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volume 5, issue 1, article 11, 2004.

Received 27 October, 2003;

accepted 11 December, 2003.

Communicated by:H. Silverman

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Journal of Inequalities in Pure and Applied Mathematics

CONVOLUTION CONDITIONS FOR SPIRALLIKENESS AND CONVEX SPIRALLIKENESS OF CERTAIN MEROMORPHIC p-VALENT FUNCTIONS

V. RAVICHANDRAN, S. SIVAPRASAD KUMAR AND K.G. SUBRAMANIAN

Department of Computer Applications Sri Venkateswara College of Engineering Pennalur, Sriperumbudur 602 105, India.

EMail:vravi@svce.ac.in Department of Mathematics Sindhi College

123 P. H. Road, Numbal Chennai 600 077, India.

EMail:sivpk71@yahoo.com Department of Mathematics Madras Christian College

Tambaram, Chennai 600 059, India.

EMail:kgsmani@vsnl.net

c

2000Victoria University ISSN (electronic): 1443-5756 153-03

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Convolution Conditions for Spirallikeness and Convex Spirallikeness of Certain

Meromorphicp-valent Functions

V. Ravichandran,S. Sivaprasad Kumar and K.G. Subramanian

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Abstract

In the present investigation, the authors derive necessary and sufficient conditions for spirallikeness and convex spirallikeness of a suitably normalized meromorphicp-valent function in the punctured unit disk, using convolution. Also we give an application of our result to obtain a convolution condition for a class of meromorphic functions defined by a linear operator.

2000 Mathematics Subject Classification:Primary 30C45, Secondary 30C80.

Key words: Meromorphicp-valent functions, Analytic functions, Starlike functions, Convex functions, Spirallike functions, Convex Spirallike functions, Hadamard product (or Convolution), Subordination, Linear operator.

1. Introduction

LetΣ_p be the class of meromorphic functions (1.1) f(z) = 1

z^p +

∞

X

n=1−p

a_nzⁿ (p∈N:={1,2,3, . . .}), which are analytic andp-valent in the punctured unit disk

E^∗ :={z :z ∈C and 0<|z|<1}=E\ {0}, whereE:={z :z ∈C and |z|<1}.

For two functions f and g analytic in E, we say that the function f(z) is subordinate tog(z)inEand write

f ≺g or f(z)≺g(z) (z ∈E), if there exists a Schwarz functionw(z), analytic inEwith

w(0) = 0 and |w(z)|<1 (z ∈E), such that

(1.2) f(z) =g(w(z)) (z ∈E).

In particular, if the functiongis univalent inE, the above subordination is equiv- alent to

f(0) =g(0) and f(E)⊂g(E).

We define two subclasses of meromorphicp-valent functions in the following:

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Definition 1.1. Let|λ|< ^π₂ andp∈N. Letϕbe an analytic function in the unit diskE. We define the classesS_p^λ(ϕ)andC_p^λ(ϕ)by

S_p^λ(ϕ) :=

f ∈Σ_p : zf⁰(z)

f(z) ≺ −pe^−iλ[cosλ ϕ(z) +isinλ]

, (1.3)

C_p^λ(ϕ) :=

f ∈Σ_p : 1 + zf⁰⁰(z)

f⁰(z) ≺ −pe^−iλ[cosλ ϕ(z) +isinλ]

. (1.4)

Analogous to the well known Alexander equivalence [2], we have (1.5) f ∈C_p^λ(ϕ)⇔ −1

pzf⁰ ∈S_p^λ(ϕ) (p∈N).

Remark 1.1. For

(1.6) ϕ(z) = 1 +Az

1 +Bz (−1≤B < A≤1), we set

S_p^λ(ϕ) =: S_p^λ[A, B] and C_p^λ(ϕ) =:C_p^λ[A, B].

Forλ = 0, we write

S_p⁰(ϕ) =:S_p^∗(ϕ) and C_p⁰(ϕ) =: C_p(ϕ), S_p⁰[A, B] =: Sp[A, B] and C_p⁰[A, B] =:Cp[A, B].

