New sufficient condi- tions for these operators to bep-valently starlike,p-valently close-to-convex, uniformlyp-valent close-to-convex and strongly starlike of orderγ(0&lt

NEW GENERAL INTEGRAL OPERATORS OFp-VALENT FUNCTIONS

B.A. FRASIN

DEPARTMENT OFMATHEMATICS

AL AL-BAYTUNIVERSITY

P.O. BOX: 130095 MAFRAQ, JORDAN

bafrasin@yahoo.com

Received 10 May, 2009; accepted 14 October, 2009 Communicated by N.E. Cho

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, uniformlyp-valent close-to-convex and strongly starlike of orderγ(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. INTRODUCTION ANDDEFINITIONS

LetA_pdenote the class of functions of the form:

f(z) = z^p+

∞

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 function f ∈ A_pis said to bep-valently starlike of orderβ(0≤β < p)if and only if

Re

zf⁰(z) f(z)

> β (z ∈ U).

We denote byS_p^?(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).

LetK_p(β)denote the class of all those functions which are p-valently convex of order β inU. Furthermore, a function f(z) ∈ A_pis said to be in the subclass C_p(β) of p-valently close-to- convex functions of orderβ(0≤β < p)inU if and only if

Re

f⁰(z) z^p−1

> β (z∈ U).

129-09

Note thatS_p^?(0) =S_p^?,K_p(0) =K_pandC_p(0) =C_pare, respectively, the classes ofp-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 inU.

A function f ∈ A_p is said to be in the class U C_p(β)of uniformly p-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 S_p(β), whereU S_p(β)is the class of uniformlyp-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) F_p(z) =

Z z

0

pt^p−1

f₁(t) t^p

α1

. . .

f_n(t) t^p

αn

dt and

(1.3) G_p(z) =

Z z

0

pt^p−1

f₁⁰(t) pt^p−1

α1

. . .

f_n⁰(t) pt^p−1

αn

dt.

If we takep = 1, we obtain of the general integral operatorsF1(z) = Fn(z) andG1(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

dt studied in [12] and for p = 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 starlikeness have been obtained. In this paper, we derive new sufficient conditions for the operatorsF_p(z) andG_p(z)to bep-valently starlike,p-valently close-to-convex and uniformlyp-valent close-to- convex inU. Furthermore, we give new sufficient conditions for these two general operators to be strongly starlike of orderγ(0< γ≤1)inU.

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 ∈ A_p 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 ∈ A_p satisfies

Re

1 + zf⁰⁰(z) f⁰(z)

< p+ 1

3 (z ∈ U), thenf is uniformlyp-valent close-to-convex inU.

Lemma 1.5 ([17]). Iff ∈ A_p 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.

2. SUFFICIENTCONDITIONS FOR THEOPERATORF_p We begin by establishing sufficient conditions for the operatorF_p to be inS_p^?.

Theorem 2.1. Let α_i > 0 be real numbers for all i = 1,2, . . . , n. If f_i ∈ A_pfor all i = 1,2, . . . , nsatisfies

(2.1) Re

zf_i⁰(z) f_i(z)

< p+ 1 4Pn

i=1α_i (z ∈ U), thenF_pisp-valently starlike inU.

Proof. From the definition (1.2), we observe thatF_p(z) ∈ A_p.On the other hand, it is easy to see that

(2.2) F_p⁰(z) =pz^p−1

f₁(z) z^p

α1

. . .

f_n(z) z^p

αn

. Differentiating (2.2) logarithmically and multiplying byz, we obtain

zF_p⁰⁰(z)

F_p⁰(z) = (p−1) +

n

X

i=1

α_i

zf_i⁰(z) f_i(z) −p

.

Thus we have

(2.3) 1 + zF_p⁰⁰(z)

F_p⁰(z) =p 1−

n

X

i=1

α_i

! +

n

X

i=1

α_i

zf_i⁰(z) fi(z)

.

Taking the real part of both sides of (2.3), we have

(2.4) Re

1 + zF_p⁰⁰(z) F_p⁰(z)

=p 1−

n

X

i=1

α_i

! +

n

X

i=1

α_iRe

zf_i⁰(z) fi(z)

.

From (2.4) and (2.1), we obtain Re

1 + zF_p⁰⁰(z) F_p⁰(z)

< p 1−

n

X

i=1

α_i

! +

n

X

i=1

α_i

p+ 1

4Pn i=1α_i

=p+ 1 4. (2.5)

Hence by Lemma 1.1, we getF_p ∈ S_p^?.This completes the proof.

Lettingn =p= 1, α₁ =α andf₁ =f in Theorem 2.1, 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 for p-valently starlike functions inU.

