Certain preserving properties of the generalized Alexander operator

Certain preserving properties of the generalized Alexander operator

Mugur Acu

Abstract

In this paper we give certain preserving properties of the general- ized Alexander operator on some subclasses ofn-uniformly functions.

2000 Mathematical Subject Classification: 30C45

1 Introduction

LetH(U) be the set of functions which are regular in the unit disc U, A={f ∈ H(U) :f(0) =f⁰(0)−1 = 0}

and S ={f ∈A:f is univalent in U}.

In [8] the subfamily T of S consisting of functions f of the form f(z) = z−

X∞

j=2

a_jz^j, a_j ≥0, j = 2,3, ..., z ∈U (1)

39

was introduced.

Let define the Alexander operator I^p :A→ H(U) I⁰f(z) = f(z)

I¹f(z) =If(z) = Z _z

0

f(t) t dt I^pf(z) = I(I^p−1f(z)), p = 1,2,3, ...

We have for f(z) =z+ P^∞

j=2

ajz^j,

I^pf(z) = z+ X∞

j=2

1

j^pa_jz^j, p= 1,2,3, ...

Now we can define the generalized Alexander operator I^λ :A→ H(U), I^λf(z) = z+

X∞

j=2

1 j^λa_jz^j, (2)

with λ∈R, λ≥0, where f(z) = z+P^∞

j=2

a_jz^j.

The purpose of this paper is to show that the class of n-uniform star- like functions of type α and order γ with negative coefficients, the class of n-uniform close to convex functions of typeα and orderγ with negative co- efficients and some subclasses of functions with negative coefficients, which derive from the above mentioned classes, are preserved by the generalized Alexander operator.

2 Preliminary results

Let Dⁿ be the S˘al˘agean differential operator (see [6]) Dⁿ : A→ A, n ∈ N, defined as:

D⁰f(z) = f(z)

D¹f(z) =Df(z) =zf⁰(z) Dⁿf(z) =D(Dⁿ⁻¹f(z)) Remark 2.1 If f ∈T, f(z) =z−P^∞

j=2

a_jz^j, a_j ≥0, j = 2,3, ..., z ∈U then Dⁿf(z) =z− P^∞

j=2

jⁿa_jz^j.

Theorem 2.1 [6] If f(z) = z− P^∞

j=2

a_jz^j, a_j ≥ 0, j = 2,3, ..., z ∈ U then the next assertions are equivalent:

(i) P^∞

j=2

ja_j ≤1 (ii) f ∈T

(iii) f ∈ T^∗, where T^∗ = T T

S^∗ and S^∗ is the well-known class of starlike functions.

Definition 2.1 [6] Let γ ∈[0,1) and n∈N, then Sn(γ) =

½

f ∈A:ReDⁿ⁺¹f(z)

Dⁿf(z) > γ , z ∈U

¾

is the set of n-starlike functions of order γ.

We denote by T_n(γ) the subclass T T

S_n(γ).

Definition 2.2 [3] Let f ∈ A. We say that f is n-close to convex of order γ with respect to a half-plane, and denote by CC_n(γ) the set of these functions, if there exists g ∈S_n(0) so that

ReDⁿ⁺¹f(z)

Dⁿg(z) > γ, z ∈U, where n ∈N, γ ∈[0,1).

Definition 2.3 [1] Let f ∈ T, f(z) = z − P^∞

j=2

a_jz^j, a_j ≥ 0, j = 2,3, ..., z ∈ U. We say that f is in the class CCT_n(γ), γ ∈ [0,1), n ∈ N, with respect to the function g ∈T_n(0), if:

ReDⁿ⁺¹f

Dⁿg > γ , z ∈U.

Theorem 2.2 [1] Letγ ∈[0,1)andn∈N. The function f ∈T of the form (1) is inCCTn(γ), with respect to the functiong ∈Tn(0), g(z) = z−P^∞

j=2

bjz^j, b_j ≥0, j = 2,3, ..., if and only if

X∞

j=2

jⁿ[jaj + (2−α)bj]<1−γ . (3)

Definition 2.4 [4] Let f ∈A, we say thatf is n-uniform starlike function of type α if

Re

µDⁿ⁺¹f(z) Dⁿf(z)

¶

≥α·

¯¯

¯¯Dⁿ⁺¹f(z) Dⁿf(z) −1

¯¯

¯¯, z ∈U

where α ≥0, n ∈N. We denote this class with US_n(α).

We denote by UTn(α) the subclass TT

USn(α).

