Subclasses of starlike functions associated with some hyperbola

Subclasses of starlike functions associated with some hyperbola

Mugur Acu

Abstract

In this paper we define some subclasses of starlike functions asso- ciated with some hyperbola by using a generalized S˘al˘agean operator and we give some properties regarding these classes.

2000 Mathematics Subject Classification: 30C45

Key words and phrases: Starlike functions, Libera-Pascu integral operator, Briot-Bouquet differential subordination, generalized S˘al˘agean

operator

1 Introduction

Let H(U) be the set of functions which are regular in the unit disc U, A = {f ∈ H(U) : f(0) = f⁰(0) − 1 = 0}, H_u(U) = {f ∈ H(U) : f is univalent in U} and S ={f ∈A:f is univalent in U}.

1Received April 4, 2006

Accepted for publication (in revised form) April 10, 2006

31

Let Dⁿ be the S˘al˘agean differential operator (see [12]) defined as:

Dⁿ :A →A , n∈N and D⁰f(z) = f(z)

D¹f(z) =Df(z) = zf⁰(z) , Dⁿf(z) = D(Dⁿ⁻¹f(z)).

Remark 1.1. If f ∈ S , f(z) = z + P^∞

j=2

a_jz^j, z ∈ U then Dⁿf(z) =z+P^∞

j=2

jⁿa_jz^j.

We recall here the definition of the well - known class of starlike functions S^∗ =

½

f ∈A:Rezf⁰(z)

f(z) >0 , z ∈U

¾ .

Let consider the Libera-Pascu integral operator L_a:A→A defined as:

f(z) =L_aF(z) = 1 +a z^a

Zz

0

F(t)·t^a−1dt , a ∈C, Re a≥0.

(1)

Generalizations of the Libera-Pascu integral operator was studied by many mathematicians such are P.T. Mocanu in [7], E. Dr˘aghici in [6] and D. Breaz in [5].

Definition 1.1.[4] Let n∈ N and λ ≥0. We denote with D_λⁿ the operator defined by

D_λⁿ:A→A ,

D⁰_λf(z) = f(z) , D_λ¹f(z) = (1−λ)f(z) +λzf⁰(z) = D_λf(z), D_λⁿf(z) = D_λ¡

D_λⁿ⁻¹f(z)¢ .

Remark 1.2.[4] We observe that Dⁿ_λ is a linear operator and for f(z) = z+

X∞

j=2

a_jz^j we have

Dⁿ_λf(z) =z+ X∞

j=2

(1 + (j−1)λ)ⁿa_jz^j.

Also, it is easy to observe that if we consider λ = 1 in the above definition we obtain the S˘al˘agean differential operator.

The next theorem is result of the so called ”admissible functions method”

introduced by P.T. Mocanu and S.S. Miller (see [8], [9], [10]).

Theorem 1.1. Let h convex in U and Re[βh(z) + γ] > 0, z ∈ U. If p ∈ H(U) with p(0) = h(0) and p satisfied the Briot-Bouquet differential subordination

p(z) + zp⁰(z)

βp(z) +γ ≺h(z), then p(z)≺h(z).

In [1] is introduced the following operator:

Definition 1.2. Let β, λ ∈R, β ≥ 0, λ ≥ 0 and f(z) = z+ X∞

j=2

a_jz^j. We denote by D_λ^β the linear operator defined by

D_λ^β :A→A ,

D^β_λf(z) = z+ X∞

j=2

(1 + (j−1)λ)^βa_jz^j.

Remark 1.3. It is easy to observe that for β = n ∈ N we obtain the Al- Oboudi operator Dⁿ_λ and for β = n ∈ N, λ = 1 we obtain the S˘al˘agean operator Dⁿ.

The purpose of this note is to define some subclasses of starlike functions associated with some hyperbola by using the operatorD^β_λ defined above and to obtain some properties regarding these classes.

