Subclasses of Meromorphic Functions Associated with Convolution

Journal of Inequalities and Applications Volume 2009, Article ID 190291,9pages doi:10.1155/2009/190291

Research Article

Subclasses of Meromorphic Functions Associated

with Convolution

Maisarah Haji Mohd,

_{Rosihan M. Ali,}

_{Lee See Keong,}

and V. Ravichandran

1_{School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia} 2_{Department of Mathematics, University of Delhi, Delhi 110 007, India}

Correspondence should be addressed to Lee See Keong,[email protected]

Received 12 December 2008; Accepted 11 March 2009

Recommended by Narendra Kumar Govil

Several subclasses of meromorphic functions in the unit disk are introduced by means of convolution with a given fixed meromorphic function. Subjecting each convoluted-derived function in the class to be subordinated to a given normalized convex function with positive real part, these subclasses extend the classical subclasses of meromorphic starlikeness, convexity, close-to-convexity, and quasi-convexity. Class relations, as well as inclusion and convolution properties of these subclasses, are investigated.

Copyrightq2009 Maisarah Haji Mohd et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetHbe the set of all analytic functions defined in the unit diskU{z:|z|<1}. We denote byAthe class of normalized analytic functionsfz z_n∞₂anzndefined inU. For two functionsfandganalytic inU, the functionfis subordinate tog, written as

fz≺gz, 1.1

if there exists a Schwarz functionw, analytic inUwith w0 0 and|wz| < 1 such that

fz gwz.In particular, if the functiongis univalent inU, thenfz≺gzis equivalent tof0 g0andfU⊂gU.

of these classes and introduced the classes

S∗_{h f∈ A} zfz

fz ≺hz

,

Ch f∈ A1zf

_z

f_z ≺hz

,

1.2

wherehis an analytic function with positive real part,h0 1, andhmaps the unit diskU onto a region starlike with respect to 1.

The convolution or the Hadamard product of two analytic functions fz z

_∞

n2anznandgz z

_∞

n2 bnznis given by

f∗gz ∞

n1

anbnzn. 1.3

In term of convolution, a functionf is starlike iff ∗z/1−zis starlike, and convex if

f∗z/1−z2_{is starlike. These ideas led to the study of the class of all functionsfsuch that}

f∗gis starlike for some fixed functionginA.In this direction, Shanmugam2introduced and investigated various subclasses of analytic functions by using the convex hull method3– 5and the method of differential subordination. Ravichandran6introduced certain classes of analytic functions with respect ton-ply symmetric points, conjugate points, and symmetric conjugate points, and also discussed their convolution properties. Some other related studies were also made in7–9, and more recently by Shamani et al.10.

LetMdenote the class of meromorphic functionsfof the form

fz 1

z

∞

n0

anzn, 1.4

which are analytic and univalent in the punctured unit disk U∗ {z : 0 < |z| < 1}. For 0 ≤ α < 1, we recall that the classes of meromorphic starlike, meromorphic convex, meromorphic close-to-convex, meromorphic γ-convex Mocanu sense, and meromorphic quasi-convex functions of orderα, denoted byMs,Mk,Mc,Mk_γ,and Mq,respectively, are defined by

Ms_{f∈ M −R}zfz

fz > α

,

Mk_{f∈ M −R1}zfz

f_z

> α

,

Mc_{f∈ M −R}zfz

gz > α, g∈ Ms

,

Mk γ

f∈ M −R 1−γzfz

fz γ

1zf

_z

f_z

> α

,

Mq_{f∈ M −R}

zf_z

g_z > α, g∈ Mk

.

The convolution of two meromorphic functionsfandg, wherefis given by1.4andgz 1/z∞_n0bnzn, is given by

f∗gz 1

z

∞

n0

anbnzn. 1.6

Motivated by the investigation of Shanmugam2, Ravichandran6, and Ali et al. 7,11, several subclasses of meromorphic functions defined by means of convolution with a given fixed meromorphic function are introduced in Section 2. These new subclasses extend the classical classes of meromorphic starlike, convex, close-to-convex, γ-convex, and quasi-convex functions given in 1.5. Section 3is devoted to the investigation of the class relations as well as inclusion and convolution properties of these newly defined classes.

We will need the following definition and results to prove our main results.

LetS∗αdenote the class of starlike functions of orderα. The classRαof prestarlike functions of orderαis defined by

Rα

f ∈ Af∗ z

1−z2−2α ∈S∗α

1.7

forα <1, and

R1

f ∈ ARf_zz > 1

2

. 1.8

Theorem 1.1see12, Theorem 2.4. Letα≤1,f ∈ Rα, andg ∈S∗α. Then, for any analytic

functionH∈ HU,

f∗Hg

f∗g U⊂co

HU, 1.9

wherecoHUdenotes the closed convex hull ofHU.

