Integral operators on new families of meromorphic functions of complex order

R E S E A R C H

Open Access

Integral operators on new families of

meromorphic functions of complex order

Aabed Mohammed and Maslina Darus

*

* Correspondence: maslina@ukm. my

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia

Abstract

In this article, we define and investigate new families of certain subclasses of meromorphic functions of complex order. Considering the new subclasses, several properties for certain integral operators are derived.

2010 Mathematics Subject Classification: 30C45.

Keywords:Analytic function, meromorphic function, integral operator

1 Introduction

LetΣ denote the class of functions of the form:

f(z) = 1 z +

∞

n=0

anzn, (1:1)

which are analytic in the punctured open unit disk

U∗_{={z∈C: 0<|z|<1}=U\{0}, (1:2)}

whereUis the open unit diskU={z∈C:|z|<1}..

We say that a function fÎ Σ is meromorphic starlike of order δ(0 ≤ δ< 1), and belongs to the classΣ*(δ), if it satisfies the inequality:

−

zf(z) f(z)

> δ. (1:3)

A functionfÎΣ is a meromorphic convex function of orderδ(0≤δ< 1), iff satis-fies the following inequality:

−

1 + zf _(z)

f(z)

> δ, (1:4)

and we denote this class byΣk(δ).

ForfÎ Σ, Wang et al. [1] and Nehari and Netanyahu [2] introduced and studied the subclassΣN(b) of Σconsisting of functionsf(z) satisfying

−

zf(z) f(z) + 1

< β (β >1, z∈U). (1:5)

Let Adenote the class of functionsfof the form f(z) =z+∞

n=2

anzn, which are

analy-tic in the open unit disk_U.

Analogous to several subclasses [3-10] of analytic functions ofA, we define the fol-lowing subclasses ofΣ.

Definition 1.1 Let a functionfÎ Σbe analytic in_U∗. Thenfis in the class_b(δ)if, and only if,fsatisfies

1−1 b

zf(z) f(z) + 1

> δ, (1:6)

whereb∈C\{0}, 0 ≤δ< 1.

Definition 1.2Let a function fÎΣbe analytic in_U∗. Thenfis in the classΣKb(δ) if,

and only if,fsatisfies

1−1 b

zf(z) f(z) + 2

> δ, (1:7)

wherebÎ C\{0}, 0 ≤δ< 1. We note thatfÎΣKb(δ) if, and only if,−zf∈b(δ).

Furthermore, the classes

1(δ)≡(δ), K1(δ)≡k(δ)

are the classes of meromorphic starlike functions of order δand meromorphic

con-vex functions of orderδin_U∗, respectively. Moreover, the classes

1(0)≡(0), K1(0)≡k(0)

are familiar classes of starlike and convex functions inU∗, respectively.

Definition 1.3 Let a function f Î Σ be analytic in _U∗. Then f is in the class

_{U(α,δ,b)if, and only if,fsatisfies}

1−1 b

zf(z) f(z) + 1

> α1

b

zf(z) f(z) + 1

+δ, (1:8)

wherea≥0,δÎ[-1,1),a+δ≥0,b∈C\{0}.

Definition 1.4 Let a function f Î Σ be analytic in U∗. Then f is in the class

KU(α,δ,b)if, and only if,fsatisfies

1−1 b

zf(z) f(z) + 2

> α1

b

zf(z) f(z) + 2

+δ, (1:9)

wherea≥0,δÎ[-1,1),a+δ≥0,b∈C\{0}.

We note that f ∈KU(α,δ,b)if, and only if,−zf∈U(α,δ,b).

Fora= 0, we have

_U(0,δ,b)≡

b(δ), KU(0,δ,b)≡Kb(δ).

Definition 1.5 Let a function f Î Σ be analytic in _U∗. Then f is in the class

_{UH(α,b)if, and only if,fsatisfies}

1−1

b

zf(z) f(z) + 1

−2α(√2−1)< _√

2

1−1 b

zf(z) f(z) + 1

+ 2α(√2−1), (1:10)

Definition 1.6 Let a function f Î Σ be analytic in U∗. Then f is in the class

KUH(α,b)if, and only if,fsatisfies

1−1 b

zf(z) f(z) + 2

−2α(√2−1)< _√

2

1−1 b

zf(z) f(z) + 2

+ 2α(√2−1). (1:11)

wherea> 0,b∈C\{0}.

