zfi0(z) fi(z)
< p+ 1 4Pn
i=1αi (z ∈ U), thenFpisp-valently starlike inU.
Proof. From the definition (1.2), we observe thatFp(z)∈ Ap.On the other hand, it is easy to see that
(2.2) Fp0(z) =pzp−1
f1(z) zp
α1
. . .
fn(z) zp
αn
. Differentiating (2.2) logarithmically and multiplying byz, we obtain
zFp00(z)
Fp0(z) = (p−1) +
n
X
i=1
αi
zfi0(z) fi(z) −p
.
Thus we have
(2.3) 1 + zFp00(z)
Fp0(z) =p 1−
n
X
i=1
αi
! +
n
X
i=1
αi
zfi0(z) fi(z)
.
Taking the real part of both sides of (2.3), we have (2.4) Re
1 + zFp00(z) Fp0(z)
=p 1−
n
X
i=1
αi
! +
n
X
i=1
αiRe
zfi0(z) fi(z)
.
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From (2.4) and (2.1), we obtain Re
1 + zFp00(z) Fp0(z)
< p 1−
n
X
i=1
αi
! +
n
X
i=1
αi
p+ 1
4Pn i=1αi
(2.5)
=p+1 4.
Hence by Lemma1.1, we getFp ∈ Sp?.This completes the proof.
Lettingn=p= 1, α1 =α and f1 =f in Theorem2.1, we have:
Corollary 2.2. Iff ∈ A satisfies
Re
zf0(z) f(z)
<1 + 1
4α (z ∈ U), whereα >0, thenRz
0
f(t) t
α
dtis starlike inU.
In the next theorem, we derive another sufficient condition forp-valently starlike functions inU.
Theorem 2.3. Letαi >0be real numbers for alli= 1,2, . . . , n.Iffi ∈ Ap for all i= 1,2, . . . , n satisfies
(2.6)
zfi0(z) fi(z) −p
< p+ 1 Pn
i=1αi (z ∈ U), thenFpisp-valently starlike inU.
Proof. From (2.3) and the hypotheses (2.6), we have
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1 + zFp00(z) Fp0(z) −p
=
n
X
i=1
αi
zfi0(z) fi(z) −p
<
n
X
i=1
αi
zfi0(z) fi(z) −p
<
n
X
i=1
αi
p+ 1 Pn
i=1αi
=p+ 1.
Now using Lemma1.2, we immediately getFp ∈ Sp?.
Lettingn=p= 1, α1 =α and f1 =f in Theorem2.3, we have:
Corollary 2.4. Iff ∈ A satisfies
zf0(z) f(z) −1
< 2
α (z ∈ U), whereα >0, thenRz
0
f(t) t
α
dtis starlike inU.
Applying Lemmas 1.3and1.4, we obtain the following sufficient conditions for Fpto bep-valently close-to-convex and uniformlyp-valent close-to-convex inU. Theorem 2.5. Letαi >0be real numbers for alli= 1,2, . . . , n.Iffi ∈ Ap for all i= 1,2, . . . , n satisfies
(2.7) Re
zfi0(z) fi(z)
< p+ (a+b) (1 +a)(1−b)Pn
i=1αi (z ∈ U),
wherea >0, b≥0anda+ 2b≤1,then Fpisp-valently close-to-convex inU. Proof. From (2.4) and the hypotheses (2.7) and applying Lemma1.3, we haveFp ∈ Cp(β).
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Lettingn=p= 1, α1 =α and f1 =f in Theorem2.5, we have:
Corollary 2.6. Iff ∈ A satisfies
Re
zf0(z) f(z)
<1 + (a+b)
(1 +a)(1−b)α (z ∈ U), whereα >0,a >0, b ≥0anda+ 2b ≤1,thenRz
0
f(t) t
α
dt is close-to-convex in U.
Theorem 2.7. Letαi >0be real numbers for alli= 1,2, . . . , n.Iffi ∈ Ap for all i= 1,2, . . . , n satisfies
(2.8) Re
zfi0(z) fi(z)
< p+ 1 3Pn
i=1αi
(z ∈ U),
thenFpis uniformlyp-valent close-to-convex inU.
Proof. The proof of the theorem follows by applying Lemma 1.4 and using (2.4), (2.8) to getFp ∈ U Cp(β).