2 Preliminary results

Definition 2.1. [13] A function f ∈ S is said to be in the class SH(α) if it satisfies

¯¯

¯¯zf⁰(z) f(z) −2α

³√

2−1´¯¯

¯¯< Re

½√

2zf⁰(z) f(z)

¾ + 2α

³√ 2−1

´ ,

for some α (α >0) and for all z ∈U . Remark 2.1. Geometric interpretation:

Let Ω(α) =

½zf⁰(z)

f(z) : z ∈U , f ∈SH(α)

¾ .

Then Ω(α) = {w=u+i·v : v² <4αu+u², u >0} . Note that Ω(α) is the interior of a hyperbola in the right half-plane which is symmetric about the real axis and has vertex at the origin.

Definition 2.2. [3] Let f ∈S and α >0. We say that the functionf is in the class SH_n(α), n ∈N, if

¯¯

¯¯Dⁿ⁺¹f(z)

Dⁿf(z) −2α³√

2−1´¯¯

¯¯< Re

½√

2Dⁿ⁺¹f(z) Dⁿf(z)

¾

+2α³√

2−1´

, z ∈U .

Remark 2.2. Geometric interpretation: If we denote with p_α the ana- lytic and univalent functions with the properties p_α(0) = 1, p⁰_α(0) > 0 and p_α(U) = Ω(α) (see Remark 2.1), then f ∈ SH_n(α) if and only if

Dⁿ⁺¹f(z)

Dⁿf(z) ≺ pα(z), where the symbol ≺ denotes the subordination in U . We havep_α(z) = (1 + 2α)

r1 +bz

1−z −2α , b=b(α) = 1 + 4α−4α²

(1 + 2α)² and the branch of the square root √

w is chosen so that Im√

w ≥0.

Theorem 2.1. [3] If F(z) ∈SH_n(α), α > 0, n ∈ N, and f(z) = L_aF(z), whereL_a is the integral operator defined by (1), thenf(z)∈SH_n(α), α >0, n∈N.

Theorem 2.2. [3] Let n ∈ N and α > 0. If f ∈ SH_n+1(α) then f ∈SH_n(α).

3 Main results

Definition 3.1.Letβ ≥0,λ≥0,α >0andp_α(z) = (1 + 2α)

r1 +bz

1−z −2α , whereb=b(α) = 1 + 4α−4α²

(1 + 2α)² and the branch of the square root√

wis cho- sen so that Im√

w ≥ 0. We say that a function f(z) ∈ S is in the class SH_β,λ(α) if

D^β+1_λ f(z)

D^β_λf(z) ≺p_α(z), z ∈U .

Remark 3.1. Geometric interpretation: f(z) ∈ SHβ,λ(α) if and only if D_λ^β+1f(z)

D_λ^βf(z) take all values in the domain Ω(α) which is the interior of a hyperbola in the right half-plane which is symmetric about the real axis and has vertex at the origin (see Remark 2.1 and Remark 2.2).

Remark 3.2.It is easy to observe that for β =n∈N and λ= 1 we obtain in the above definition we obtain the class SH_n(α) studied in [3] and for λ= 1, β= 0 we obtain the class SH(α) studied in [13].

Theorem 3.1. Let β ≥0, α >0 and λ >0. We have SH_β+1,λ(α)⊂SH_β,λ(α). Proof. Let f(z)∈SH_β+1,λ(α).

With notation

p(z) = D^β+1_λ f(z)

D^β_λf(z) , p(0) = 1,

we obtain

D^β+2_λ f(z)

D^β+1_λ f(z) = D_λ^β+2f(z)

D_λ^βf(z) · D^β_λf(z)

D^β+1_λ f(z) = 1

p(z)· D^β+2_λ f(z) D^β_λf(z) (2)

Also, we have

D_λ^β+2f(z) D^β_λf(z) =

z+ X∞

j=2

(1 + (j−1)λ)^β+2a_jz^j

z+ X∞

j=2

(1 + (j−1)λ)^βajz^j and

zp⁰(z) = z

³

D^β+1_λ f(z)

´₀

D_λ^βf(z) − D_λ^β+1f(z) D_λ^βf(z) · z

³

D_λ^βf(z)

´₀ D_λ^βf(z) =

= z

à 1 +

X∞

j=2

(1 + (j−1)λ)^β+1ja_jz^j−1

!