We note that f ∈KUH(α,b)if, and only if,−zf∈UH(α,b).

Let us also introduce the following families of new subclassesF1(δ,b),F2(α,δ,b),

andF3(α,b)as follows.

Definition 1.7 Let a functionfÎΣbe analytic in_U∗. Thenfis in the classF1(δ,b),

if, and only if,fsatisfies

1−1

b

zzf(z) + 3f(z) zf(z) + 2f(z) + 1

> δ. (1:12)

whereb∈C\{0}, 0 ≤δ< 1.

We note that f ∈F1(δ,b)if, and only if,zf(z) + 2f(z)∈b(δ).

Definition 1.8 Let a function f Î Σ be analytic in U∗. Then f is in the class

F2(α,δ,b)if, and only if,fsatisfies

1−1

b

zzf(z) + 3f(z) zf(z) + 2f(z) + 1

> α1

b

zzf(z) + 3f(z) zf(z) + 2f(z) + 1

+δ,(1:13)

wherea≥0,δÎ[-1,1),a+δ≥0,b∈C\{0}.

We note that f ∈F2(α,δ,b)if, and only if,zf(z) + 2f(z)∈U(α,δ,b).

Definition 1.9 Let a functionfÎΣ be analytic in_U∗. Then fis in the classF3(α,b)

if, and only if,fsatisfies

1−

1 b

zzf(z) + 3f(z) zf(z) + 2f(z) + 1

−2α(√2−1)

<

√

2

1−1 b

zzf(z) + 3f(z) zf(z) + 2f(z) + 1

+ 2α(√2−1),

(1:14)

wherea> 0,b∈C\{0}.

We note that f ∈F3(α,b)if, and only if, zf(z) + 2f(z)∈UH(α,b).

Recently, many authors introduced and studied various integral operators of analytic and univalent functions in the open unit disk_U[11-21].

Most recently, Mohammed and Darus [22,23] introduced the following two general integral operators of meromorphic functionsΣ:

Hn(z) =

1 z2

z

0

uf1(u) γ1

. . .ufn(u) γn

du, (1:15)

and

Hγ1,...,γn(z) =

1 z2

z

0

−u2f₁(u)γ1

. . .−u2f_n(u)γn

du. (1:16)

Corollary 1.1 If fÎΣ satisfies the following inequality

zf_f((_zz₎)− 2zf(z)

f(z)

< (1−δ) (3−δ)

2−δ , 0≤δ <1, (1:17)

then fÎ Σ*(δ).

Corollary 1.2If fÎΣ satisfies the following inequality

zf_f((_zz₎)− zf(z)

f(z) + 1

< 1

2, (1:18)

then f ÎΣ*.

Corollary 1.3If fÎΣ satisfies the following inequality

zf(z) f(z)

2zf(z) f(z) −

zf(z) f(z) −1

>−1

2, (1:19)

then f ÎΣ*.

In this article, we derive several properties for the integral operators Hn(z)and Hγ1,...,γn(z)of the subclasses given by (1.5) and Definitions 1.1 to 1.6.

2 Some properties for Hn(z)

In this section, we investigate some properties for the integral operatorHn(z)defined

by (1.15) of the subclasses given by (1.5), Definitions 1.1, 1.3, and 1.5. Theorem 2.1For iÎ {1,...,n},letgi> 0,fiÎΣand

zf_i(z) f_i(z) −2

zf_i(z) fi(z)

<

(1−δ) (3−δ)

2−δ , (0≤δ <1). (2:1)

If

n

i=1 γi>

2

1−δ, (2:2)

thenHn(z)∈N(β),whereb> 1.

Proof On successive differentiation ofHn(z), which is defined in (1.5), we get

z2H_n(z) + 2zHn(z) =

zf1(z) γ1

. . .zfn(z) γn

, (2:3)

and

z2_H

n(z) + 4zH

n(z) + 2Hn(z) = n i=1γ

i

zf_i(z) +fi(z)

zfi(z)

zf1(z) γ1

. . .zfn(z) γn

. (2:4)

Then from (2.3) and (2.4), we obtain

z2_H

n(z) + 4zH

n(z) + 2Hn(z)

z2_H

n(z) + 2zHn(z)

=

n

i=1 γi

f_i(z) fi(z)

+1 z

. (2:5)

By multiplying (2.5) with zyield

z2_H

n(z) + 4zH

n(z) + 2Hn(z)

zH_n(z) + 2Hn(z)