Theorem 2.3. Let α_i > 0 be real numbers for all i = 1,2, . . . , n. If f_i ∈ A_p for all i = 1,2, . . . , n satisfies

(2.6)

zf_i⁰(z) f_i(z) −p

< p+ 1 Pn

i=1α_i (z ∈ U), thenF_pisp-valently starlike inU.

Proof. From (2.3) and the hypotheses (2.6), we have

1 + zF_p⁰⁰(z) F_p⁰(z) −p

=

n

X

i=1

α_i

zf_i⁰(z) f_i(z) −p

<

n

X

i=1

α_i

zf_i⁰(z) f_i(z) −p

<

n

X

i=1

α_i

p+ 1 Pn

i=1α_i

=p+ 1.

Now using Lemma 1.2, we immediately getF_p ∈ S_p^?.

Lettingn =p= 1, α₁ =α andf₁ =f in Theorem 2.3, 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 1.3 and 1.4, we obtain the following sufficient conditions for F_pto be p-valently close-to-convex and uniformlyp-valent close-to-convex inU.

Theorem 2.5. Let α_i > 0 be real numbers for all i = 1,2, . . . , n. If f_i ∈ A_p for all i = 1,2, . . . , n satisfies

(2.7) Re

zf_i⁰(z) f_i(z)

< p+ (a+b) (1 +a)(1−b)Pn

i=1α_i (z ∈ U), wherea >0, b ≥0anda+ 2b≤1,then F_pisp-valently close-to-convex inU.

Proof. From (2.4) and the hypotheses (2.7) and applying Lemma 1.3, we haveF_p ∈ C_p(β).

Lettingn =p= 1, α₁ =α andf₁ =f in Theorem 2.5, 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 inU.

Theorem 2.7. Let α_i > 0 be real numbers for all i = 1,2, . . . , n. If f_i ∈ A_p for all i = 1,2, . . . , n satisfies

(2.8) Re

zf_i⁰(z) f_i(z)

< p+ 1 3Pn

i=1α_i (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 get

Fp ∈ U Cp(β).

Lettingn =p= 1, α₁ =α andf₁ =f in Theorem 2.7, 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 Lemma 1.5, we obtain the next result

Theorem 2.9. Let α_i > 0 be real numbers for all i = 1,2, . . . , n. If f_i ∈ A_p for all i = 1,2, . . . , n satisfies

(2.9) Re

zf_i⁰(z) f_i(z)

> p− 3p+ 4 4Pn

i=1α_i (z ∈ U), then

Re s

zF_p⁰(z) Fp(z) >

√p

2 (z ∈ U).

Proof. It follows from (2.4) and (2.9) that

Re

1 + zF_p⁰⁰(z) F_p⁰(z)

> p 1−

n

X

i=1

α_i

! +

n

X

i=1

α_i

p− 3p+ 4 4Pn

i=1αi

= p 4 −1.

By Lemma 1.5, we conclude that Re

szF_p⁰(z) F_p(z) >

√p

2 (z ∈ U).

Lettingn =p= 1, α₁ = 1 and f₁ =f in Theorem 2.9, we have:

Corollary 2.10. Iff ∈ A satisfies

(2.10) Re

zf_i⁰(z) f_i(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).

3. SUFFICIENT CONDITIONS FOR THEOPERATORG_p

The first two theorems in this section give a sufficient condition for the integral operatorGpto be in the classS_p^?.

Theorem 3.1. Let α_i > 0 be real numbers for all i = 1,2, . . . , n. If f_i ∈ A_p for all i = 1,2, . . . , n satisfies

(3.1) Re

1 + zf_i⁰⁰(z) f_i⁰(z)

< p+ 1 4Pn

i=1α_i (z ∈ U), thenG_pisp-valently starlike inU.

Proof. From the definition (1.3), we observe thatGp(z)∈ Apand zG⁰⁰_p(z)

G⁰_p(z) = (p−1) +

n

X

i=1

α_i

zf_i⁰⁰(z)

f_i⁰(z) −(p−1)

or

(3.2) 1 + zG⁰⁰_p(z)

G⁰_p(z) =p 1−

n

X

i=1

α_i

! +

n

X

i=1

α_i

1 + zf_i⁰⁰(z) f_i⁰(z)

.

Taking the real part of both sides of (3.2), we have

(3.3) Re

1 + zG⁰⁰_p(z) G⁰_p(z)

=p 1−

n

X

i=1

α_i

! +

n

X

i=1

α_iRe

1 + zf_i⁰⁰(z) f_i⁰(z)

.

From (3.3) and the hypotheses (3.1), we obtain Re

1 + zG⁰⁰_p(z) G⁰_p(z)

< p 1−

n

X

i=1

αi

! +

n

X

i=1

αi

p+ 1

4Pn i=1α_i

=p+ 1 4. (3.4)

Therefore, using Lemma 1.1, it follows that the integral operator G_pbelongs to the class

S_p^?.

