Univalent Functions Defined by a Generalized Multipler Differential Operator

Univalent Functions Defined by a Generalized

Multipler Differential Operator

Adnan Ghazy Alamoush

Faculty of Science, Taibah University, Saudi Aarabia; adnan–[email protected]

Abstract:In this paper, we investigate a new subclass of univalent functions defined by a generalized differential operator, and obtain some interesting properties of functions belonging to the class Rm_λ,µ(α,β,γ,ϑ).

Keywords:analytic functions; univalent function; differential operator; differential subordination; convex functions; structural formula

1. Introduction

LetAdenote the class of the functions f of the form

f(z) =z+ ∞

∑

akzk (1)

which are analytic in the open unit discU={z∈C:|z|<1}. LetH(U)be the space of holomorphic functions inU. BySandKwe denote the subclasses of functions inAwhich are univalent and convex inU, respectively. LetPbe the well-known Caratheodory class of normalized functions with positive real part inU.

The Hadamard product or convolution of the functions

f(z) =z+_∑∞_k=2akzk and g(z) =z+∑∞k=2bkzk is given by

(f ∗g)(z) =z+_∑∞_k=2akbkzk, (z∈U) We now define a new generalized multiplier differential operator.

Definition 1. Let m ∈ _N0 = N∪ {0}, α,β,λ,µ,ρ ≥ 0, 0≤ γ ≤ λ, 0 < ϕ≤ 1, α+β >0. Then for

f ∈A, we define a new generalized multiplier operator Dm

λ,µ(α,β,γ,ϑ)by D0_λ,µ(α,β,γ,ϑ)f(z) = f(z),

D1_λ,µ(α,β,γ,ϑ)f(z) = [α+β+γ−(2ϑ−1)(λ+µ)]f(z)+[(2ϑ−1)(λ+µ)−γ]z f 0_(z)+

γλz2f00(z)

α+β ,

... ,

D_λ,µm (α,β,γ,ϑ)f(z) =D_λ,µ(α,β,γ,ϑ)

Dm_λ,µ−1(α,β,γ,ϑ)

.

Remark 1. If f(z)is given by (1), then from Definition1, we obtain

D_λ,µm (α,β,γ,ϑ)f(z) =z+ ∞

∑

Ωm

λ,µ(α,β,γ,ϑ)akzk, (2) where

Ωm

λ,µ(α,β,γ,ϑ) =

α+ [(2ϑ−1)(λ+µ) +γ(kλ−1)] (k−1) +β α+β

m

. (3)

From (2) it follows thatD_λ,µm (α,β,γ,ϑ)f(z)can be written in terms of convolution as

D_λ,µm (α,β,γ,ϑ)f(z) = (f ∗g)(z), (4)

where

g(z) =z+ ∞

∑

Ωm

λ,µ(α,β,γ,ϑ)akzk. (5) The classDm

λ,µ(α,β,γ,ϑ)includes many earlier classes (see also [1]), which are mentioned below:

• D_1,0m(0, 1, 0, 1):=Dkf(z), has been studied by Salagean [2].

• D_1,0m(1, 1, 0, 1):=Lmf(z), has been studied by Uralegaddi and Somanatha [3].

• Dm

1,0(α, 1, 0, 1):=Lmβ, has been studied by Cho and Srivastava [4].

• Dρ

λ,0(0, 1, 0, 1):=D ρ

λf(z), has been studied by Acu and Owa [5].

• D_λ,0m (0, 1, 0, 1):=Dm_λf(z), has been studied by Al-Oboudi [6].

• D_λ,0ρ (1,β,γ, 1):=L1(ρ,λ,β)f(z), has been studied by Catas et al. [7].

• D_λ,0m (α, 0, 0, 1):=Dm_λ(α), has been studied by Aouf et al. [8].

• Dm

λ,0(0, 1, 0, α+β

2 ):=Dmα,β,0,λf(z), has been studied AlAmoush and Darus [9].

• D_λ,0m (0, 1,γ, 1):=Dm_λ,γf(z), has been studied by R ˘aducanu and Orhan [10].

• D_λ,µm (α,β, 0, 1):=D_λm(α,β,µ), has been studied by Darus and Faisal [11].