=

n

i=1 γi

zf_i(z) fi(z)

+ 1

This is equivalent to

z2_H

n(z) + 3zH n(z)

zH_n(z) + 2Hn(z)

+ 1 =

n

i=1 γi

zf_i(z) fi(z)

+ 1

. (2:7)

Therefore, we have

−

zH_n(z) H

n(z)

+ 1

−2

1 +2Hn(z) zHn(z)

=−

n

i=1 γi

zf_i(z) fi(z)

+ 1

+ 1. (2:8)

Then, we easily get

−

zH_n(z) H

n(z)

+ 1

=

2Hn(z)

zH_n(z) −

n i=1

γi

zf_i(z) fi(z)

+ 1

+ 1

+

n i=1γ

i

−zf i(z)

fi(z)

+ 3−

n i=1γ

i.

(2:9)

Taking real parts of both sides of (2.9), we obtain

−

zH_n(z) H

n(z)

+ 1

= 2Hn(z) zH_n(z) −

n i=1

γi

zf_i(z) fi(z)

+ 1

+ 1

+

n i=1γ

i

−zf i(z)

fi(z)

+ 3−

n i=1γ

i

≤

2Hn(z)

zH_n(z) −

n i=1

γi

zf_i(z) fi(z)

+ 1

+ 1

+

n i=1

γi

−zf i(z)

fi(z)

+ 3−

n i=1

γi.

(2:10)

Let

β =

2Hn(z)

zH_n(z) −

n i=1γ

i

zf_i(z) fi(z)

+ 1

+ 1

+

n i=1

γi

−zf i(z)

fi(z)

+ 3−

n i=1

γi.

(2:11)

Since

2Hn(z)

zH_n(z) −

n i=1γ

i

zf_i(z) fi(z)

+ 1

+ 1

>0, applying Corollary 1.1, we have

β > δ n

i=1

γi+ 3− n

i=1

γi= 3−(1−δ) n

i=1

γi. (2:12)

Then, by the hypothesis (2.2), we have b> 1. Therefore,Hn(z)∈N(β), whereb>

1. This completes the proof. □

Corollary 2.2For iÎ{1,...,n},letgi> 0,fiÎΣand

zf_i(z) f_i(z) −2

zf_i(z) fi(z)

<

3

2. (2:13)

If

n

i=1

γi>2, (2:14)

thenHn(z)∈N(β),whereb> 1.

Making use of (2.11), Corollary 1.2 and Corollary 1.3, one can prove the following results.

Theorem 2.3 For iÎ{1,...,n},letgi> 0,fiÎΣand

zf_i(z) f_i(z) −

zf_i(z) fi(z)

+ 1

<

1

2. (2:15)

If

n

i=1

γi>2, (2:16)

thenHn(z)∈N(β),whereb> 1.

Theorem 2.4For iÎ {1,...,n},letgi> 0,fiÎΣand

zf

i(z)

fi(z)

2zf_i(z) fi(z) −

zf_i(z) f_i(z) −1

>−1

2. (2:17)

If

n

i=1

γi>2, (2:18)

thenHn(z)∈N(β),whereb> 1.

Theorem 2.5For iÎ {1,...,n},letgi> 0and fi∈_b(δi)(0≤δ< 1andb∈C\{0}).

If

0<

n

i=1

γi(1−δi)≤1, (2:19)

thenHn(z)is in the classF1(μ,b), μ= 1− n i=1

γi(1−δi).

Proof From (2.7), we have

zzH_n(z) + 3H_n(z) zH_n(z) + 2Hn(z)

+ 1 =

n

i=1 γi

zf_i(z) fi(z)

+ 1

. (2:20)

Equivalently, (2.20) can be written as

1−1 b

zzH_n(z) + 3H_n(z) zH_n(z) + 2Hn(z) + 1

= n

i=1

γi 1− 1 b

zf_i(z) fi(z) + 1

+ 1− n

i=1

Taking the real part of both terms of the last expression, we have

1−1 b

zzH_n(z) + 3H_n(z) zH_n(z) + 2Hn(z)

+ 1

= n

i=1

γi 1−1 b

zf_i(z)

fi(z) + 1

+1−

n

i=1

γi. (2:22)

Since fi∈_b(δi), hence

1−1

b

zzH_n(z) + 3H_n(z) zH_n(z) + 2Hn(z)

+ 1

>

n

i=1

γiδi+ 1− n

i=1

γi. (2:23)

so that

1−1

b

zzH_n(z) + 3H_n(z) zHn(z) + 2Hn(z)

+ 1

>1−

n

i=1

γi(1−δi). (2:24)

ThenHn(z)∈F1(μ,b), μ= 1− n i=1

γi(1−δi).