Lettingn =p= 1, α₁ =αand f₁ =f in Theorem 3.1, 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. Let α_i > 0 be real numbers for all i = 1,2, . . . , n. If f_i ∈ A_p for all i = 1,2, . . . , n satisfies

(3.5)

zf_i⁰⁰(z) f_i⁰(z)

< p+ 1 Pn

i=1α_i −p+ 1 (z ∈ U), wherePn

i=1α_i >1,thenG_pisp-valently starlike inU. Proof. It follows from (3.2) and (3.5) that

1 + zG⁰⁰_p(z) G⁰_p(z) −p

=

n

X

i=1

αi

zf_i⁰⁰(z) f_i⁰(z)

−(p−1)

n

X

i=1

αi

<(p−1)

n

X

i=1

α_i+

n

X

i=1

α_i

p+ 1 Pn

i=1αi

−p+ 1

< p+ 1.

Therefore, it follows from Lemma 1.2 thatG_pis in the classS_p^?. Lettingn =p= 1, α₁ =αand f₁ =f in Theorem 3.3, 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 1.3 and 1.4, we obtain the following sufficient conditions for G_pto be p-valently close-to-convex and uniformlyp-valent close-to-convex inU.

Theorem 3.5. Let α_i > 0 be real numbers for all i = 1,2, . . . , n. If f_i ∈ A_p for all i = 1,2, . . . , n satisfies

(3.6) Re

1 + zf_i⁰⁰(z) f_i⁰(z)

< p+ a+b

(1 +a)(1−b)Pn

i=1α_i (z ∈ U), where a >0, b ≥0 anda+ 2b ≤1,thenG_pisp-valently close-to-convex inU.

Proof. In view of (3.3) and (3.6) and by using Lemma 1.3, we haveG_p ∈ C_p(β).

Lettingn =p= 1, α₁ =αand f₁ =f in Theorem 3.5, 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.

Theorem 3.7. Let α_i > 0 be real numbers for all i = 1,2, . . . , n. If f_i ∈ A_p for all i = 1,2, . . . , n satisfies

(3.7) Re

1 + zf_i⁰⁰(z) f_i⁰(z)

< p+ 1 3Pn

i=1α_i (z ∈ U), thenG_pis uniformlyp-valent close-to-convex inU.

Proof. In view of (3.3) and (3.7) and by using Lemma 1.4, we haveG_p ∈ U C_p(β).

Lettingn =p=α= 1and f₁ =f in Theorem 3.7, 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 Lemma 1.5, we obtain the next result.

Theorem 3.9. Let α_i > 0 be real numbers for all i = 1,2, . . . , n. If f_i ∈ A_p for all i = 1,2, . . . , n satisfies

(3.8) Re

1 + zf_i⁰⁰(z) f_i⁰(z)

> p− 3p+ 4 4Pn

i=1α_i (z ∈ U), then

(3.9) Re

s

zG⁰_p(z) G_p(z) >

√p

2 (z ∈ U).

Proof. It follows from (3.3) and (3.8) that

Re

1 + zG⁰⁰_p(z) G⁰_p(z)

> p 1−

n

X

i=1

αi

! +

n

X

i=1

αi

p− 3p+ 4 4Pn

i=1α_i

= p 4 −1.

By Lemma 1.5, we get the result (3.9).

Lettingn =p= 1, α₁ = 1 and f₁ =f in Theorem 3.9, we have Corollary 3.10. Iff ∈ A satisfies

(3.10) Re

1 + zf_i⁰⁰(z) f_i⁰(z)

>−3

4 (z ∈ U), then

(3.11) Re

s zf⁰(z) Rz

0 f⁰(t)dt > 1

2 (z ∈ U).

4. STRONGSTARLIKENESS OF THE OPERATORSF_pANDG_p

Applying Lemma 1.6 and using (2.4), we obtain the following sufficient condition for the operatorF_pto be strongly starlike of orderγinU.

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

Re

zf_i⁰(z) f_i(z)

> p− γ 2Pn

i=1α_i (z ∈ U), thenFpis strongly starlike of orderγ (0< γ≤1)inU.

Lettingn =p= 1, α₁ =α andf₁ =f in Theorem 4.1, 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 Lemma 1.6 and using (3.3), we obtain the following sufficient condition for the operatorG_pto be strongly starlike of orderγinU.

Theorem 4.3. Let α_i > 0 be real numbers for all i = 1,2, . . . , n. If f_i ∈ A_p for all i = 1,2, . . . , n satisfies

Re

1 + zf_i⁰⁰(z) f_i⁰(z)

> p− γ 2Pn

i=1α_i (z ∈ U), thenG_p is strongly starlike of orderγ (0< γ ≤1)inU.

Lettingn =p=α₁ = 1andf₁ =f in Theorem 4.3, 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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