Definition 2.5 [3] Let f ∈A, we say that f is n-uniform close to convex function of type α in respect to the function n-uniform starlike of type α g(z), where α≥0, n ∈N, if

Re

µDⁿ⁺¹f(z) Dⁿg(z)

¶

≥α·

¯¯

¯¯Dⁿ⁺¹f(z) Dⁿg(z) −1

¯¯

¯¯, z ∈U

where α ≥0, n ∈N. We denote this class with UCC_n(α).

Definition 2.6 [5] Let f ∈A, we say that f is n-uniform starlike function of order γ and type α if

Re

µDⁿ⁺¹f(z) Dⁿf(z)

¶

≥α·

¯¯

¯¯Dⁿ⁺¹f(z) Dⁿf(z) −1

¯¯

¯¯+γ, z ∈U

where α ≥ 0, γ ∈ [−1,1), α+γ ≥ 0, n ∈ N. We denote this class with USn(α, γ).

We denote by UT_n(α, γ) the subclass T T

US_n(α, γ).

Theorem 2.3 [5] Let α ≥ 0, γ ∈ [−1,1), α+γ ≥ 0 and n ∈ N. The function f of the form (1) is in UT_n(α, γ) if and only if

X∞

j=2

jⁿ[(α+ 1)j −(α+γ)]aj ≤1−γ.

Definition 2.7 [3] Let f ∈ A, we say that f is n-uniform close to convex function of order γ and type α in respect to the function n-uniform starlike of order γ and type α, g(α), whereα ≥0, γ ∈[−1,1), α+γ ≥0, n ∈N, if

Re

µDⁿ⁺¹f(z) Dⁿg(z)

¶

≥α·

¯¯

¯¯Dⁿ⁺¹f(z) Dⁿg(z) −1

¯¯

¯¯+γ, z ∈U

where α ≥ 0, γ ∈ [−1,1), α +γ ≥ 0, n ∈ N. We denote this class with UCC_n(α, γ).

Definition 2.8 [2] Let f ∈ T, f(z) = z − P^∞

j=2

ajz^j, aj ≥ 0, j = 2,3, ..., z ∈U. We say thatf is in the class UCCT_n(α), α≥0, n∈N, with respect to the function g(z) ∈ UTn(α) g(z) = z − P^∞

j=2

bjz^j, bj ≥ 0, j = 2,3, ..., z ∈U, if:

Re

µDⁿ⁺¹f(z) Dⁿg(z)

¶

> α·

¯¯

¯¯Dⁿ⁺¹f(z) Dⁿg(z) −1

¯¯

¯¯z ∈U.

Definition 2.9 [2] Let f ∈T, f(z) = z− P^∞

j=2

a_jz^j, a_j ≥0, j = 2,3, ..., z ∈ U. We say that f is in the class UCCT_n(α, γ), α ≥0, γ ∈[−1,1), α+γ ≥ 0, n ∈N, with respect to the function g(z)∈UT_n(α, γ), g(z) =z−P^∞

j=2

b_jz^j, b_j ≥0, j = 2,3, ..., z ∈U, if

Re

µDⁿ⁺¹f(z) Dⁿg(z)

¶

≥α·

¯¯

¯¯Dⁿ⁺¹f(z) Dⁿg(z) −1

¯¯

¯¯+γ, z ∈U.

Theorem 2.4 [2] Let n ∈ N, α ≥ 0, γ ∈ [−1,1), with α+γ ≥ 0. The function f ∈ T of the form (1) is in UCCTn(α, γ), with respect to the function g ∈ UT_n(α, γ), g(z) = z − P^∞

j=2

b_jz^j, b_j ≥ 0, j = 2,3, ..., z ∈ U, if and only if

X∞

j=2

jⁿ[(α+ 1)|ja_j −b_j|+ (1−γ)b_j]≤1−γ . (4)

3 Main results

Theorem 3.1 If F ∈ UTn(α, γ), α ≥ 0, γ ∈ [−1,1), α +γ ≥ 0, n ∈ N and f = I^λ(F), where I^λ is defined by (2), then f ∈ UT_n(α, γ), α ≥ 0, γ ∈[−1,1), α+γ ≥0, n∈N.

Proof. From F(z) = z−P^∞

j=2

ajz^j, aj ≥0, j = 2,3, ... and f(z) =I^λ(F)(z) we have:

f(z) = z− X∞

j=2

α_jz^j, where α_j = 1

j^λa_j ≥0, j = 2,3, ...