Now, adopting the techniques used by Breaz et al. [11] and Bulut [21], we prove the following two theorems.

Theorem 2.6 For iÎ{1,...,n},letgi> 0and fi∈U(α,δ,b)(a≥0,δÎ [-1,1),a+

δ≥0andb∈C\{0}).If

n

i=1

γi≤1, (2:25)

thenHn(z)is in the classF2(α,δ,b).

Proof Since fi∈U(α,δ,b), it follows from Definition 1.3 that

1−1

b

zf_i(z) fi(z)

+ 1

> α1

b

zf_i(z) fi(z)

+ 1

+δ. (2:26)

Considering Definition 1.8 and with the help of (2.21), we obtain

1−1 b

zzH_n(z) + 3H_n(z) zH_n(z) + 2Hn(z) + 1

−α1

b

zzH_n(z) + 3H_n(z) zH_n(z) + 2Hn(z) + 1

−δ

= 1− n

i=1γ i+

n

i=1γ

i 1−

1 b

zf_i(z) fi(z) + 1

−αn i=1γ

i 1 b

zf_i(z) fi(z) + 1

−δ

≥1− n

i=1

γi+ n

i=1

γi 1− 1 b

zf_i(z) fi(z) + 1

−αn i=1

γi

1 b

zf_i(z) fi(z) + 1

−δ

>1− n

i=1

γi+ n

i=1

γi α

1 b

zf_i(z) f(z) + 1

+δ

−αn i=1

γi

1 b

zf_i(z) fi(z) + 1

−δ

=₍1−δ)

1− n

i=1γ i

≥0.

(2:27)

This completes the proof. □

Theorem 2.7For iÎ {1,...,n}, letgi> 0 and fi∈UH(α,b)(a > 0andb∈C\{0}).

n

i=1

γi≤1, (2:28)

thenHn(z)is in the classF3(α,b).

Proof Since fi∈UH(α,b), it follows from Definition 1.5 that

√2

1−1 b

zfi(z) fi(z)

+ 1

+2α(√2−1)− 1− 1 b

zfi(z) fi(z)

+ 1

−2α(√2−1)

>0. (2:29)

Considering this inequality and (2.21), we obtain

√2

1−1 b

zzH

n(z) + 3Hn(z)

zHn(z) + 2Hn(z) + 1

+ 2α

_√

2−1

− 1− 1 b

zzH

n(z) + 3Hn(z)

zHn(z) + 2Hn(z) + 1

−2α

_√

2−1

= √2

1−n

i=1 γi1

b

zf_i(z) fi(z) + 1

+ 2α√2−1

−1−n

i=1 γi1

b

zf_i(z) fi(z) + 1

−2α√2−1

=√2−√2n i=1

γi 1 b

zfi(z) fi(z) + 1

+ 2α

_√

2−1

− 1− n i=1 γi 1 b

zfi(z) fi(z) + 1

−2α

_√

2−1

=√2 +√2n i=1

γi 1−1 b

zf_i(z) fi(z) + 1

−√2n

i=1 γi+ 2α

_√

2−1

− 1 + n i=1 γi

1−1 b

zf_i(z) fi(z) + 1

−2α

_√

2−1

−n

i=1 γi+ 2α

_√

2−1

n

i=1 γi

−2α

_√

2−1

=√2

1−n

i=1 γi

+ 2α√2−1+√2n i=1

γi 1−1 b

zf_i(z) fi(z) + 1

−

1−2α

_√

2−1 1−

n i=1 γi + n i=1 γi

1−1 b

zfi(z) fi(z) + 1

−2α

_√

2−1

≥√2

1−n

i=1 γi

+ 2α√2−1+√2n i=1

γi 1− 1 b

zf_i(z) fi(z) + 1

−n

i=1 γi 1− 1 b

zfi(z) fi(z) + 1

−2α

_√

2−1−1−2α

_√

2−1 1− n i=1 γi

=n i=1

γi

√

2

1−1

b

zf_i(z) fi(z) + 1

+ 2α√2−1

−1−1

b

zf_i(z) fi(z) + 1

−2α√2−1

+√2

1−n

i=1 γi

+ 2α√2−1

−2α√2−1

n

i=1

γi−1−2α

_√

2−1

1−n

i=1 γi

>√2 + 2α√2−1

−1−2α√2−1 1−n

i=1 γi > 1− n i=1 γi min _√

2−1

(1 + 4α),√2 + 1

≥0.