Theorem 1.2see13. Lethbe convex inUandβ, γ ∈CwithRβhz γ>0. Ifpis analytic inUwithp0 h0, then

pz zpz

βpz γ ≺hz impliespz≺hz. 1.10

2. Definitions

Definition 2.1. The classMs_ghconsists of functionsf∈ Msatisfyingg∗fz/0 inU∗and the subordination

−z_gg_∗∗_ff_zz ≺hz. 2.1

Remark 2.2. Ifgz 1/z1/1−z, thenMs_ghcoincides withMsh, where

Ms_{h f∈ M −}zfz

fz ≺hz

. 2.2

Definition 2.3. The classMk_ghconsists of functionsf∈ Msatisfyingg∗fz/0 inU∗and the subordination

−

1zg∗f

_z

g∗f_z

≺hz. 2.3

Definition 2.4. The classMc_ghconsists of functionsf ∈ Msuch thatg∗ψz/0 inU∗for someψ∈ Ms_ghand satisfying the subordination

−z

g∗fz

g∗ψz ≺hz. 2.4

Definition 2.5. Forγreal, the classMk_g,γhconsists of functionsf∈ Msatisfyingg∗fz/0, g∗f_{z/0 inU}∗_{and the subordination}

−

γ

1z

g∗fz

g∗fz

1−γ

_zg∗fz

g∗fz

≺hz. 2.5

Definition 2.6. The classMqghconsists of functionsf∈ Msuch thatg∗ϕz/0 inU∗for someϕ∈ Mk_ghand satisfying the subordination

−zg∗fz

g∗ϕz ≺hz. 2.6

3. Main Results

This section is devoted to the investigation of class relations as well as inclusion and convolution properties of the new subclasses given inSection 2.

Theorem 3.1. Lethbe a convex univalent function satisfyingRhz<2−α,0≤α <1, andg ∈ M withz2_{g∈Rα. Iff∈ M}s_{h, thenf∈ M}s

Proof. Define the functionFby

Fz −zfz

fz . 3.1

Forf∈ Msh, it follows that

−R

_zfz fz

<2−α, 3.2

and therefore,

R

_zz2_fz

z2_fz

> α. 3.3

Hencez2_f∈S∗_{α. A computation shows that}

−z

g∗fz g∗fz

g∗ −zf_z

g∗fz

g∗fFz g∗fz

z2_g∗z2_fFz

z2_g∗z2_fz . 3.4

Theorem 1.1yields

−z

g∗fz g∗fz

z2_g∗z2_fFz

z2_g∗z2_fz ∈co

FU, 3.5

and becauseFz≺hz, it follows that

−z

g∗fz

g∗fz ≺hz. 3.6

Theorem 3.2. The functionf ∈ Mk_ghif and only if−zf∈ Ms_gh.

Proof. The results follow from the equivalence relations

−

1z

g∗fz

g∗fz

≺hz⇐⇒ −

zg∗fz

g∗fz ≺hz⇐⇒ −

zg∗ −zf_z

g∗ −zf_z ≺hz.

3.7

Theorem 3.3. Lethbe a convex univalent function satisfyingRhz<2−α, 0≤α <1,andφ∈ M withz2_{φ∈Rα. Iff ∈ M}s

Proof. Sincef ∈ Ms_gh, it follows that

−R

_zg∗fz

g∗fz

<2−α, 3.8

and thus

R

_zz2_g∗fz

z2_g∗fz

> α. 3.9

Let

Pz −zg∗fz

g∗fz . 3.10

A similar computation as in the proof ofTheorem 3.1yields

−z_φφ_∗∗_gg_∗∗_ff_zz z2φ_z2z_φz∗z_∗2_zg₂∗_gf_∗fzP_zz. 3.11

Inequality3.9shows thatz2g∗f∈S∗α. ThereforeTheorem 1.1yields

−z_φφ_∗∗_gg_∗∗_ff_zz ≺hz, 3.12

henceφ∗f ∈ Ms_gh.

Corollary 3.4. Ms

gh⊂ Msφ∗ghunder the conditions ofTheorem 3.3.

Proof. The proof follows from3.12.

In particular, whengz 1/z1/1−z, the following corollary is obtained.

Corollary 3.5. Lethandφsatisfy the conditions ofTheorem 3.3. Iff∈ Msh, thenf∈ Ms_φh.

Theorem 3.6. Lethandφsatisfy the conditions ofTheorem 3.3. Iff∈ Mk_gh, thenφ∗f∈ Mk_gh. EquivalentlyMkgh⊂ Mkφ∗gh.