From Theorem 2.3 we need only to show that:

X∞

j=2

jⁿ[(α+ 1)j−(α+γ)]α_j ≤1−γ . (5)

For λ∈R, λ≥0,a_j ≥0 and j = 2,3, ... , we have:

α_j = 1

j^λa_j ≤a_j and thus

X∞

j=2

jⁿ[(α+ 1)j−(α+γ)]αj ≤ X∞

j=2

jⁿ[(α+ 1)j−(α+γ)]aj

(6)

where α≥0,γ ∈[−1,1), α+γ ≥0 and n∈N. From F ∈UT_n(α, γ), using Theorem 2.3, we have

X∞

j=2

jⁿ[(α+ 1)j−(α+γ)]a_j ≤1−γ

and thus from (6) we obtain the condition (5).

Theorem 3.2 If F ∈UCCT_n(α, γ), α≥0, γ ∈[−1,1), α+γ ≥0, n ∈N, with respect to the function G ∈ UTn(α, γ) and f = I^λ(F), g = I^λ(G), where I^λ is defined by (2), then f ∈ UCCT_n(α, γ), α ≥ 0, γ ∈ [−1,1), α+γ ≥0, n ∈N, with respect to the function g ∈UTn(α, γ).

Proof. From the above Theorem we have g ∈UT_n(α, γ). ForF(z) =z−P^∞

j=2

ajz^j, aj ≥0, j = 2,3, ...andG(z) = z−P^∞

j=2

bjz^j, bj ≥ 0, j = 2,3, ... we have f(z) = I^λ(F)(z) = z− P^∞

j=2

αjz^j, where αj = 1 j^λaj ≥ 0, j = 2,3, ... and g(z) = I^λ(G)(z) = z − P^∞

j=2

βjz^j, where βj = 1 j^λbj ≥ 0, j = 2,3, ... .

From Theorem 2.4 we need only to show that:

X∞

j=2

jⁿ[(α+ 1)|jα_j−β_j|+ (1−γ)β_j]≤1−γ . (7)

It is easy to see that

(α+ 1)|jα_j −β_j|+ (1−γ)β_j = 1

j^λ [(α+ 1)|ja_j−b_j|+ (1−γ)b_j]≤ (8)

≤(α+ 1)|ja_j−b_j|+ (1−γ)b_j

for λ ∈ R, λ ≥ 0, aj ≥ 0, bj ≥ 0, j = 2,3, ... , α ≥ 0, γ ∈ [−1,1) and α+γ ≥0.

From F ∈ UCCTn(α, γ), with respect to the function G ∈ UTn(α, γ), we have (see Theorem 2.4):

X∞

j=2

jⁿ[(α+ 1)|ja_j−b_j|+ (1−γ)b_j]≤1−γ and thus from (8) we obtain the condition (7).

If we takeγ = 0 in Theorem 3.1 and Theorem 3.2 we obtain:

Theorem 3.3 If F ∈ UTn(α), α ≥ 0, n ∈ N and f = I^λ(F), where I^λ is defined by (2), then f ∈UT_n(α), α ≥0, n∈N.

Theorem 3.4 If F ∈UCCTn(α), α≥ 0, n ∈N, with respect to the func- tionG∈UT_n(α)andf =I^λ(F),g =I^λ(G), whereI^λis defined by (2), then f ∈UCCTn(α), α≥0, n∈N, with respect to the function g ∈UTn(α).

If we take γ ∈ [0,1) and α = 0 in Definition 2.6 we have UT_n(0, γ) = T_n(γ) and thus from Theorem 3.1, with γ ∈[0,1) and α= 0, we obtain:

Theorem 3.5 If F ∈T_n(γ), γ ∈[0,1), n ∈N and f =I^λ(F), where I^λ is defined by (2), then f ∈Tn(γ), γ ∈[0,1), n ∈N.

Remark 3.1 From Theorem 3.2 with γ ∈ [0,1) and α = 0, we obtain the preserving property of the generalized Alexander operator on the subclass UCCT_n(0, γ), γ ∈[0,1), which is not the same with the class CCT_n(γ).

In a similarly way with the proof of the Theorem 3.2, using the condition (3) instead of the condition (4), we obtain:

Theorem 3.6 If F ∈ CCT_n(γ), γ ∈ [0,1), n ∈ N, with respect to the function G∈ T_n(0) and f =I^λ(F), g =I^λ(G), where I^λ is defined by (2), thenf ∈CCT_n(γ), γ ∈[0,1),n ∈N, with respect to the functiong ∈T_n(0).

References

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University ”Lucian Blaga” of Sibiu Department of Mathematics

Str. Dr. I. Rat¸iu, Nr. 5–7, 550012 - Sibiu, Romania.

E-mail address: acu [email protected]