• D_λ,µm (α, 0, 0, 1):=Dm_λ(α,µ), has been studied by Darus and Faisal [12].

Definition 2. Let m∈_N0=N∪ {0},v∈[0, 1), α,β,λ,µ≥0, 0≤γ≤λ, 0<ϕ≤1, α+β>0. Then

a function f ∈A is said to be in the class Rm_λ,µ(α,β,γ,ϑ), if it satisfies the condition <hDm

λ,µ(α,β,γ,ϑ)

i0

>v, z∈U.

The main object of this paper is to present a systematic investigation for the classRm_λ,µ(α,β,γ,ϑ).

In particular, for this function class, we derive an inclusion result, structural formula, extreme points and other interesting properties.

2. Preliminaries

In order to prove our results, we will make use of the following lemmas.

Lemma 1. [13] Let h∈K, and Let A≥0. Suppose that B(z)and D(z)are analytic inU, with D(0) =0and <(B(z))≥A+4

D(z) h0₍₀₎

, z∈U.

If an analytic function p with p(0) =h(0)satisfies

Az2p00(z) +B(z)zp0(z) +p(z) +D(z)≺h(z), z∈U, then

p(z)≺h(z), z∈_U. Lemma 2. [14] Let q be a convex function inUand let

h(z) =q(z) +vzq0(z),

wherev>0. If p∈ H(U)with

p(z)≺q(z), z∈_U, and this result is sharp.

Lemma 3. [15] If p(z)is analytic inU, p(0) = 1and−1(z)and ,<(p(z)) > 12, then for any function F analytic inU, the function F∗p takes values in the convex hull of F(U).

Note that the symbol ”≺” stands for subordination throughout this paper.

3. Main Results

Theorem 1. Let m∈ N0=N∪ {0},v∈[0, 1), α,β,λ,µ≥0, 0≤γ≤λ, 0< ϕ≤1, α+β>0, then

Rm_λ,µ+1(α,β,γ,ϑ)⊂Rm_λ,µ(α,β,γ,ϑ).

Proof.Let f ∈Rm_λ,µ+1(α,β,γ,ϑ). By using the properties of the operatorDm_λ,µ(α,β,γ,ϑ), we get

Dm+1

λ,µ (α,β,γ,ϑ)f(z)

=[α+β+γ−(2ϑ−1)(λ+µ)]D

m

λ,µ(α,β,γ,ϑ)f(z) + [(2ϑ−1)(λ+µ)−γ]z(D m

λ,µ(α,β,γ,ϑ)f(z))

0_+γλz2_(Dm

λ,µ(α,β,γ,ϑ)f(z))

00

α+β .

(6)

Differentiating (6) with respect toz, we obtain

(D_λ,µm+1(α,β,γ,ϑ)f(z))0 =p(z) +

(2ϑ−1)(λ+µ) +2λγ

α+β

p0(z) +

γλ

α+β

p00(z), (7)

where

p(z) = (D_λ,µm (α,β,γ,ϑ)f(z))0.

Sincef ∈Rm_λ,µ+1(α,β,γ,ϑ), by using Definition2and (7), we have <np(z) +h(2ϑ−1)(_αλ₊+_βµ)+2λγip0(z) +h γλ

α+β

i

p00(z)o>v,z∈U, which is equivalent to

n

p(z) +h(2ϑ−1)(_αλ₊+_βµ)+2λγip0(z) +h γλ α+β

i

p00(z)o≺ 1+(₁2v₋−_z1)z ≡h(z).

From Lemma1, with A= γλ

α+β, B(z) =

(2ϑ−1)(λ+µ)+2λγ

α+β , and D(z) =0, we havep(z) ≺ h(z), which implies that<n(Dm

λ,µ(α,β,γ,ϑ)f(z))0

o

>v, z∈U. Hencef ∈Rmλ,µ(α,β,γ,ϑ)and the proof of the theorem is complete.

ClearlyRm_λ,µ(α,β,γ,ϑ)⊂Rm_λ,µ−1(α,β,γ,ϑ)⊂...⊂R0_λ,µ(α,β,γ,ϑ)⊂S(see [16,17]).