Lettingn=p= 1, α1 =α and f1 =f in Theorem2.7, we have:
Corollary 2.8. Iff ∈ A satisfies
Re
zf0(z) f(z)
<1 + 1
3α (z ∈ U), whereα >0, thenRz
0
f(t) t
α
dt is uniformly close-to-convex inU. Using Lemma1.5, we obtain the next result
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Theorem 2.9. Letαi >0be real numbers for alli= 1,2, . . . , n.Iffi ∈ Ap for all i= 1,2, . . . , n satisfies
(2.9) Re
zfi0(z) fi(z)
> p− 3p+ 4 4Pn
i=1αi (z ∈ U), then
Re s
zFp0(z) Fp(z) >
√p
2 (z ∈ U).
Proof. It follows from (2.4) and (2.9) that
Re
1 + zFp00(z) Fp0(z)
> p 1−
n
X
i=1
αi
! +
n
X
i=1
αi
p− 3p+ 4 4Pn
i=1αi
= p 4 −1.
By Lemma1.5, we conclude that Re
szFp0(z) Fp(z) >
√p
2 (z ∈ U).
Lettingn=p= 1, α1 = 1 and f1 =f in Theorem2.9, we have:
Corollary 2.10. Iff ∈ A satisfies
(2.10) Re
zfi0(z) fi(z)
>−3
4 (z ∈ U), then
(2.11) Re
v u u t
f(z) Rz
0
f(t) t
dt
> 1
2 (z ∈ U).
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3. Sufficient Conditions for the Operator G
pThe first two theorems in this section give a sufficient condition for the integral operatorGpto be in the classSp?.
Theorem 3.1. Letαi >0be real numbers for alli= 1,2, . . . , n.Iffi ∈ Ap for all i= 1,2, . . . , n satisfies
(3.1) Re
1 + zfi00(z) fi0(z)
< p+ 1 4Pn
i=1αi (z ∈ U), thenGpisp-valently starlike inU.
Proof. From the definition (1.3), we observe thatGp(z)∈ Ap and zG00p(z)
G0p(z) = (p−1) +
n
X
i=1
αi
zfi00(z)
fi0(z) −(p−1)
or
(3.2) 1 + zG00p(z)
G0p(z) =p 1−
n
X
i=1
αi
! +
n
X
i=1
αi
1 + zfi00(z) fi0(z)
.
Taking the real part of both sides of (3.2), we have (3.3) Re
1 + zG00p(z) G0p(z)
=p 1−
n
X
i=1
αi
! +
n
X
i=1
αiRe
1 + zfi00(z) fi0(z)
.
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From (3.3) and the hypotheses (3.1), we obtain Re
1 + zG00p(z) G0p(z)
< p 1−
n
X
i=1
αi
! +
n
X
i=1
αi
p+ 1
4Pn i=1αi
(3.4)
=p+1 4.
Therefore, using Lemma 1.1, it follows that the integral operatorGpbelongs to the classSp?.
Lettingn=p= 1, α1 =αand f1 =f in Theorem3.1, we obtain Corollary 3.2. Iff ∈ A satisfies
Re
1 + zf00(z) f0(z)
<1 + 1
4α (z ∈ U), whereα >0, thenRz
0(f0(t))αdt is starlike inU.
Theorem 3.3. Letαi >0be real numbers for alli= 1,2, . . . , n.Iffi ∈ Ap for all i= 1,2, . . . , n satisfies
(3.5)
zfi00(z) fi0(z)
< p+ 1 Pn
i=1αi −p+ 1 (z ∈ U), wherePn
i=1αi >1,thenGpisp-valently starlike inU. Proof. It follows from (3.2) and (3.5) that
1 + zG00p(z) G0p(z) −p
=
n
X
i=1
αi
zfi00(z) fi0(z)
−(p−1)
n
X
i=1
αi
<(p−1)
n
X
i=1
αi +
n
X
i=1
αi
p+ 1 Pn
i=1αi −p+ 1
< p+ 1.
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Therefore, it follows from Lemma1.2thatGp is in the classSp?. Lettingn=p= 1, α1 =αand f1 =f in Theorem3.3, we obtain:
Corollary 3.4. Iff ∈ A satisfies
zf00(z) f0(z)
< 2
α (z ∈ U), whereα >0, thenRz
0(f0(t))αdt is starlike inU.