For0 ≤ α <1, the classes S_p^λ[1−2α,−1]andC_p^λ[1−2α,−1]reduces to the classesS_p^λ(α)andC_p^λ(α)of meromorphicp-valentlyλ-spirallike functions

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of order αand meromorphic p-valentlyλ-convex spirallike functions of order αinE^∗respectively:

S_p^λ(α) :=

f ∈Σ_p : <

e^iλzf⁰(z) f(z)

<−pαcosλ (0≤α <1;|λ|< π 2)

,

C_p^λ(α) :=

f ∈Σ_p :<

e^iλ

1 + zf⁰⁰(z) f⁰(z)

<−pαcosλ (0≤α <1;|λ|< π 2)o

. The classesS_p⁰[1−2α,−1]andC_p⁰[1−2α,−1]reduces to the classesS_p^∗(α) andC_p(α)of meromorphicp-valently starlike functions of orderαand mero- morphicp-valently convex functions of orderαinE^∗ respectively.

For two functionsf(z)given by (1.1) and

(1.7) g(z) = 1

z^p +

∞

X

n=1−p

b_nzⁿ (p∈N),

the Hadamard product (or convolution) off andgis defined by (1.8) (f ∗g)(z) := 1

z^p +

∞

X

n=1−p

a_nb_nzⁿ=: (g∗f)(z).

Many important properties of certain subclasses of meromorphic p-valent functions were studied by several authors including Aouf and Srivastava [1],

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Joshi and Srivastava [3], Liu and Srivastava [4], Liu and Owa [5], Liu and Sri- vastava [6], Owa et al. [7] and Srivastava et al. [9]. Motivated by the works of Silverman et al. [8], Liu and Owa [5] have obtained the following Theorem1.1 withλ = 0for the classS_p^∗(α)and Liu and Srivastava [6] have obtained it for the classesS_p^λ(α)(with a slightly different definition of the class).

Theorem 1.1. [6, Theorem 1, p. 14] Letf(z)∈Σ_p. Then f ∈S_p^λ(α) (0≤α <1;|λ|< π

2;p∈N) if and only if

f(z)∗

1−Ωz z^p(1−z)²

6= 0 (z ∈E^∗), where

Ω := 1 +x+ 2p(1−α) cosλe^−iλ

2p(1−α) cosλe^−iλ , |x|= 1.

In the present investigation, we extend the Theorem 1.1 for the above defined classS_p^λ(ϕ). As a consequence, we obtain a convolution condition for the functions in the classC_p^λ(ϕ). Also we apply our result to obtain a convolution condition for a class of meromorphic functions defined by a linear operator.

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2. Convolution condition for the class S

_p^λ

(ϕ)

We begin with the following result for the general classS_p^λ(ϕ):

Theorem 2.1. Letϕbe analytic in Eand be defined on ∂E:= {z ∈ C :|z| = 1}. The functionf ∈Σ_p is in the classS_p^λ(ϕ)if and only if

(2.1) f(z)∗ 1−Ψz

z^p(1−z)² 6= 0 (z ∈E^∗) where

(2.2) Ψ := 1 +p

1−e^−iλ[cosλ ϕ(x) +isinλ]

p{1−e^−iλ[cosλ ϕ(x) +isinλ]} (|x|= 1;|λ|< π 2).

Proof. In view of (1.3),f(z)∈S_p^λ(ϕ)if and only if zf⁰(z)

f(z) 6=−pe^−iλ[cosλ ϕ(x) +isinλ] (z ∈E^∗;|x|= 1;|λ|< π 2) or

(2.3) zf⁰(z) +pe^−iλ[cosλ ϕ(x) +isinλ]f(z)6= 0 (z ∈E^∗;|x|= 1;|λ|< π 2).

Forf ∈Σ_pgiven by (1.1), we have

(2.4) f(z) =f(z)∗ 1

z^p(1−z) (z ∈E^∗)

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and

(2.5) zf⁰(z) =f(z)∗

1

z^p(1−z)² − p+ 1 z^p(1−z)

(z ∈E^∗).

By making use of the convolutions (2.5) and (2.4) in (2.3), we have f(z)∗

1

z^p(1−z)² − p+ 1

z^p(1−z) +pe^−iλ[cosλ ϕ(x) +isinλ]

z^p(1−z)

6= 0 (z ∈E^∗;|x|= 1;|λ|< π

2) or

f(z)∗

"

p

e^−iλ[cosλϕ(x) +isinλ]−1 z^p(1−z)²

+

1 +p

1−e^−iλ[cosλϕ(x) +isinλ] z z^p(1−z)²

# 6= 0 (z ∈E^∗;|x|= 1;|λ|< π

2), which yields the desired convolution condition (2.1) of Theorem2.1.