This completes the proof. □

3 Some properties for Hγ1,...,γn(z)

In this section, we investigate some properties for the integral operator Hγ1,...,γn(z)

defined by (1.16) of subclasses given by (1.5), Definitions 1.2, 1.4, and 1.6. Theorem 3.1For iÎ {1,...,n},letgi> 0,fiÎΣand

n

i=1 γi>

2

1−δ, (0≤δ <1). (3:1)

Proof On successive differentiation ofHγ1,...,γn(z), which is defined in (1.16), we have

2zH_γ1,...,γn(z) +z 2_H

γ1,...,γn(z) =

−z2f₁(z)γ1. . .

−z2f_n(z)γn_{, (3}:₂₎

and

z2_H

γ1,...,γn(z) + 4zH

γ1,...,γn(z) + 2Hγ1,...,γn(z)

=

n i=1γ

i

f_i(z) f_i(z) +

2

z −z

2_f 1(z)

γ1

. . .−z2_f n(z)

γn

. (3:3)

Then from (3.2) and (3.3), we obtain

z2H_γ1,...,γn(z) + 4zH

γ1,...,γn(z) + 2Hγ1,...,γn(z)

z2H

γ1,...,γn(z) + 2zHγ1,...,γn(z)

=

n

i=1 γi

fi(z)

fi(z)

+2 z

. (3:4)

By multiplying (3.4) with zyields,

z2_H

γ1,...,γn(z) + 4zH

γ1,...,γn(z) + 2Hγ1,...,γn(z)

zH_γ1,...,γn(z) + 2H_γ1,...,γn(z)

=

n

i=1 γi

zfi(z)

fi(z)

+ 2

. (3:5)

that is equivalent to

z2H_γ1,...,γn(z) + 3zH

γ1,...,γn(z)

zH_γ1,...,γn(z) + 2H_γ1,...,γn(z)

+ 1 =

n

i=1 γi

zfi(z)

fi(z)

+ 2

. (3:6)

Therefore, we have

−

zH_γ1,...,γn(z) H

γ1,...,γn(z)

+ 1

−2

1 +2Hγ1,...,γn(z)

zH_γ1,...,γn(z)

=−

n

i=1 γi

zf_i(z) f_i(z) + 2

+ 1. (3:7)

so that

−

zH_γ1,...,γn(z) H

γ1,...,γn(z)

+ 1

=

2Hγ1,...,γn(z)

zH_γ1,...,γn(z) −

n i=1

γi

zf_i(z) f_i(z) + 2

+ 1

+

n i=1

γi −

zf_i(z) f_i(z) + 1

+ 3−

n i=1

γi.

(3:8)

Taking the real parts of both terms of the last expression, we obtain

−

zH_γ1,...,γn(z) H

γ1,...,γn(z)

+ 1

= 2Hγ1,...,γn(z)

zH_γ1,...,γn(z)

−n

i=1 γi

zf_i(z) f_i(z) + 2

+ 1

+

n i=1

γi −

zf_i(z) f_i(z) + 1

+ 3−

n i=1

γi

≤2Hγ1,...,γn(z)

zH_γ1,...,γn(z)

−n

i=1 γi

zf_i(z) f_i(z) + 2

+ 1

+

n i=1γ

i −

zf_i(z) f_i(z) + 1

+ 3−

n i=1γ

i.

Let

β=

2H_γ1,...,γn(z) zH_γ1,...,γn(z)

−

n

i=1 γi

zf_i(z)

f_i(z) + 2

+ 1

+

n

i=1 γi −

zf_i(z)

f_i(z) + 1

+3− n

i=1

γi. (3:10)

Since

2Hγ1,...,γn(z)

zH_γ1,...,γn(z)

−n

i=1 γi

zf_i(z) f_i(z) + 2

+ 1

>0,fiÎΣk(δ), we get

β >3−(1−δ)

n

i=1

γi, (3:11)

which, in light of the hypothesis (3.1), yieldsb> 1.