Applying Lemmas 1.3and1.4, we obtain the following sufficient conditions for Gpto bep-valently close-to-convex and uniformlyp-valent close-to-convex inU. Theorem 3.5. Letαi >0be real numbers for alli= 1,2, . . . , n.Iffi ∈ Ap for all i= 1,2, . . . , n satisfies
(3.6) Re
1 + zfi00(z) fi0(z)
< p+ a+b
(1 +a)(1−b)Pn
i=1αi (z ∈ U), where a >0, b≥0 anda+ 2b≤1,thenGpisp-valently close-to-convex inU. Proof. In view of (3.3) and (3.6) and by using Lemma1.3, we haveGp∈ Cp(β).
Lettingn=p= 1, α1 =αand f1 =f in Theorem3.5, we obtain Corollary 3.6. Iff ∈ A satisfies
Re
1 + zf00(z) f0(z)
<1 + a+b
(1 +a)(1−b)α (z ∈ U), whereα > 0, a > 0, b ≥ 0 anda+ 2b ≤ 1,thenRz
0(f0(t))αdt is close-to-convex inU.
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Theorem 3.7. Letαi >0be real numbers for alli= 1,2, . . . , n.Iffi ∈ Ap for all i= 1,2, . . . , n satisfies
(3.7) Re
1 + zfi00(z) fi0(z)
< p+ 1 3Pn
i=1αi (z ∈ U), thenGpis uniformlyp-valent close-to-convex inU.
Proof. In view of (3.3) and (3.7) and by using Lemma1.4, we haveGp ∈ U Cp(β).
Lettingn=p=α= 1and f1 =f in Theorem3.7, we have:
Corollary 3.8. Iff ∈ A satisfies
Re
1 + zf00(z) f0(z)
<1 + 1
3α (z ∈ U), whereα >0,thenRz
0(f0(t))αdt is uniformly close-to-convex inU. Using Lemma1.5, we obtain the next result.
Theorem 3.9. Letαi >0be real numbers for alli= 1,2, . . . , n.Iffi ∈ Ap for all i= 1,2, . . . , n satisfies
(3.8) Re
1 + zfi00(z) fi0(z)
> p− 3p+ 4 4Pn
i=1αi (z ∈ U), then
(3.9) Re
s
zG0p(z) Gp(z) >
√p
2 (z ∈ U).
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Proof. It follows from (3.3) and (3.8) that
Re
1 + zG00p(z) G0p(z)
> p 1−
n
X
i=1
αi
! +
n
X
i=1
αi
p− 3p+ 4 4Pn
i=1αi
= p 4 −1.
By Lemma1.5, we get the result (3.9).
Lettingn=p= 1, α1 = 1 and f1 =f in Theorem3.9, we have Corollary 3.10. Iff ∈ A satisfies
(3.10) Re
1 + zfi00(z) fi0(z)
>−3
4 (z ∈ U), then
(3.11) Re
s
zf0(z) Rz
0 f0(t)dt > 1
2 (z ∈ U).
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4. Strong Starlikeness of the Operators F
pand G
pApplying Lemma1.6 and using (2.4), we obtain the following sufficient condition for the operatorFpto be strongly starlike of orderγinU.
Theorem 4.1. Letαi >0be real numbers for alli= 1,2, . . . , n.Iffi ∈ Ap for all i= 1,2, . . . , n satisfies
Re
zfi0(z) fi(z)
> p− γ 2Pn
i=1αi (z ∈ U), thenFpis strongly starlike of orderγ (0< γ ≤1)inU.
Lettingn=p= 1, α1 =α andf1 =f in Theorem4.1, we have Corollary 4.2. Iff ∈ A satisfies
Re
zf0(z) f(z)
>1− γ
2α (z ∈ U), whereα >0, thenRz
0
f(t) t
α
dt is strongly starlike of orderγ (0< γ ≤1)inU. Applying once again Lemma1.6 and using (3.3), we obtain the following suffi- cient condition for the operatorGpto be strongly starlike of orderγinU.
Theorem 4.3. Letαi >0be real numbers for alli= 1,2, . . . , n.Iffi ∈ Ap for all i= 1,2, . . . , n satisfies
Re
1 + zfi00(z) fi0(z)
> p− γ 2Pn
i=1αi (z ∈ U), thenGp is strongly starlike of orderγ(0< γ ≤1)inU.
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Lettingn=p=α1 = 1andf1 =f in Theorem4.3, we have Corollary 4.4. Iff ∈ A satisfies
Re
1 + zf00(z) f0(z)
>1− γ
2α (z ∈ U), whereα >0,thenRz
0(f0(t))αdt is strongly starlike of orderγ(0< γ ≤1)inU.
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