Σ Σ Σ Σ Differential Subordination for Certain Analytic Function in the Upper Half-Plane C

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563

− ∈

Σ Σ

Differential Subordination for Certain Analytic Function in the Upper Half-Plane

C. Ramachandran¹ and T. Soupramanien²

1Department of Mathematics, University College of Engineering Villupuram, Anna Uni- versity, Villupuram - 605 602, Tamilnadu, india.

E-mail address: [email protected]

2Department of Mathematics, IFET College of Engineering, Gangarampalayam, Villupu- ram - 605 108, Tamilnadu, India.

E-mail address: [email protected]

Abstract

Many articles focus with differential subordination for analytic function in the unit disk, but only a few article deals with the upper half-plane. There has been no work in this area for the past one decade.

The present paper aim is to investigate differential subordination for certain analytic function in the upper half-plane associated by suitable class of admissible functions. Though this concept is an unique path in the field of Geometric function theory, it will prove to be an ebullient future study for young researchers on upper half plane.

1. INTRODUCTION

Let ∆ denote the upper half-plane; that is,

∆ = {z ∈ C : Im(z) > 0} (1)

and let H[∆] denote the class of function f which are holomorphic in ∆ and which satisfy the so-called hydrodynamic normalization (see [1, 4, 6]):

lim [f (z) z] = 0, (z ∆) (2)

z→∞

Also let S[∆] denote the class of all functions in H[∆] which are univalent in ∆. Stankie- wicz [6] introduced the following in a series of articles (see [2, 6–8]):

A function f ∈ H[∆], with f (z) ƒ= 0, is said to be starlike in ∆ if and only if f ^J(z)

Im < 0 (3)

f (z)

We denote by S^∗[∆] the subclass of H[∆] which consists of functions which are starlike in ∆.

A function f ∈ H[∆], with f (z) ƒ= z, is said to be convex in ∆ if and only if f ^JJ(z)

Im > 0 (4)

f ^J(z)

Also we denote by K[∆] the subclass of H[∆] which consists of functions which are convex in ∆.

We first need to recall the notion of subordination in the upper half-plane.

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564

z→ξ

If f and g are in H[∆], then the function f is subordinate to g, if there exists a function ϕ ∈ H[∆] with ϕ[∆] ⊂ ∆ such that f (z) = g(ϕ(z)). Furthermore, if the function g is univalent in ∆, then we have the following equivalence:

f (z) ≺ g(z) (z ∈ ∆) ⇔ f (∆) ⊂ g(∆) (5) We are extending the results of Ra˘ducanu and Pascu [5], Tang et al. [9] in the theory of differential subordinations to the upper half-plane.

Definition 1. [3] Denote by Q(∆) the set of functions q ∈ H[∆] that are analytic and injective on ∆¯ \ E(q), where

E(q) = .

ξ ∈ ∂∆; lim q(z) = ∞ Σ

(6) and are such that q^J(ξ) ƒ= 0 for ξ ∈ ∂∆ \ E(q).

Definition 2. [5] Let Ω be a set in C and q ∈ Q(∆). The class of admissible functions Ψ∆[Ω, q] consists of those functions ψ : C³×∆ → C that satisfy the following admissibility condition

whenever

ψ(r, s, t; z) ∈/ Ω, (7)

r = q(ξ), s = kq^J(ξ), Im

. t Σ

≥ k²I m

.q^JJ(ξ)Σ

(8) where z ∈ ∆, ξ ∈ ∂∆ \ E(q), and k ≥ 0.

q^J(ξ) q^J(ξ)

If ψ : C²× ∆ → C, then the admissibility condition reduces to

ψ(q(ξ), kq^J(ξ); z) ∈/ Ω, (9)

where z ∈ ∆, ξ ∈ ∂∆ \ E(q), and k ≥ 0.