By takingλ = 0in the Theorem2.1, we obtain the following result for the classS_p^∗(ϕ).

Corollary 2.2. Let ϕ be analytic in E and be defined on ∂E. The function f ∈Σ_pis in the classf ∈S_p^∗(ϕ)if and only if

(2.6) f(z)∗ 1−Υz

z^p(1−z)² 6= 0 (z ∈E^∗)

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where

(2.7) Υ := 1 +p(1−ϕ(x))

p(1−ϕ(x)) (|x|= 1).

By takingϕ(z) = (1 +Az)/(1 +Bz),−1 ≤ B < A≤ 1in Theorem2.1, we obtain the following result for the classS_p^λ[A, B].

Corollary 2.3. The functionf ∈Σp is in the classS_p^λ[A, B]if and only if

(2.8) f(z)∗ 1−Υz

z^p(1−z)² 6= 0 (z ∈E^∗) where

(2.9) Υ := x−B+p(A−B) cosλe^−iλ

p(A−B) cosλe^−iλ (|x|= 1).

Remark 2.1. By takingA = 1−2α, B = −1in the above Corollary2.3, we obtain Theorem1.1of Liu and Srivastava [6].

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3. Convolution condition for the class C

_p^λ

(ϕ)

By making use of Theorem2.1, we obtain a convolution condition for functions in the classC_p^λ(ϕ)in the following:

Theorem 3.1. Letϕbe analytic inEand be defined on∂E. The functionf ∈Σ_p is in the classC_p^λ(ϕ)if and only if

(3.1) f(z)∗ p−[2 +p+ (p−1)Ψ]z+ (p+ 1)Ψz²

z^p(1−z)² 6= 0 (z ∈E^∗) whereΨis given by (2.2).

Proof. In view of the Alexander-type equivalence (1.5), we find from Theo- rem2.1thatf ∈C_p^λ(ϕ)if and only if

zf⁰(z)∗ 1−Ψz

z^p(1−z)² =f(z)∗z

1−Ψz z^p(1−z)²

⁰

6= 0 (z ∈E^∗) which readily yields the desired assertion (3.1) of Theorem3.1.

By taking λ = 0in the Theorem3.1, we obtain the following result of the classC_p(ϕ).

Corollary 3.2. Let ϕ be analytic in E and be defined on ∂E. The function f ∈Σ_pis in the classf ∈C_p(ϕ)if and only if

(3.2) f(z)∗ p−[2 +p+ (p−1)Υ]z+ (p+ 1)Υz²

z^p(1−z)² 6= 0 (z ∈E^∗) whereΥis given by (2.7).

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4. Convolution conditions for a class of function defined by linear operator

We begin this section by defining a classTn+p−1(ϕ). First of all for a function f(z)∈Σp, defineD^n+p−1f(z)by

D^n+p−1f(z) = f(z)∗

1 z^p(1−z)^n+p

= (z^n+2p−1f(z))^(n+p−1) (n+p−1)!z^p

= 1 z^p +

∞

X

m=1−p

(m+n+ 2p−1)!

(n+p−1)!(m+p)!a_mz^m.

By making use of the operatorD^n+p−1f(z), we define the classTn+p−1(ϕ)by Tn+p−1(ϕ) =

f(z)∈Σp : D^n+pf(z)

D^n+p−1f(z) ≺ϕ(z)

. When

ϕ(z) = 1 + (1−2γ)z 1−z where

γ = n+p(2−α)

n+p (0≤α <1),

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the class Tn+p−1(ϕ) reduces to the following class Tn+p−1(α) studied by Liu and Owa [5]:

Tn+p−1(α) =

f(z)∈Σ_p : <

D^n+pf(z)

D^n+p−1f(z)− n+ 2p n+p

<− pα n+p

. By making use of Corollary2.2, we prove the following result for the class Tn+p−1(ϕ):

Theorem 4.1. The functionf(z)∈Σp is in the classTn+p−1(ϕ)if and only if (4.1) f(z)∗ 1 + [(n+p)(1−Ω)−1]z

z^p(1−z)^n+p+1 6= 0 (z ∈E^∗;|x|= 1), whereΩis given by

(4.2) Ω := 1 + (n+p)(ϕ(x)−1)

(n+p)(ϕ(x)−1) (|x|= 1).

