p-VALENT MEROMORPHIC FUNCTIONS WITH ALTERNATING COEFFICIENTS BASED ON INTEGRAL OPERATOR

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p-valent Meromorphic Functions Sh. Najafzadeh, A. Ebadian

and S. Shams vol. 9, iss. 2, art. 40, 2008

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p-VALENT MEROMORPHIC FUNCTIONS WITH ALTERNATING COEFFICIENTS BASED ON

INTEGRAL OPERATOR

SH. NAJAFZADEH

Department of Mathematics, Faculty of Science Maragheh University, Maragheh, Iran

EMail:najafzadeh1234@yahoo.ie

A. EBADIAN AND S. SHAMS

Department of Mathematics, Faculty of Science Urmia University, Urmia, Iran

EMail:a.ebadian@mail.urmia.ac.ir

Received: 11 April, 2007

Accepted: 15 January, 2008

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 30C45, 30C50.

Key words: Meromorphic Functions, Alternating Coefficients, Distortion Bounds.

Abstract: By using a linear operator, a subclass of meromorphicallyp−valent functions with alternating coefficients is introduced. Some important properties of this class such as coefficient bounds, distortion bounds, etc. are found.

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1. Introduction

LetΣ_pbe the class of functions of the form

(1.1) f(z) =Az^−p +

∞

X

n=p

a_nzⁿ, A≥0

that are regular in the punctured disk∆^∗ ={z : 0<|z|<1}andσ_pbe the subclass ofΣ_p consisting of functions with alternating coefficients of the type

(1.2) f(z) = Az^−p+

∞

X

n=p

(−1)ⁿ⁻¹a_nzⁿ, a_n≥0, A≥0.

Let

(1.3) Σ^∗_p(β) =

f ∈Σ_p : Re

z[J(f(z))]⁰ J(f(z))

<−β,0≤β < p

and letσ^∗_p(β) = Σ^∗_p(β)∩σ_pwhere (1.4) J(f(z)) = (γ −p+ 1)

Z 1 0

(u^γ)f(uz)du, p < γ is a linear operator.

With a simple calculation we obtain

(1.5) J(f(z)) =











Az^−p+

∞

P

n=p

(−1)ⁿ⁻¹

γ−p+1 γ+n+1

anzⁿ, f(z)∈σp;

Az^−p+

∞

P

n=p

_γ−p+1

γ+n+1

a_nzⁿ, f(z)∈Σ_p.

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For more details about meromorphic p-valent functions, we can see the recent works of many authors in [1], [2], [3].

Also, Uralegaddi and Ganigi [4] worked on meromorphic univalent functions with alternating coefficients.

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2. Coefficient Estimates

Theorem 2.1. Let

f(z) =Az^−p+

∞

X

n=p

a_nzⁿ ∈Σ_p.

If

(2.1)

∞

X

n=p

(n+β)

γ −p+ 1 γ+n+ 1

|a_n| ≤A(p−β),

thenf(z)∈Σ^∗_p(β).

Proof. It is sufficient to show that

M =

z[Jf(z))]⁰ Jf(z)) +p

z[Jf(z))]⁰

Jf(z)) −(p−2β)

<1 for |z|<1.

However, by (1.5)

M =

−pAz^−p+

∞

P

n=p

n_γ−p+1

γ+n+1

a_nzⁿ+pAz^−p+

∞

P

n=p

p_γ−p+1

γ+n+1

a_nzⁿ

−pAz^−p+

∞

P

n=p

n

γ−p+1 γ+n+1

a_nzⁿ−(p−2β)Az^−p−

∞

P

n=p

(p−2β)

γ−p+1 γ+n+1

a_nzⁿ

≤

∞

P

n=p

h

(n+p)

γ−p+1 γ+n+1

i

|an| 2A(p−β)−

∞

P

n=p

(n−p+ 2β)

γ−p+1 γ+n+1

|a_n| .

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The last expression is less than or equal to 1 provided

∞

X

n=p

(n+p)

γ−p+ 1 γ+n+ 1

|a_n| ≤2A(p−β)−

∞

X

n=p

(n−p+2β)

γ−p+ 1 γ+n+ 1

|a_n|,

which is equivalent to

∞

X

n=p

(n+β)

γ−p+ 1 γ+n+ 1

|a_n| ≤A(p−β)

which is true by (2.1) so the proof is complete.

The converse of Theorem 2.1 is also true for functions in σ_p^∗(β), where p is an odd number.

Theorem 2.2. A functionf(z)inσ_pis inσ_p^∗(β)if and only if

(2.2)

∞

X

n=p

(n+β)

γ −p+ 1 γ+n+ 1

an≤A(p−β).

Proof. According to Theorem2.1it is sufficient to prove the “only if” part. Suppose that

(2.3) Re

z(Jf(z))⁰ (Jf(z))

= Re





−Apz^−p+P∞

n=pn(−1)ⁿ⁻¹

γ−p+1 γ+n+1

anzⁿ Az^−p+P∞

n=p(−1)ⁿ⁻¹

γ−p+1 γ+n+1

a_nzⁿ



<−β.

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By choosing values of zon the real axis so that ^(z(J_(J_f(z))^f(z))⁰ is real and clearing the denominator in (2.3) and lettingz → −1through real values we obtain

Ap−

∞

X

n=p

n

γ−p+ 1 γ+n+ 1

a_n≥β A+

∞

X

n=p

γ−p+ 1 γ+n+ 1

a_n

! ,

which is equivalent to

∞

X

n=p

(n+β)

γ −p+ 1 γ+n+ 1

a_n≤A(p−β).