Theorem 1. [5] Let ψ ∈ Ψ∆[Ω, q] and p ∈ H[∆]. If

ψ(p(z), p^J(z), p^JJ(z); z) ∈ Ω, (10) for z ∈ ∆, then

p(z) ≺ q(z) (11)

Definition 3. [9] Let Ω be a set in C and q ∈ Q(∆) ∩ H[∆]. The class of admissible functions Φ∆[Ω, q] consists of those functions φ : C³× ∆ → C that satisfy the following admissibility condition

φ(u, v, w; z) ∈/ Ω (12)

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565

q(ξ)

. Σ . Σ

−

2

. Σ

u = q(ξ), v = ^kq^J^(ξ), (q(ξ) ƒ= 0) and u(wv + v²)

Im

q^J(ξ)

≥ k Im q^JJ(ξ)

q^J(ξ) , (z ∈ ∆, ξ ∈ ∂∆ \ E(q); k ≥ 0) . (13) Definition 4. [9] Let Ω be a set in C. The class of admissible functions Φ∆[Ω, z] consists of those functions φ : C³× ∆ → C such that

k φ ξ, ,

ξ

Lξ k²

; z kξ

∈/ Ω (14)

whenever z ∈ ∆, Im(L) = 0, ξ ∈ R \ {0}, and k > 0.

In the main results, we determine some classes of admissible functions and investigate some differential subordination properties of analytic functions in the upper half-plane.

2. MAIN RESULts

Theorem 2. Let φ ∈ Φ∆[Ω, q]. If f ∈ H[∆] satisfy .

φ .f (z)

, f ^J(z) − 1 , zf ^JJ(z) − f ^J(z)

− 1

; z Σ

: z ∈ ∆ Σ

⊂ Ω (15)

then

z f (z) z zf ^J(z) − f (z) f (z) z f (z)

z ≺ q(z) (z ∈ ∆) (16)

Proof. Define the function p(z) in ∆ by p(z) = A simple calculation yields

f (z)

z (17)

f ^J(z) 1 p^J(z) and

f (z) −

z = (18)

p(z)

zf ^JJ(z) f ^J(z) 1 p^JJ(z) p^J(z) zf ^J(z) − f (z) −

f (z) − z =

p^J(z) − (19)

p(z) We now define the transformation from C³to C by

s rt − s²

u(r, s, t) = r, v(r, s, t) = , w(r, s, t) =

r rs

(20)

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566

z

Let

ψ(r, s, t; z) = φ(u, v, w; z) = φ r, s , r

rt − s² rs ; z

Σ

(21) Using (17)-(19), and from (21), we obtain

ψ(p(z), p^J(z), p^JJ(z); z) = φ .f (z)

, f ^J(z) − 1

, zf ^JJ(z) − f ^J(z) − 1

; z Σ

(22) z f (z) z zf ^J(z) − f (z) f (z) z

Hence (15) becomes

From (20), we easily get ψ(p(z), p^J(z), p^JJ(z); z) ∈ Ω (23)

t = u(wv + v²) (24)

Thus, the admissibility condition for φ ∈ Φ∆[Ω, q] in Definition 3 is equivalent to the admissibility condition for ψ as given in Definition 2. Therefore ψ ∈ Ψ∆[Ω, q] and by Theorem 1, we have p(z) ≺ q(z), or, equivalently, ^f(z) ≺ q(z). Which completes the

proof Q

If Ω ƒ= C is a simply connected domain, then Ω = h(∆) for some conformal mapping h(z) of ∆ onto Ω. In this case, the class Φ∆[h(∆), q] is written as Φ∆[h, q]. The following result is an immediate consequence of Theorem 2.

Theorem 3. Let φ ∈ Φ∆[h, q]. If f ∈ H[∆] satisfies φ

.f (z)

, f ^J(z) − 1 , zf ^JJ(z) − f ^J(z) − 1

; z Σ

≺ h(z) (25)

then

z f (z) z zf ^J(z) − f (z) f (z) z .

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567

. Σ

q(z) q^J(z) q(z) f (z)

z ≺ q(z) (z ∈ ∆) (26)

Our next result is an extension of Theorem 2 to the case where the behavior of q(z) on

∂∆ is not known.

Theorem 4. Let h and q be univalent in ∆ with q ∈ Q(∆), and set qρ(z) = q(ρz) and hρ(z) = h(ρz). Let φ : C³× ∆ → C satisfy one of the following conditions:

(1) φ ∈ Φ∆[h, qρ], for some ρ ∈ (0, 1), or

(2) there exists ρ0 ∈ (0, 1) such that φ ∈ Φ∆[hρ, qρ] for all ρ ∈ (ρ0, 1).

If f ∈ H[∆] satisfies (25), then f (z)

z ≺ q(z) (z ∈ ∆). (27)

Proof. Case(i):

By applying Theorem 2 we obtain p ≺ qρ. Since qρ ≺ q. we deduce that p ≺ q.

