SOME PROPERTIES OF CERTAIN SUBCLASSES OF MEROMORPHIC P-VALENT INTEGRAL OPERATORS

(1)

SOME PROPERTIES OF CERTAIN SUBCLASSES OF MEROMORPHIC P-VALENT INTEGRAL OPERATORS

DEBORAH OLUFUNMILAYO MAKINDE1

Abstract. For meromorphic p-valent function of the formfi(z) = ₍_z1₋−_wα₎p+

∑_∞

n=2a

i n(z−

w)n_{, α <}_{1, which are analytic in the punctured unit disk}_z_{: 0}_<_|_z₋_w_|_<_{1 with a pole} of orderp atw, a class Γp_β(ζ1, ζ2;γ) is introduced and some properties for Γpα(ζ1, ζ2;γ) offi(z) in relatioon to the coeﬃcient bounds, convex combination and convolution were discussed.

Keywords: Analytic,Coeﬃcient bound, Convex combination, Meromorphic, p-valent, In-tegral operator.

AMS Subject Classiﬁcation: Primary 30C45.

1. Introduction

Let A denotes the class of functions of the form

f(z) =z+ ∞

∑

n=2 anzn

analytic and normalized withf(0) =f′(0)−1 = 0 in the open diskU ={z∈C:|z|<1}. In [6], Seenivasagan gave a condition of the univalence of the integral operator

Fα,β(z) = {

β

∫ z

0 tβ−1

k ∏

i=1

(

fi(s)

s

)1/α

ds

}1/β

wherefi(z) is deﬁned by

fi(z) =z+

∞

∑

n=2

ai_nzn (1)

while Makinde in [5] gave a condition for the starlikeness for the function:

Fα(z) = ∫ z

0

k ∏

i=1

(

fi(s)

s

)1/α

ds, α∈C (2)

1 _{Department of Mathematics, Faculty of Science, Obafemi Awolowo University, Ile Ife 220005, Nigeria.}

e-mail: [email protected]

§ Submitted for GFTA’13, held in I¸sık University on October 12, 2013.

(2)

wherefi(z) is deﬁned by (1).

Also, Kanas and Ronning [2] introduced the class of function of the form

f(z) = (z−w) + ∞

∑

n=2

ai_n(z−w)n

wherewis a ﬁxed point in the unit disk normalized with f(w) =f′(w)−1 = 0.

We deﬁnefi(z) by

fi(z) =

1−α

(z−w)p +

∞

∑

n=2

ai_n(z−w)n, α <1 (3)

wherewis an arbitrary ﬁxed point in the D, and Fw,α(z) is deﬁned by

Fw,α(z) = ∫ z

0

k ∏

i=1

(

fi(s−w)

s−w

)1/α

ds, α∈C (4)

Furthermore, Xiao-Feili et al [7] denoteL∗₁(β1, β2, λ) as a subclass of A such that:

L∗₁(β1, β2, γ) =

{

f ∈A: f

′₍_z₎₋₁

β1f′(z) +β2 ≤λ

}, 0≤β1≤1; 0< β1 ≤1; 0< λ≤1

for someβ1, β2 and for some realλ. Also, he denotedT to be the subclass ofAconsisting of functions of the form:

f(z) =z−

∞

∑

n=2

anzn, an≥0

andL∗(β1, β2, λ) denotes the subclass ofL∗1(β1, β2, λ) deﬁned by: L∗(β1, β2, λ) =L∗₁(β1, β2, λ)∩T

for some real number, 0≤β1 ≤1; 0< β2 ≤1; 0< λ≤1

The classL∗(β1, β2, λ) was studied by Kim and Lee in [3], see also [1], [2], [7].

LetFα(z) be deﬁned by (2), then

zF_α′′(z)

F_α′(z) =

k ∑

i=1

1

α

(

zf_i′(z)

fi(z) −

1

)

LetG(z) be denoted by

G(z) =

k ∑

i=1

1

α

(

zf_i′(z)

fi(z) −

1

)

(5)

The class

Γα(ζ1, ζ2, γ) =

{

fi∈A

G(z) +α1 −1

ζ1(G(z) +_α1) +ζ2

≤γ

}

was studied by Makinde and Oladipo [sientia magna accepted] We deﬁne

Γp_α(ζ1, ζ2, γ) =

{

fi∈A

G(z) +α1 −1

ζ1(G(z) +_α1) +ζ2

≤γ

}

(6)

for some complex ζ1, ζ2, αand for some real γ, 0≤ |ζ1| ≤ 1, 0<|ζ2| ≤ 1, |α| ≤ 1 and 0< γ≤1 with G(z) as in (5) andfi(z) as in (3).

Letfi(z) =z+ ∑_∞

n=2ainzn and gi(z) =z+ ∑_∞

n=2binzn, we deﬁne the convolution offi(z)

andgi(z) by

fi(z)∗gi(z) = (fi∗gi)(z) =z+

∞

∑

n=2

(3)

We shall now present the main results of this paper.

