On new subclasses of analytic functions involving generalized differential and integral operators

Vol. 4, No. 1, 2011, 59-66

ISSN 1307-5543 – www.ejpam.com

On New Subclasses of Analytic Functions Involving

Generalized Differential and Integral Operators

Maslina Darus

_{, Rabha W. Ibrahim}

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

Abstract. We define a generalized differential and integral operators on the classA of analytic func-tions f(z) =z+P∞_n=2anznin the unit diskU:={z∈C:|z|<1}involving k−th Hadamard product

(convolution) as follows

D_αk_,λf(z) =z+

∞

X

n=2

[(n−1)(λ−α) +n]k_a

nzn, (z∈U).

These operators are generalized for some of well known operators for example Sˇalˇagean operator. New classes containing these operators are investigated. Characterization and other properties of these classes are studied.

2000 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Hadamard product; Integral operator; Differential operator; Sˇalˇagean oper-ator.

1. Introduction and Preliminaries

Let H be the class of functions analytic in U := {z ∈C _:|_z| _{< 1}} _and H[a,n] be the subclass ofH consisting of functions of the form f(z) =a+a_nzn+a_n+1zn+1+. . .. LetA be

the subclass ofH consisting of functions of the form

f(z) =z+

∞

X

n=2

a_nzn, (z ∈U). (1)

Given two functions f,g∈ A, f(z) =z+P∞_n=2a_nzn andg(z) =z+P∞_n=2b_nzn their convolu-tion or Hadamard product f(z)∗g(z)is defined by

f(z)∗g(z) =z+

∞

X

n=2

a_nb_nzn, (z ∈U).

∗_{Corresponding author.}

Email addresses:maslinaukm.my(M. Darus),rabhaibrahimyahoo.om (R. Ibrahim)

And for several functions f₁(z), . . . ,f_m(z)∈ A

f₁(z)∗. . .∗f_m(z) =z+

∞

X

n=2

(a_1n...a_mn)zn, (z ∈U).

Our aim is to use the Hadamard product ofk−th order to define generalized differential and integral operators.

For a function f inA of the form (1) first, we define the following generalized differential operator:

D0f(z) =f(z)

=z+

∞

X

n=2 a_nzn,

D1_α,λf(z) = (α−λ)f(z) + (λ−α+1)z f′(z) =z+

∞

X

n=2

[(n−1)(λ−α) +n]a_nzn,

.. . D_αk_,λf(z) =D1_α,λD_αk−_,λ1_,f(z)

=z+

∞

X

n=2

[(n−1)(λ−α) +n]ka_nzn

(2)

forα ≥ 0,λ≥ 0 and k ∈N_{0 =} N∪ {₀}_{with D}k

α,λf(0) = 0. Note that when α = λ we get

Sˇalˇagean’s differential operator[see 11].

Let M(µ) be the subclass of the class A consisting of functions f(z) which satisfy the inequality

ℜ{z f ′_(z)

f(z) }< µ, (z∈U)

for someµ(µ >1). And letN(µ)be the subclass of the classA consisting of functions f(z)

which satisfy the inequality

ℜ{z f ′′_(z)

f′(z) }< µ, (z∈U)

for someµ(µ >1). Then f ∈ N(µ)if and only ifz f′∈ M(µ). In this paper we define and study the following subclasses involving the generalized differential operator (2). LetM_αk_,λ(µ)

be the subclass of the classA consisting of functions f(z)which satisfy the inequality

ℜ{z[D

k

α,λf(z)]

′

for someµ(µ >1). And letN_αk_,λ(µ)be the subclass of the classA consisting of functionsf(z)

which satisfy the inequality

ℜ{z[D

k

α,λf(z)]

′′

[D_αk_,λf(z)]′ }< µ, (z∈U)

for someµ(µ >1). Then f ∈ Nk

α,λ(µ)if and only ifz f

′_{∈ M}k

α,λ(µ)(µ).

Remark 1. When k=0,then the classes

M0

α,λ(µ)≡ M(µ)andN 0

α,λ(µ)≡ N(µ)

were introduced by

(i) For1< µ≤ 4

3,k=0,Uralegaddi et al.[13, 12].

(ii) Forµ >1,k=0,Owa and Srivastava[9]and Owa and Nishiwaki[10]. (iii) Forµ >1,α=1Bulut[2].

2. Coefficient estimates.

In this section we derive sufficient conditions for f(z) to belongs to the classesM_αk_,λ(µ)

andN_αk_,λ(µ), which are obtained by using coefficient inequalities. Theorem 1. If f(z)∈ A satisfies the inequality

∞

X

n=2

|[(n−1)(λ−α) +n]k|n(n−κ) +|n+κ−2µ|o|an| ≤2(µ−1) (3)

for some0≤κ≤1andµ >1,then f ∈ M_αk_,λ(µ).

Proof. Assume that the inequality (3) holds. It suffices to show that

z[D_αk_,λf(z)]′

Dk α,λf(z)

−κ

z[Dk α,λf(z)]′

D_αk_,λf(z) −(2µ−κ)

<1, (z∈U).