Now we will show that the setRm_λ,µ(α,β,γ,ϑ)is convex (see [18]).

Theorem 2. The set Rm_λ,µ(α,β,γ,ϑ)is convex.

Proof. Let the functions

fi(z) =z+∑∞k=2aizk (i=1, 2)

be in the classRm_λ,µ(α,β,γ,ϑ). It is sufficient to show that the functionh(z) =χ1f1(z) +χ2f2(z), with

χ1andχ2nonnegative andχ1+χ2=1, is in the classRm_λ,µ(α,β,γ,ϑ).

Since

h(z) =z+∑∞_k=2(χ1ak1+χ2ak2)z k_,

[Dm_λ,µ(α,β,γ,ϑ)h(z)]0=1+∑∞_k=2k(χ1ak1+χ2ak2)[v m

λ,µ(α,β,γ,ϑ)]zk

−1_,

hence

<Dmλ,µ(α,β,γ,ϑ)h(z)

0

=< 1+χ1

∞

∑

k[vm_λ,µ(α,β,γ,ϑ)]ak1z k−1

!

+< 1+χ2

∞

∑

k[vm_λ,µ(α,β,γ,ϑ)]ak2z k−1

!

−1 (8)

Sincef1,f2∈Rm_λ,µ(α,β,γ,ϑ), this implies that

< 1+χi ∞

∑

k[v_λ,µm (α,β,γ,ϑ)]akiz k−1

!

>1+χi(v−1). (9)

Using (9) in (8), we obtain

<Dm

λ,µ(α,β,γ,ϑ)h(z)

0

>1+v(χ₁+χ2)−(χ1+χ2)

and sinceχ1+χ2=1, the theorem is proved.

Theorem 3. Let q be convex function with q(0) = 1and let h be a function of the form h(z) = q(z) +

zq0(z), z ∈ U.If f ∈ A satisfies the differential subordination(Dm_λ,µ(α,β,γ,ϑ))0 ≺ h(z), z ∈ U, then Dm_λ,µ(α,β,γ,ϑ)/z≺q(z)and the result is sharp.

Proof. If we let p(z) = D_λ,µm (α,β,γ,ϑ)/z, z∈ U, then we get(D_λ,µm (α,β,γ,ϑ))0 = p(z) +zp0(z). So

the subordination(Dm_λ,µ(α,β,γ,ϑ))0 ≺ h(z), z ∈ U, becomesp(z) +zp0(z) = q(z) +zq0(z), z ∈ U. Hence from Lemma2, we have(Dm_λ,µ(α,β,γ,ϑ))/z≺q(z). The result is sharp.

4. Structural Formula

In this section, a structural formula, extreme points and coefficient bounds for functions in Rm_λ,µ(α,β,γ,ϑ)are obtained.

Theorem 4. A function f ∈ A is in the class Rm_λ,µ(α,β,γ,ϑ)if and only if it can be expressed as

f(z) = "

z+ ∞

∑

1 Ωm λ,µ(α,β,γ,ϑ) zk # ∗ Z

|ζ|=1

"

z+2(1−v)ζ ∞

∑

(ζz)k

k

#

dσ(ζ), (10)

whereΩm_λ,µ(α,β,γ,ϑ)is given by (3) andσis a positive probability measure defined on the unit circleT = {ζ∈_C:|ζ|<1}.

Proof. From (4) it follows that, f ∈Rm

λ,µ(α,β,γ,ϑ)if and only if

(Dm

λ,µ(α,β,γ,ϑ)f(z)) 0₋

v 1−v ∈ P.