Proof. By making use of the familiar identity

z(D^n+p−1f(z))⁰ = (n+p)D^n+pf(z)−(n+ 2p)D^n+p−1f(z), we have

z(D^n+p−1f(z))⁰

D^n+p−1f(z) = (n+p) D^n+pf(z)

D^n+p−1f(z) −(n+ 2p),

and therefore, by using the definition of the classTn+p−1(ϕ), we see thatf(z)∈ T_n+p−1(ϕ)if and only if

D^n+p−1f(z)∈S_p^∗

n+ 2p

p − n+p p ϕ(z)

.

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Then, by applying Corollary2.2for the functionD^n+p−1f(z), we have

(4.3) D^n+p−1f(z)∗ 1−Ωz

z^p(1−z)² 6= 0, where

Ω =

1 +ph 1−n

n+2p

p − ^n+p_p ϕ(x)oi ph

1−n

n+2p

p − ^n+p_p ϕ(x)oi

= 1 + (n+p) (ϕ(x)−1)

(n+p) (ϕ(x)−1) , |x|= 1.

Since

D^n+p−1f(z) =f(z)∗

1 z^p(1−z)^n+p

, the condition (4.3) becomes

(4.4) f(z)∗

g(z)∗ 1−Ωz z^p(1−z)²

6= 0 where

g(z) = 1 z^p(1−z)^n+p.

By making use of the convolutions (2.5) and (2.4), it is fairly straight forward to show that

(4.5) g(z)∗ 1−Ωz

z^p(1−z)² = 1 + [(n+p)(1−Ω)−1]z z^p(1−z)^n+p+1 .

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By using (4.5) in (4.4), we see that the assertion in (4.1) follows and thus the proof of our Theorem4.1is completed.

By taking

ϕ(z) = 1 + (1−2γ)z 1−z where

γ = n+p(2−α)

n+p (0≤α <1)

in our Theorem4.1, we obtain the following result of Liu and Owa [5]:

Corollary 4.2. The functionf(z)∈Σ_p is in the classTn+p−1(α)if and only if f(z)∗ 1 + [(n+p)(1−Ω)−1]z

z^p(1−z)^n+p+1 6= 0 (z ∈E^∗;|x|= 1), whereΩis given by

Ω = 1 +x+ 2p(1−α)

2p(1−α) (|x|= 1).

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References

[1] M.K. AOUF AND H.M. SRIVASTAVA, A new criterion for meromorphi- callyp-valent convex functions of orderα, Math. Sci. Res. Hot. Line, 1(8) (1997), 7–12.

[2] P.L. DUREN, Univalent Functions, In Grundlehren der Mathematischen Wissenschaften, Bd., Volume 259, Springer-Verlag, New York, (1983).

[3] S.B. JOSHIANDH.M. SRIVASTAVA, A certain family of meromorphically multivalent functions, Computers Math. Appl. 38(3/4) (1999), 201–211.

[4] J.-L. LIU ANDH.M. SRIVASTAVA, A linear operator and associated fam- ilies of meromorphically multivalent functions, J. Math. Anal. Appl., 259 (2001), 566–581.

[5] J.-L. LIU AND S. OWA, On a class of meromorphicp-valent functions in- volving certain linear operators, Internat. J. Math. Math. Sci., 32 (2002), 271–180.

[6] J.-L. LIUANDH.M. SRIVASTAVA, Some convolution conditions for star- likeness and convexity of meromorphically multivalent functions, Applied Math. Letters, 16 (2003), 13–16.

[7] S. OWA, H.E. DARWISH AND M.K. AOUF, Meromorphic multivalent functions with positive and fixed second coefficients, Math. Japon., 46 (1997), 231–236.

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[8] H. SILVERMAN, E.M. SILVIAANDD. TELAGE, Convolution conditions for convexity, starlikeness and spiral-likeness, Math. Zeitschr., 162 (1978), 125–130.

[9] H.M. SRIVASTAVA, H.M. HOSSENANDM.K. AOUF, A unified presenta- tion of some classes of meromorphically multivalent functions, Computers Math. Appl. 38 (11/12) (1999), 63–70.