We observe

z[Dk α,λf(z)]

′

D_αk_,λf(z) −κ

z[Dk_α,λf(z)]′

Dk α,λf(z)

−(2µ−κ) =

1−κ+P∞_n=2(n−κ)[(n−1)(λ−α) +n]ka_nzn−1 1+κ−2µ+P∞_n=2(n+κ−2µ)[(n−1)(λ−α) +n]k_a

nzn−1

≤ 1−κ+

P∞

n=2(n−κ)|[(n−1)(λ−α) +n]k||an||z|n−1 2µ−1−κ−P_n∞₌₂|(n+κ−2µ)||[(n−1)(λ−α) +n]k_||a

n||z|n−1 < 1−κ+

P∞

n=2(n−κ)|[(n−1)(λ−α) +n] k_||a

n| 2µ−1−κ−P_n∞₌₂|(n+κ−2µ)||[(n−1)(λ−α) +n]k_||a

The last expression is bounded above by 1 if

1−κ+

∞

X

n=2

(n−κ)|[(n−1)(λ−α)+n]k||a_n|<2µ−1−κ− ∞

X

n=2

|(n+κ−2µ)||[(n−1)(λ−α)+n]k||a_n|

which is equivalent to assertion (3), hence the proof. Whenk=0, the next result can found in[10]. Corollary 1. If f(z)∈ A satisfies the inequality

∞

X

n=2 n

(n−κ) +|n+κ−2µ|o|an| ≤2(µ−1) (4)

for some0≤κ≤1andµ >1,then f ∈ M_α0_,λ(µ)≡ M(µ). Whenκ=1, we obtain the next result

Corollary 2. If f(z)∈ A satisfies the inequality ∞

X

n=2

(n−µ)|[(n−1)(λ−α) +n]k||an| ≤µ−1 (5)

for1< µ≤ 3₂,then f ∈ M_αk_,λ(µ).

Whenk=0,κ=1 the next result can found in[10]. Corollary 3. If f(z)∈ A satisfies the inequality

∞

X

n=2

(n−µ)|a_n| ≤µ−1 (6)

for1< µ≤ 3₂,then f ∈ M(µ).

Theorem 2. If f(z)∈ A satisfies the inequality ∞

X

n=2

n|[(n−1)(λ−α) +n]k|nn−κ+1+|n+κ−2µ|o|an| ≤2(µ−1) (7)

for some0≤κ≤1andµ >1,then f ∈ N_αk_,λ(µ). Whenk=0, the next result can be found in[10]. Corollary 4. If f(z)∈ A satisfies the inequality

∞

X

n=2

nnn−κ+1+|n+κ−2µ|o|a_n| ≤2(µ−1) (8)

3. Integral operator.

Analogous to the generalized differential operator (2), we define and study a new integral operator I_αk_,λ:A → A as follows. Let

φ(z):= (λ−α)z (1−z)2 −

(λ−α)z 1−z +

z

(1−z)2

and

F(z) =φ(z)∗. . .∗φ(z)

| {z }

k−t imes

=z+

∞

X

n=2

[(n−1)(λ−α) +n]kzn

Now we define the integral operatorI_αk_,λ such that

I_αk_,λ:= [F(z)]−1∗f(z), (z∈U)

where f ∈ A and

F(z)∗[F(z)]−1= z

1−z =z+ ∞

X

n=2

zn, (z∈U).

Implies

[F(z)]−1=z+

∞

X

n=2

1

[(n−1)(λ−α) +n]kz

n_{, (z∈U)}

thus we have

I_αk_,λf(z) =z+

∞

X

n=2

an

[(n−1)(λ−α) +n]kz

n_{, (z∈U). (9)}

Remark 2. Note that whenα=λ,the integral operator (9) reduces to the integral operator

I_αk_,αf(z) =z+

∞

X

n=2 a_n

nkz

n_{, (z∈U),}

which defined and studied by Sˇalˇagean[see 11].

Lemma 1. Let f ∈ A.Then (i) I_α0_,λf(z) = f(z),

(ii) I_α1_,αf(z) =Rz

0 f(t)

t d t.

(i)

I_α0_,λf(z) =z+

∞

X

n=2

a_nzn= f(z),

(ii)

Z z

0 f(t)

t d t=

Z z

0

[1+

∞

X

n=2

antn−1]d t

=z+

∞

X

n=2 an

nz n

=I_α1_,αf(z).

Define the subclasses involving the generalized integral operator (9). LetS_αk_,β,λ(µ)be the subclass of the classA consisting of functions f(z)which satisfy the inequality

ℜ{z[I

k

α,λf(z)]

′

I_αk_,λf(z) }< µ, (z∈U)

for someµ(µ >1). It is clear that

S0

α,λ(µ)≡ M(µ).

And letK_αk_,β,λ(µ)be the subclass of the classA consisting of functions f(z)which satisfy the inequality

ℜ{z[I

k

α,λf(z)]

′′

[I_αk_,λf(z)]′ }< µ, (z∈U)

for someµ(µ >1). Then f ∈ K_αk_,λ(µ)if and only ifz f′∈ S_αk_,λ(µ)(µ). Also we have K0

α,λ(µ)≡ N(µ).

In the same manner of Theorem 1 and Theorem 2, we have the following results. Theorem 3. If f(z)∈ A satisfies the inequality

∞

X

n=2 n

(n−κ) +|n+κ−2µ|o

|[(n−1)(λ−α) +n]k_| |an| ≤2(µ−1) (10) for some0≤κ≤1andµ >1,then f ∈ S_αk_,λ(µ).

Theorem 4. If f(z)∈ A satisfies the inequality ∞

X

n=2

nnn−κ+1+|n+κ−2µ|o

4. Conclusion.

This work is a generalization for well known differential and integral operators of univa-lent functions. Moreover, the classes which are studied here also generalized the ones studied by different authors

M0

α,λ(µ)≡ S 0

α,λ(µ)≡ M(µ)

and

N0

α,λ(µ)≡ K 0

α,λ(µ)≡ N(µ).

In fact, many other operators can be seen in[1,3-8,14]for different problems.

ACKNOWLEDGEMENTS The work presented here was supported by UKM-ST-06-FRGS0107-2009.

References

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