2. Main Results

Theorem 2.1. Let fi(z) be as in (3) and Fw,α be as in (4). Then fi(z) is in the class

Γpα(ζ1, ζ2, γ)if and only if

k ∑ i=1 ∞ ∑ n=2 [

n(1−γζ1)−α(1 +γζ2)]|ai_n| ≤γ|(1−α)(−pζ1+αζ2)| − |(1−α)(−p−α)|, (8)

0≤ζ1≤1, 0< ζ2 ≤1, 0< α≤1

Proof. From (6), we have

G(z) +_α1 −1

ζ1(G(z) +_α1) +ζ2

= ∑k i=1 (

−p₍_z−w1−α₎p

)

+∑∞_n₌₂nai n(z−w)n

∑k i=1α

( 1−α

(z−w)p+

∑_∞

n=2nain(z−w)n

) −1

∑k i=1ζ1

(

−p₍_z1₋−_wα₎p

)

+∑∞_n₌₂nai n(z−w)n

∑k i=1α

( 1−α

(z−w)p+

∑∞

n=2nain(z−w)n

) +ζ2

≤ |(1−α)(−p−α)|+ ∑k

i=1

∑_∞

n=2(n−α)|ain| |(1−α)(−pζ1+αζ2)−∑_ik₌₁∑∞_n₌₂(n+αζ2)|ai

n||

Let fi(z) satisfy the inequality (8), the fi(z) ∈ Γpα(ζ1, ζ2, γ). Conversely, let the function fi(z)∈Γpα(ζ1, ζ2, γ), then

k ∑ i=1 ∞ ∑ n=2 [

n(1−γζ1)−α(1 +γζ2)]|ai_n| ≤γ|(1−α)(−pζ1+αζ2)| − |(1−α)(−p−α)|

Corollary 2.1. Let fi(z)∈Γpα(ζ1, ζ2, γ), then k ∑ i=1 ∞ ∑ n=2

|ai_n| ≤ γ|(1−α)([−pζ1+αζ2)| − |(1−α)(−p−α)| n(1−γζ1)−α(1 +γζ2)] .

Theorem 2.2. Let fi(z) ∈ Γpα(ζ1, ζ2, γ) and the function gi(z) defined by gi(z) = z + ∑_∞

n=2binzi be in the same Γ p

α(ζ1, ζ2, γ). Then the function Ωi(z) defined by

Ωi(z) = (1−λ)fi(z) +λgi(z) =z+

∞

∑

n=2 C_nizi

is also in the classΓpα(ζ1, ζ2, γ), where

C_ni = (1−λ)ai_n+λbi_n, 0≤λ≤1.

Proof. Letfi(z), gi(z) be in Γpα(ζ1, ζ2, γ). Then by (8) and following the proof of Theorem

2.1, we have

k ∑ i=1 ∞ ∑ n=2 [

(4)

This shows that the convex combinationfi(z), gi(z) is in the class Γpα(ζ1, ζ2, γ)

This concludes the proof of the Theorem 2.1.

Theorem 2.3. Let fi(z) belong to the class Γpα(ζ1, ζ2, γ) and gi(z) belong to the class

Γpα(β1, β2, γ),then (fi∗gi)(z) belong to the classΓpα(ζ1, ζ2, γ)⊂Γpα(β1, β2, γ).

Proof. fi(z) belong to the class Γpα(ζ1, ζ2, γ) implies

k ∑

i=1

∞

∑

n=2

[

n(1−γζ1)−α(1 +γζ2)

]

|ai_n| ≤γ|(1−α)(−pζ1+αζ2)| − |(1−α)(−p−α)|,

0≤ζ1 ≤1, 0< ζ2 ≤1, 0< α≤1

Similarly,gi(z) belong to the class Γpα(β1, β2, γ) implies

k ∑

i=1

∞

∑

n=2

[

n(1−γβ1)−α(1 +γβ2)]|ai_n| ≤γ|(1−α)(−pβ1+αβ2)| − |(1−α)(−p−α)|,

0≤β1 ≤1, 0< β2 ≤1, 0< α≤1

But

(fi∗gi)(z) = k ∑

i=1

∞

∑

n=2

[

n(1−γβ1)−α(1 +γβ2)

]

|ai_n||bi_n|

≤ k ∑

i=1

∞

∑

n=2

[

n(1−γζ1)−α(1 +γζ2)

] |ai_n|

≤ γ|(1−α)(−pζ1+αζ2)| − |(1−α)(−p−α)|

which implies that

(fi∗gi)(z)∈Γαp(ζ1, ζ2, γ)⊂Γpα(β1, β2, γ).

Theorem 2.4. Let Ψi(z)∈Γαp(ζ1, ζ2, γ) and the function υi(z) defined by

υi(z) =z+

∞

∑

n=2

Ai_nB_nizi

be in the same Γpα(ζ1, ζ2, γ). Then the function Φi(z) defined by

Φi(z) = (1−λ)Ψi(z) +λυi(z) =z+

∞

∑

n=2 C_nizi

is also in the classΓpα(ζ1, ζ2, γ), where

C_ni = (1−λ)ai_nbi_n+λAi_nB_ni, 0≤λ≤1.

(5)

References

[1] O. Altintas, and S. Owa, On soem subclassess of univalent functions with negative coeﬃcient, Pusan Kyongnam Math. J. 4 (1988) 41-46.

[2] D. Hur and G.H. Oh, On certain class of analytic functions with negative cieﬃcients, Pusan Kyongnam Math. J. 5(19890), 69-80.

[3] S. Kanas and F.Ronnig, Unifromly Starlike and conves functions and other related classes of univalent functions, Ann Universitym Narue Curie-Sklodowska Section A,53(1999), 95-105.

[4] H.S. Kim and S.K. Lee, On soem classes of univalent functions, Math Japan, 32(987), 781-796. [5] D.O Makinde and T.O. Opoola, On suﬃcient condition fo starlikeness, General Mathematics 18(3)

2010, 30-35.

[6] D.O. Makinde, On a certain family of meromorphic functions with positive coeﬃcients, Acta Univer-sitatis Apulensis (submitted).

[7] D.O. Makinde and A.T. Oladipo, Some properties of certain subclasses of univalent integral operator. Journal of Sientia Magna (accepted)

[8] N. Seenivasagan, Suﬃcient conditions for univalence, Applied Mathemaitics E-notes, 8,2008, 30-35. [9] F. Silverman, Univalent functions with negative coeﬃcients, Proc. Amer. Math. Soc, 51

(1975),109-116.