Therefore, Hγ1,...,γn(z)∈N(β), b> 1. This completes the proof. □

Theorem 3.2For iÎ {1,...,n},letgi> 0,fiÎΣand

1<

n

i=1

γi<2. (3:12)

IfHγ1,...,γn(z)∈(δ),thenHγ1,...,γn(z)∈N(β),b> 1.

Proof It follows from (3.7) that

−

zH_γ1,...,γn(z) H

γ1,...,γn(z)

+ 1

=

1 + 2Hγ1,...,γn(z)

zH_γ1,...,γn(z) −

n i=1γ

i

zf_i(z) f_i(z) + 1

+2

_n

i=1

γi−1 −Hγ 1,...,γn(z)

zH_γ1,...,γn(z)

+ 3−

n i=1

γi.

(3:13)

Taking the real parts of both terms of the last expression, we obtain

−

zH_γ1,...,γn(z) H

γ1,...,γn(z)

+ 1

= 1 + 2Hγ1,...,γn(z)

zH_γ1,...,γn(z) −

n i=1

γi

zf_i(z) f_i(z) + 1

+2

_n

i=1 γi−1

−Hγ1,...,γn(z)

zH_γ1,...,γn(z)

+ 3−

n i=1

γi.

(3:14)

Thus, we have

−

zH_γ1,...,γn(z) H

γ1,...,γn(z)

+ 1

= 1 + 2Hγ1,...,γn(z)

zH_γ1,...,γn(z) −

n i=1

γi

zf_i(z) f_i(z) + 1

+2

_n

i=1 γi−1

⎛ ⎜ ⎜ ⎜ ⎝−

1 zH_γ1,...,γn(z)

Hγ1,...,γn(z) ⎞ ⎟ ⎟ ⎟ ⎠+ 3−

n i=1

γi

≤

1 +2Hγ1,...,γn(z)

zH_γ1,...,γn(z) −

n i=1γ

i

zf_i(z) f_i(z) + 1

+2

_n

i=1 γi−1

−zH

γ1,...,γn(z) Hγ1,...,γn(z)

zH_γ1,...,γn(z) H_γ1,...,γn(z)

2 + 3− n i=1

γi.

Let

β =

1 +2Hγ1,...,γn(z)

zH_γ1,...,γn(z) −

n i=1

γi

zf_i(z) f_i(z) + 1

+2

_n

i=1γ i−1

−zH

γ1,...,γn(z) Hγ1,...,γn(z)

zH_γ1,...,γn(z) Hγ1,...,γn(z)

2 + 3− n i=1γ

i.

(3:16)

Since

1 +2Hγ1,...,γn(z)

zH_γ1,...,γn(z) −

n i=1

γi

zf_i(z) f_i(z) + 1

>0 and Hγ1,...,γn(z)∈(δ), we

have

β >2

_n

i=1 γi−1

δ

zH_γ1,...,γn(z) Hγ1,...,γn(z)

2 + 3− n

i=1

γi. >3− n

i=1 γi.

(3:17)

Then, by the hypothesis (3.12), we see that b> 1. Therefore,Hγ1,...,γn(z)∈N(β), b

> 1. This completes the proof. □

Now, using the method given in the proofs of Theorems 2.5, 2.6, and 2.7, one can prove the following results:

Theorem 3.3For iÎ {1,...,n},letgi> 0fiÎΣKb(δi) (0≤δ< 1andb∈C\{0})).If

0<

n

i=1

γi(1−δi)≤1, (3:18)

thenHγ1,...,γn(z)is in the classF1(μ,b), μ= 1− n i=1

γi(1−δi).

Theorem 3.4 For iÎ{1,..., n},letgi> 0and fi∈KU(α,δ,b)(a≥0,δÎ [-1,1),a+

δ≥0andb∈C\{0})).If

n

i=1

γi≤1, (3:19)

thenHγ1,...,γn(z)is in the classF2(α,δ,b).

Theorem 3.5 For i Î {1,..., n}, let gi > 0 and fi∈KUH(α,b) (a ≥ 0, and

b∈C\{0})).If

n

i=1

γi≤1, (3:20)

thenHγ1,...,γn(z)is in the classF3(α,δ,b).

Acknowledgements

The above study was supported by MOHE grant: UKM-ST-06-FRGS0244-2010.

Authors_’contributions

Competing interests

The authors declare that they have no competing interests.