D_λ^βf(z) −

−p(z)· z

à 1 +

X∞

j=2

(1 + (j −1)λ)^βjajz^j−1

!

D_λ^βf(z) or

zp⁰(z) = z+

X∞

j=2

j(1 + (j −1)λ)^β+1a_jz^j D_λ^βf(z) − (3)

−p(z)· z+

X∞

j=2

j(1 + (j−1)λ)^βajz^j D^β_λf(z) . We have

z+ X∞

j=2

j(1 + (j−1)λ)^β+1a_jz^j =

=z+ X∞

j=2

((j−1) + 1) (1 + (j−1)λ)^β+1a_jz^j =

=z+ X∞

j=2

(1 + (j−1)λ)^β+1a_jz^j+ X∞

j=2

(j −1) (1 + (j −1)λ)^β+1a_jz^j =

=z+D_λ^β+1f(z)−z+ X∞

j=2

(j−1) (1 + (j−1)λ)^β+1a_jz^j =

=D_λ^β+1f(z) + 1 λ

X∞

j=2

((j−1)λ) (1 + (j −1)λ)^β+1a_jz^j =

=D^β+1_λ f(z) + 1 λ

X∞

j=2

(1 + (j −1)λ−1) (1 + (j−1)λ)^β+1a_jz^j =

=D^β+1_λ f(z)− 1 λ

X∞

j=2

(1 + (j−1)λ)^β+1ajz^j + 1 λ

X∞

j=2

(1 + (j−1)λ)^β+2ajz^j =

=D_λ^β+1f(z)− 1 λ

³

D^β+1_λ f(z)−z

´ + 1

λ

³

D_λ^β+2f(z)−z

´

=

=D^β+1_λ f(z)− 1

λD^β+1_λ f(z) + z λ + 1

λD_λ^β+2f(z)− z λ =

= λ−1

λ D_λ^β+1f(z) + 1

λD^β+2_λ f(z) =

= 1 λ

³

(λ−1)D_λ^β+1f(z) +D^β+2_λ f(z)

´ . Similarly we have

z+ X∞

j=2

j(1 + (j−1)λ)^βa_jz^j = 1 λ

³

(λ−1)D_λ^βf(z) +D^β+1_λ f(z)´ .

From (3) we obtain

zp⁰(z) =

= 1 λ

Ã(λ−1)D_λ^β+1f(z) +D^β+2_λ f(z)

D_λ^βf(z) −p(z)(λ−1)D_λ^βf(z) +D_λ^β+1f(z) D_λ^βf(z)

!

=

= 1 λ

Ã

(λ−1)p(z) + D_λ^β+2f(z)

D_λ^βf(z) −p(z) ((λ−1) +p(z))

!

=

= 1 λ

ÃD_λ^β+2f(z)

D_λ^βf(z) −p(z)²

!

Thus

λzp⁰(z) = D^β+2_λ f(z)

D^β_λf(z) −p(z)²

or D^β+2_λ f(z)

D^β_λf(z) =p(z)²+λzp⁰(z). From (2) we obtain

D^β+2_λ f(z) D^β+1_λ f(z) = 1

p(z)

¡p(z)²+λzp⁰(z)¢

=p(z) +λzp⁰(z) p(z) , whereλ >0.

From f(z)∈SH_β+1,λ(α) we have p(z) +λzp⁰(z)

p(z) ≺pα(z),

with p(0) = p_α(0) = 1, α > 0, β ≥ 0, λ > 0, and Re p_α(z) > 0 from here construction. In this conditions from Theorem 1.1, we obtain

p(z)≺pα(z)

or D^β+1_λ f(z)

D^β_λf(z) ≺p_α(z). This means f(z)∈SH_β,λ(α).