Using Hergoltz integral representation of functions in Caratheodory classP(see [19] and [20]), there exists a positive Borel probability measureσsuch that

(Dm_λ,µ(α,β,γ,ϑ)f(z))0−v

1−v =

R

|ζ|=1 1+ζz

1−ζzdσ(ζ), z∈U, which is equivalent to

(Dm_λ,µ(α,β,γ,ϑ)f(z))0 =

Z

|ζ|=1

1+ (1−2v)ζz

1−ζz dσ(ζ), z∈U. (11)

Integrating (11), we obtain

Dm_λ,µ(α,β,γ,ϑ)f(z) =R₀z h_R

|ζ|=1

1+(1−2v)ζu 1−ζu dσ(ζ)

i

=R

|ζ|=1

h_Rz

0

1+(1−2v)ζu 1−ζu du

i

dσ(ζ)

that is

Dm_λ,µ(α,β,γ,ϑ) =

Z

|ζ|=1

"

z+2(1−v)ζ ∞

∑

(ζz)k

k

#

dσ(ζ). (12)

Equality (10) follows now, from (4), (5) and (12). Since the converse of this deductive process is also true, we have proved our theorem.

Corollary 1. The extreme points of the class Dm_λ,µ(α,β,γ,ϑ)are

f(z)_ζ =z+2(1−v)ζ ∞

∑

(ζz)k

kΩm

λ,µ(α,β,γ,ϑ)

, z∈_U, |ζ|=1. (13)

Proof. Consider the functions

g_ζ(z) =z+2(1−v)ζ_∑∞_k=2(ζz)k

k and gσ(z) =

R

|ζ|=1gζ(z)dσ(ζ).

Since the mapσ→g_ζis one-to-one, making use of (5), (6) and (12), the assertion follows from (10) (see

[21]).

From Corollary2, we can obtain coefficient bounds for the functions in the classDm

λ,µ(α,β,γ,ϑ).

Corollary 2. If f ∈Rm_λ,µ(α,β,γ,ϑ)is given by (1), then |ak| ≤ _kΩm2(1−v)

λ,µ(α,β,γ,ϑ)

, k≥2. The result is sharp.

Proof. The coefficient bounds are maximized at an extreme point. Therefore, the result follows from (13).

Corollary 3. If f ∈Rm

λ,µ(α,β,γ,ϑ)then, for|z|=r<1 r−2(1−v)r2∑∞k=2_kΩm 1

λ,µ(α,β,γ,ϑ)

≤ |f(z)| ≤r+2(1−v)r2∑∞k=2_kΩm 1 λ,µ(α,β,γ,ϑ)

,

and

1−2(1−v)r∑∞k=2kΩm 1 λ,µ(α,β,γ,ϑ)

≤ |f(z)| ≤1+2(1−v)r∑∞k=2kΩm 1 λ,µ(α,β,γ,ϑ)

.

5. Convolution Property

in this part, we prove the analogue of the Polya- Schoenberg conjecture for the class Rm_λ,µ(α,β,γ,ϑ).

Theorem 5. The class Rm_λ,µ(α,β,γ,ϑ)is closed under the convolution with a convex function. That is, if

f ∈Rm

λ,µ(α,β,γ,ϑ)and g∈C then f∗g∈Rmλ,µ(α,β,γ,ϑ).

Proof. It is known that ifgis convex univalent inU, then (see [13])

Using convolution properties, we have

<Rm_λ,µ(α,β,γ,ϑ)(f ∗g)(z) 0

=<

[Rm_λ,µ(α,β,γ,ϑ)(f)]0∗g(z)

z

(14)

and the result follows by application of Lemma3.

Corollary 4. The class Rm_λ,µ(α,β,γ,ϑ)is invariant under Bernardi integral operator.

Proof. LetRm_λ,µ(α,β,γ,ϑ). The Bernardi integral operator is defined as (see [22]):

Fc(f)(z) = 1_z+kc Rz

0 tc

−1_{f(t)dt, (c∈A, c>−1).} It is easy to check thatFc(f)(z) = (f ∗g)(z)where

g(z) =_∑∞_k=1k1₊+c_czk = 1+c zk

Rz

0 t

c

1−tf(t)dt, (z∈U, c>−1).

Since the function φ(z) = ₁₋z_z , z ∈ U is convex, it follows (see [23]) that the function g is also convex. From Theorem 5, we obtain Fc(f) ∈ Rm_λ,µ(α,β,γ,ϑ). Therefore, Fc[Rm_λ,µ(α,β,γ,ϑ)] ⊂

Rm_λ,µ(α,β,γ,ϑ).

Conflicts of Interest:The author declares no conflict of interest.

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