COEFFICIENT INEQUALITIES FOR CERTAIN MEROMORPHICALLY p-VALENT FUNCTIONS

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Meromorphicallyp-valent Functions A. Ebadian and Sh. Najafzadeh

vol. 9, iss. 4, art. 114, 2008

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COEFFICIENT INEQUALITIES FOR CERTAIN MEROMORPHICALLY p-VALENT FUNCTIONS

A. EBADIAN Sh. NAJAFZADEH

Department of Mathematics Department of Mathematics

Faculty of Science, Urmia University Maragheh University

Urmia, Iran Maragheh, Iran

EMail:aebadian@yahoo.com EMail:najafzadeh1234@yahoo.ie

Received: 12 November, 2007

Accepted: 21 August, 2008

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

Key words: p-Valent, Meromorphic, Starlike, Convex, Close-to-convex.

Abstract: The aim of this paper is to prove some inequalities forp-valent meromorphic

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1. Introduction

LetΣ_pdenote the class of functionsf(z)of the form

(1.1) f(z) = z^−p +

∞

X

k=p

a_kz^k

which are analytic meromorphic multivalent in the punctured unit disk

∆^∗ ={z : 0<|z|<1}.

We say that f(z) is p-valently starlike of order γ(0 ≤ γ < p) if and only if for z ∈∆^∗

(1.2) −Re

zf⁰(z) f(z)

> γ.

Alsof(z)isp-valently convex of orderγ(0≤γ < p)if and only if

(1.3) −Re

1 + zf⁰⁰(z) f⁰(z)

> γ, z ∈∆^∗.

Definition 1.1. A functionf(z)∈Σ_pis said to be in the subclassX_p^∗(j)if it satisfies the inequality

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is thej-th differential off(z)and a functionf(z)∈Σ_p is said to be in the subclass Y_p^∗(j)if it satisfies the inequality

(1.6)

−z[f^(j)(z)]⁰

f^(j)(z) −(p+j)

< p+j.

To establish our main results we need the following lemma due to Jack [5].

Lemma 1.2. Letw(z)be analytic in ∆ = {z : |z| < 1}withw(0) = 0. If|w(z)|

attains its maximum value on the circle|z|=r <1at a pointz0then z₀w⁰(z₀) = cw(z₀),

wherecis a real number andc≥1.

Some different inequalities on p-valent holomorphic and p-valent meromorphic functions by using operators were studied in [1], [2], [3] and [4].

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2. Main Results

Theorem 2.1. Iff(z)∈Σ_p satisfies the inequality

(2.1) Re

z[f^(j)(z)]⁰

f^(j)(z) +p+j

>1− 1 2p, thenf(z)∈X_p^∗(j).

Proof. Lettingf(z)∈Σ_p, we define the functionw(z)by

(2.2) (p−1)!

(−1)^j(p+j−1)!

f^(j)(z)

z^−p−j = 1−w(z), (z ∈∆^∗).

It is easy to verify thatw(0) = 0.

From (2.2) we obtain

f^(j)(z) = −(−1)^j(p+j−1)!

(p−1)! z^−p−j +(p+j −1)!

(p−1)! z^−p−jw(z) or

[f^(j)(z)]⁰ = (−1)^j(p+j)z^−p−j−1(p+j −1)!

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Now, suppose that there exists a pointz₀ ∈∆^∗such that

|z|≤|zmax₀||w(z)|=|w(z₀)|= 1.

Then, by letting w(z₀) = e^iθ(w(z₀) 6= 1)and using Jack’s lemma in the equation (2.3), we have

−Re

z[f^(j)(z)]⁰

f^(j)(z) +p+j

= Re

z₀w⁰(z₀) 1−w(z₀)

= Re

cw(z₀) 1−w(z₀)

=cRe

e^iθ 1−e^iθ

= −c 2 < −1

2

which contradicts the hypothesis (2.1). Hence we conclude that for allz,|w(z)|<1 and from (2.2) we have

(p−1)!f^(j)(z)

(−1)^j(p+j−1)!z^−p−j −1

=|w(z)|<1

and this gives the result.

Theorem 2.2. Iff(z)∈Σ_p satisfies the inequality

(2.4) Re

z[f^(j)(z)]⁰ f^(j)(z) −

1 + z[f^(j)(z)]⁰⁰ [f^(j)(z)]⁰

> 2p+ 1 2(p+ 1), thenf(z)∈Y_p^∗(j).

Proof. Letf(z)∈Σ_p. We consider the functionw(z)as follows:

(2.5) −z[f^(j)(z)]⁰

f^(j)(z) = (p+j)(1−w(z)).

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It is easy to see thatw(0) = 0. Furthermore, by differentiating both sides of (2.5) we get

−

1 + z[f^(j)(z)]⁰⁰ [f^(j)(z)]⁰

= (p+j)(1−w(z)) + zw⁰(z) 1−w(z). Now suppose that there exists a pointz₀ ∈∆^∗such that max

|z|≤|z0||w(z)|=|w(z₀)|= 1.

Then, by lettingw(z0) = e^iθ and using Jack’s lemma we have Re

z[f^(j)(z)]⁰ f^(j)(z) −

1 + z[f^(j)(z)]⁰⁰ f^(j)(z)]⁰

= Re

z₀w⁰(z₀) 1−w(z₀)

=c Re

e^iθ 1−e^iθ

=−c 2 <−1

2, which contradicts the condition (2.4). So we conclude that|w(z)|<1for allz ∈∆^∗. Hence from (2.5) we obtain

−z[f^(j)(z)]⁰

[f^(j)(z)] −(p+j)

< p+j.

This completes the proof.

By takingj = 0in Theorems2.1and2.2, we obtain the following corollaries.

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Corollary 2.4. Iff(z)∈Σ_p satisfies the inequality

−Re zf⁰

f −

1 + zf⁰⁰ f⁰

> 2p+ 1 2(p+ 1), then

−^zf_f⁰ −p

< por equivalentlyf(z)is meromorphicallyp-valent starlike with respect to the origin.

By takingj = 1in Theorems2.1and2.2, we obtain the following corollaries.

−Re zf⁰⁰

f⁰ +p+ 1

>1− 1 2p, then

−_z^f−p−1⁰^(z) −p

< p or equivalently f(z) is meromorphically p-valent close-to- convex with respect to the origin.

−Re zf⁰⁰

f⁰ −

1 + zf⁰⁰⁰ f⁰⁰

> 2p+ 1 2(p+ 1),

then

−zf⁰⁰

f⁰ −(p+ 1)

< p+ 1, or equivalentlyf(z)is meromorphically multivalent convex.

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References

[1] H. IRMAKANDO.F. CETIN, Some theorems involving inequalities onp-valent functions, Turk. J. Math., 23 (1999), 453–459.

[2] H. IRMAK, N.E. CHO, O.F. CETINANDR.K. RAINA, Certain inequalities in- volving meromorphically functions, Hacet. Bull. Nat. Sci. Eng. Ser. B, 30 (2001), 39–43.

[3] H. IRMAK ANDR.K. RAINA, On certain classes of functions associated with multivalently analytic and multivalently meromorphic functions, Soochow J.

Math., 32(3) (2006), 413–419.

[4] H. IRMAKANDR.K. RAINA, New classes of non-normalized meromorphically multivalent functions, Sarajevo J. Math., 3(16)(2) (2007), 157–162.

[5] I.S. JACK, Functions starlike and convex of order α, J. London Math. Soc., 3 (1971), 469–474.