Theorem 3.2. Let β ≥ 0, α > 0 and λ ≥ 1. If F(z) ∈ SH_β,λ(α) then f(z) =L_aF(z) ∈SH_β,λ(α), where L_a is the Libera-Pascu integral operator defined by (1).

Proof. From (1) we have

(1 +a)F(z) =af(z) +zf⁰(z) and, by using the linear operator D^β+1_λ , we obtain

(1 +a)D^β+1_λ F(z) =aD^β+1_λ f(z) +D^β+1_λ Ã

z+ X∞

j=2

ja_jz^j

!

=

=aD_λ^β+1f(z) +z+ X∞

j=2

(1 + (j−1)λ)^β+1ja_jz^j We have (see the proof of the above theorem)

z+ X∞

j=2

j(1 + (j −1)λ)^β+1ajz^j = 1 λ

³

(λ−1)D^β+1_λ f(z) +D_λ^β+2f(z)

´

Thus

(1 +a)D^β+1_λ F(z) =aD^β+1_λ f(z) + 1 λ

³

(λ−1)D^β+1_λ f(z) +D_λ^β+2f(z)

´

=

= µ

a+λ−1 λ

¶

D^β+1_λ f(z) + 1

λD^β+2_λ f(z) or

λ(1 +a)D_λ^β+1F(z) = ((a+ 1)λ−1)D^β+1_λ f(z) +D^β+2_λ f(z). Similarly, we obtain

λ(1 +a)D_λ^βF(z) = ((a+ 1)λ−1)D^β_λf(z) +D_λ^β+1f(z). Then

D_λ^β+1F(z) D_λ^βF(z) =

D^β+2_λ f(z)

D^β+1_λ f(z) ·D_λ^β+1f(z)

D_λ^βf(z) + ((a+ 1)λ−1)· D_λ^β+1f(z) D_λ^βf(z) D_λ^β+1f(z)

D_λ^βf(z) + ((a+ 1)λ−1)

.

With notation

D^β+1_λ f(z)

D^β_λf(z) =p(z), p(0) = 1, we obtain

D^β+1_λ F(z) D^β_λF(z) =

D^β+2_λ f(z)

D^β+1_λ f(z)·p(z) + ((a+ 1)λ−1)·p(z) p(z) + ((a+ 1)λ−1) . (4)

We have (see the proof of the above theorem) λzp⁰(z) = D^β+2_λ f(z)

D^β+1_λ f(z) · D_λ^β+1f(z)

D_λ^βf(z) −p(z)² =

= D^β+2_λ f(z)

D^β+1_λ f(z) ·p(z)−p(z)².

Thus D_λ^β+2f(z)

D_λ^β+1f(z) = 1 p(z)·¡

p(z)²+λzp⁰(z)¢ .

Then, from (4), we obtain D^β+1_λ F(z)

D^β_λF(z) = p(z)²+λzp⁰(z) + ((a+ 1)λ−1)p(z) p(z) + ((a+ 1)λ−1) =

=p(z) +λ zp⁰(z)

p(z) + ((a+ 1)λ−1), wherea ∈C,Re a≥0, β≥0, and λ≥1.

From F(z)∈SH_β,λ(α) we have p(z) + zp⁰(z)

1

λ(p(z) + ((a+ 1)λ−1)) ≺pα(z),

wherea∈C, Re a≥0,α >0,β ≥0,λ≥1, and from her construction, we haveRe pα(z) >0. In this conditions we have from Theorem 1.1 we obtain

p(z)≺pα(z)

or D^β+1_λ f(z)

D_λ^βf(z) ≺p_α(z). This means f(z) = L_aF(z)∈SH_β,λ(α).

Remark 3.3. If we consider β =n∈N in the previously results we obtain the Theorem 3.1 and Theorem 3.2 from [2].

References

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University ”Lucian Blaga” of Sibiu Department of Mathematics

Str. Dr. I. Rat.iu, No. 5-7 550012 - Sibiu, Romania E-mail: acu [email protected]