Received: 21 August 2011 Accepted: 25 November 2011 Published: 25 November 2011

References

1. Wang, Z-G, Sun, Y, Zhang, Z-H: Certain classes of meromorphic multivalent functions. Comput Math Appl.58, 1408_–1417 (2009). doi:10.1016/j.camwa.2009.07.020

2. Nehari, Z, Netanyahu, E: On the coefficients of meromorphic Schlicht functions. Proc Am Math Soc.8, 15–23 (1957). doi:10.1090/S0002-9939-1957-0083038-0

3. Goodman, AW: Univalent Functions. Mariner, Tampa, FLI, II(1983)

4. Frasin, BA: Family of analytic functions of complex order. Acta Math Acad Paed Ny.22(2), 179–191 (2006) 5. Ronning, F: Uniformly convex functions. Ann Polon Math.57, 165_–175 (1992)

6. Murugusundaramoorthy, G, Maghesh, N: A new subclass of uniforlmly convex functions and a corresponding subclass of starlike functions with fixed second coefficient. J Inequal Pure Appl Math.5(4), 1_–10 (2005)

7. Stankiewicz, J, Wisniowska, A: Starlike functions associated with some hyperbola. Folia Scientiarum Universitatis Tehnicae Resoviensis 147, Matematyka.19, 117_–126 (1996)

8. Acu, M: Subclasses of convex functions associated with some hyperbola. Acta Universitatis Apulensis.12, 3–12 (2006) 9. Acu, M, Owa, S: Convex functions associated with some hyperbola. J Approx Theory Appl.1, 37_–40 (2005) 10. Nasr, MA, Aouf, MK: Starlike function of complex order. J Nat Sci Math.25(91), 1–12 (1985)

11. Baricz, Á, Frasin, BA: Univalence of integral operators involving Bessel functions. Appl Math Lett.23, 371–376 (2010). doi:10.1016/j.aml.2009.10.013

12. Mohammed, A, Darus, M: New properties for certain integral operators. Int J Math Anal.4(42), 2101–2109 (2010) 13. Frasin, BA: Univalence of two general integral operators. Filomat.23, 223_–229 (2009). doi:10.2298/FIL0903223F 14. Breaz, D, Breaz, N, Srivastava, HM: An extension of the univalent condition for a family of integral operators. Appl Math

Lett.22, 41_–44 (2009). doi:10.1016/j.aml.2007.11.008

15. Breaz, D, Breaz, N: Two integral operators. Studia Universitatis Babes-Bolyai, Mathematica, Cluj-Napoca.3, 13–21 (2002) 16. Breaz, D, Owa, S, Breaz, N: A new integral univalent operator. Acta Univ Apulensis Math Inform.16, 11_–16 (2008) 17. Breaz, D, Güney, HÖ, Salagean, GS: A new general integral operator. Tamsui Oxford, J Math Sci.25(4), 407–414 (2009) 18. Darus, M, Faisal, I: A study on Beckers univalence criteria. Abs Appl Anal2011, 13 (2011). Article, ID 759175 19. Breaz, N, Braez, D, Darus, M: Convexity properties for some general integral operators on uniformly analytic functions

classes. Comput Math Appl.60, 3105–3107 (2010). doi:10.1016/j.camwa.2010.10.012

20. Latha, S, Dileep, L: On a generalized integral operator. Int J Math Anal.3(30), 1487_–1491 (2009)

21. Bulut, S: A new general integral operator defined by Al-Oboudi differential operator. J Inequal Appl2009, 13 (2009). Article, ID 158408

22. Mohammed, A, Darus, M: A new integral operator for meromorphic functions. Acta Universitatis Apulensis.24, 231–238 (2010)

23. Mohammed, A, Darus, M: Starlikeness properties of a new integral operator for meromorphic functions. J Appl Math

2011, 8 (2011). Article, ID 804150

24. Goyal, SP, Prajapat, JK: A new class of meromorphic multivalent functions involving certain linear operator. Tamsui Oxford, J Math Sci.25(2), 167–176 (2009)

doi:10.1186/1029-242X-2011-121

Cite this article as:Mohammed and Darus:Integral operators on new families of meromorphic functions of complex order.Journal of Inequalities and Applications20112011:121.

Submit your manuscript to a

journal and benefi t from:

7Convenient online submission

7Rigorous peer review

7Immediate publication on acceptance

7Open access: articles freely available online

7High visibility within the fi eld

7Retaining the copyright to your article