Corollary 2.3. Iff(z)∈σ^∗_p(β)then

(2.4) a_n≤ A(p−β)(γ+n+ 1)

(n+β)(γ−p+ 1) for n =p, p+ 1, . . . . The result is sharp for functions of the type

(2.5) f_n(z) = Az^−p+ (−1)ⁿ⁻¹A(p−β)(γ+n+ 1) (n+β)(γ−p+ 1) zⁿ.

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3. Distortion Bounds and Important Properties of σ

_p^∗

(β)

In this section we obtain distortion bounds for functions in the classσ_p^∗(β)and prove some important properties of this class, wherepis an odd number.

Theorem 3.1. Let f(z) = Az^−p +

∞

P

n=p

(−1)ⁿ⁻¹a_nzⁿ, a_n ≥ 0be in the classσ_p^∗(β) andβ ≥γ+ 1then

(3.1) Ar^−p− A(p−β)

γ−p+ 1r^p ≤ |f(z)| ≤Ar^−p+ A(p−β) γ −p+ 1r^p. Proof. Sinceβ ≥γ+ 1,so _γ+n+1^n+β ≥1.Then

(γ−p+ 1)

∞

X

n=p

a_n≤

∞

X

n=p

n+β γ+n+ 1

(γ−p+ 1)a_n≤A(p−β),

and we have

|f(z)|=|Az^−p+

∞

X

n=p

(−1)ⁿ⁻¹a_nzⁿ|

≤ A r^p +r^p

∞

X

n=p

a_n ≤ A

r^p +r^p A(p−β) (γ−p+ 1). Similarly,

|f(z)| ≥ A r^p −

∞

X

n=p

a_nrⁿ ≥ A r^p −r^p

∞

X

n=p

a_n ≥ A

r^p − A(p−β) γ−p+ 1r^p.

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Theorem 3.2. Let f(z) = Az^−p+

∞

X

n=p

a_nzⁿ and g(z) = Az^−p+

∞

X

n=p

(−1)ⁿ⁻¹b_nzⁿ

be in the classσ_p^∗(β).Then the weighted mean off andg defined by

W_j(z) = 1

2[(1−j)f(z) + (1 +j)g(z)]

is also in the same class.

Proof. Sincef andg belong toσ^∗_p(β),then by (2.2) we have

(3.2)











∞

P

n=p

(n+β)_γ−p+1

γ+n+1

a_n ≤A(p−β),

∞

P

n=p

(n+β)_γ−p+1)

γ+n+1

b_n ≤A(p−β).

After a simple calculation we obtain Wj(z) =Az^−p +

∞

X

n=p

1−j

2 an+1 +j 2 bn

(−1)ⁿ⁻¹zⁿ.

However,

∞

X

n=p

(n+β)

γ−p+ 1) γ+n+ 1

1−j

2 a_n+1 +j 2 b_n

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=

1−j 2

^∞ X

n=p

(n+β)

γ−p+ 1) γ+n+ 1

a_n+

1 +j 2

^∞ X

n=p

(n+β)

γ −p+ 1) γ+n+ 1

b_n

by(3.2)

≤

1−j 2

A(p−β) +

1 +j 2

A(p−β)

=A(p−β).

Hence by Theorem2.2,W_j(z)∈σ^∗_p(β).

Theorem 3.3. Let f_k(z) =Az^−p+

∞

X

n=p

(−1)ⁿ⁻¹a_n,kzⁿ∈σ_p^∗(β), k = 1,2, . . . , m

then the arithmetic mean off_k(z)defined by

(3.3) F(z) = 1

m

X

k=1

f_k(z)

is also in the same class.

Proof. Sincef_k(z)∈σ^∗_p(β),then by (2.2) we have (3.4)

∞

X

n=p

(n+β)

γ−p+ 1 γ+n+ 1

a_n,k ≤A(p−β) (k = 1,2, . . . , m).

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After a simple calculation we obtain F(z) = 1

m

X

k=1

Az^−p+

∞

X

n=p

(−1)ⁿ⁻¹a_n,kzⁿ

!

=Az^−p+

∞

X

n=p

(−1)ⁿ⁻¹ 1 m

m

X

k=1

a_n,k

! zⁿ.

However,

∞

X

n=p

(n+β)

γ−p+ 1 γ+n+ 1

1 m

m

X

k=1

a_n,k

!by(3.4))

≤ 1 m

m

X

k=1

A(p−β) = A(p−β)

which in view of Theorem2.2yields the proof of Theorem3.3.

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References

[1] H. IRMAK AND S. OWA, Certain inequalities for multivalent starlike and meromorphically multivalent functions, Bulletin of the Institute of Mathematics, Academia Sinica, 31(1) (2001), 11–21.

[2] S.B. JOSHI AND H.M. SRIVASTAVA, A certain family of meromorphi- cally multivalent functions, Computers and Mathematics with Applications, 38 (1999), 201–211.

[3] Sh. NAJAFZADEH, A. TEHRANCHI AND S.R. KULKARNI, Application of differential operator onp-valent meromorphic functions, An. Univ. Oradea Fasc.

Mat., 12 (2005), 75–90.

[4] B.A. URALEGADDIANDM. D. GANIGI, Meromorphic starlike functions with alternating coefficients, Rendiconti di Mathematica, Serie VII, 11 (1991), 441–

446.