Proof. Iff ∈ Mkgh, it follows fromTheorem 3.2that−zf∈ Msgh.Theorem 3.3shows that

φ∗−zf _{−zφ∗f∈ M}s

gh.Henceφ∗f∈ Mkgh.

Theorem 3.7. Under the conditions ofTheorem 3.3, iff ∈ Mc_ghwith respect toψ ∈ Ms_gh, then φ∗f ∈ Mc

ghwith respect toφ∗ψ ∈ Msgh.

Let the functionGbe defined by

Gz −z_gg_∗∗_ψf_zz. 3.13

A similar computation as in the proof ofTheorem 3.1yields

−z_φφ_∗∗_gg_∗∗_ψf_zz z2φz∗z2g∗ψzGz

z2_φz∗z2_g∗ψz . 3.14

Sincez2_φ∈Rαandz2_g∗ψ∈S∗_{α, it follows fromTheorem 1.1that}

−z_φφ_∗∗_gg_∗∗_ψf_zz ≺hz. 3.15

Thusφ∗f ∈ Mc_ghwith respect toφ∗ψ.

Corollary 3.8. Mc

gh⊂ Mcφ∗ghunder the assumptions ofTheorem 3.3.

Proof. The subordination3.15shows thatf∈ Mc_φ∗gh.

Theorem 3.9. LetRγhz<0. Then

iMk

g,γh⊂ Msgh, iiMk

g,γh⊂ Mkg,βhforγ < β≤0.

Proof. Define the functionPby

Pz −z_gg_∗∗_ff_zz 3.16

and the functionJgγ;fby

Jgγ;fz −

γ

1zg∗f

_z

g∗fz

1−γ

_zg∗fz

g∗fz

. 3.17

Forf∈ Mk_g,γh, it follows thatJgγ;fz≺hz.Note also that

Jgγ;fz Pz− γzP

_z

Pz . 3.18

iSinceRγhz<0 and

Theorem 1.2yieldsPz≺hz.Hencef∈ Ms_gh. iiObserve that

Jgβ;fz −

β

1zg∗f

_z

g∗f_z

1−β

_zg∗fz

g∗fz

1−β

γ

Pz β

γJgγ;fz.

3.20

FurthermoreJgγ;fz ≺ hzand Pz ≺ hz fromi. Since 0 < β/γ < 1 andhUis convex, we deduce thatJgβ;fz∈hU. Therefore,Jgβ;fz≺hz.

Corollary 3.10. The classMk_ghis a subset of the classMqgh.

Proof. The proof follows from the definition of the classes by takingfϕ.

Theorem 3.11. The functionf∈ Mqghif and only if−zf∈ Mcgh.

Proof. Iff∈ Mqgh, then there existsϕ∈ Mkghsuch that

−zg∗fz

g∗ϕ_z ≺hz. 3.21

Also,

−zg∗ −zf_z

g∗ −zϕ_z

−zg∗fz

g∗ϕz ≺hz. 3.22

Sinceϕ∈ Mk_gh, byTheorem 3.2,−zϕ∈ Ms_gh. Hence−zf∈ Mc_gh. Conversely, if−zf∈ Mc_gh, then

−z

g∗ −zf_z

g∗ϕ1

z ≺hz 3.23

for someϕ1∈ Msgh. Letϕ∈ Mkghbe such that−zϕϕ1∈ Msgh. The proof is completed by observing that

−zg∗fz g∗ϕ_z −

zg∗ −zf_z

g∗ −zϕ_z ≺hz. 3.24

Corollary 3.12. Lethandφsatisfy the conditions ofTheorem 3.3. Iff∈ Mqgh, thenφ∗f∈ Mqgh.

Proof. Iff ∈ Mqgh,Theorem 3.11gives−zf ∈ Mcgh.Theorem 3.7next givesφ∗−zf

−zφ∗f∈ Mc

Corollary 3.13. Mqgh⊂ Mφq∗ghunder the conditions ofTheorem 3.3.

Proof. Iff∈ Mqgh, it follows fromCorollary 3.12thatφ∗f∈ Mqgh.The subordination

−zφ∗g∗fz

φ∗g∗ϕz ≺hz 3.25

givesf ∈ Mq_φ∗gh. ThereforeMgqh⊂ Mq_φ∗gh.

Open Problem

An analytic convex function in the unit disk is necessarily starlike. For the meromorphic case, is it true thatMk_gh⊂ Ms_gh?

Acknowledgment

The work presented here was supported in part by the USM’s Reserach University grant and the FRGS grant

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