Case(ii):

If we let pρ(z) = p(ρz), then

ψ pρ(z), p^J_ρ(z), p^J_ρ^J(z); ρz = ψ (p(ρz), p^J(ρz), p^JJ(ρz); ρz) ∈ hρ(z) By using Theorem 2, and comment associated with

ψ(p(z), p^J(z), p^JJ(z); w(z)) ∈ Ω

where w(z) = ρz, we obtain pρ(z) ≺ qρ(z), for ρ ∈ (ρ0, 1). By letting ρ → 1⁻ we obtain

p ≺ q. Q

The next theorem gives the best dominant of the differential subordination (25).

Theorem 5. Let h be univalent in ∆ and φ : C³× ∆ → C. Suppose that the following differential equation

φ .

q(z), q^J(z)

, q^JJ(z)

− q^J(z)

; z Σ

= h(z), (28)

has a solution q(z) and satisfies one of the following conditions:

(1) q ∈ Q(∆) and φ ∈ Φ∆[h, q],

(2) q is univalent in ∆ and φ ∈ Φ∆[h, qρ] for some ρ ∈ (0, 1), or

(3) q is univalent in ∆ and there exists ρ0 ∈ (0, 1) such that φ ∈ Φ∆[hρ, qρ] for all ρ ∈ (ρ0, 1).

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568

z

. Σ

∈ If f ∈ H[∆] satisfies (25), then

and q is the best dominant.

f (z)

z ≺ q(z) (z ∈ ∆), (29)

Proof. By applying Theorems 3 and 4 we deduce that q is a dominant. Since q satisfies (28), it is a solution of (25) and therefore q will be dominated by all dominants. Hence q will

be the best dominant. Q

In particular case q(z) = z, and in view of Definition 3, the class of admissible functions Φ∆[Ω, q], denoted by Φ∆[Ω, z], is described in Definition 4.

Corollary 1. Let φ ∈ Φ∆[Ω, z]. If f ∈ H[∆] satisfies φ

.f (z)

, f ^J(z) − 1 , zf ^JJ(z) − f ^J(z) − 1

; z Σ

∈ Ω (30)

then

z f (z) z zf ^J(z) − f (z) f (z) z f (z)

z ≺ z (z ∈ ∆) (31)

For the special case Ω = q(∆) = {ω : Im(ω) > 0}, the class Φ∆[Ω, z] is simply denoted by Φ∆[∆, z]. The above corollary can now be written in the following:

Corollary 2. Let φ ∈ Φ∆[∆, z]. If f ∈ H[∆] satisfies I m

. φ

.f (z)

, f ^J(z) − 1 , zf ^JJ(z) − f ^J(z) − 1

; z ΣΣ

> 0 (32) z f (z) z zf ^J(z) − f (z) f (z) z

then

Im

.f (z)Σ

> 0 (z ∈ ∆) (33)

Next, we introduce the following class of admissible functions.

Definition 5. [9] Let Ω be a set in C and q ∈ Q(∆) ∩ H[∆]. The class of admissible functions Φ∆,1[Ω, q] consists of those functions φ : C²× ∆ → C that satisfy the following admissibility condition

φ q(ξ), q(ξ) + kq^J(ξ)

; z / Ω (34)

q(ξ) where z ∈ ∆, ξ ∈ ∂∆ \ E(q), and k ≥ 0.

Theorem 6. Let φ ∈ Φ∆,1[Ω, q]. If f ∈ H[∆] satisfies

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569

z

Σ .

φ .f (z)

, f ^J(z)

+ f (z) − 1

; z Σ

: z ∈ ∆ Σ

⊂ Ω (35)

then

z f (z) z f (z)

z ≺ q(z) (z ∈ ∆) (36)

Proof. Define the function p(z) in ∆ by p(z) = A simple calculation yields

f (z)

z (37)

f ^J(z)

+ f (z) − 1

= p(z) + p^J(z) (38)

f (z) z p(z)

We now define the transformation from C²to C by u(r, s) = r, v(r, s) = r + s

. (39)

r Let

ψ(r, s; z) = φ(u, v; z) = φ .

r, r + s

; z (40)

r

The proof will make use of Theorem 1. Using (37) and (38), and from (40), we obtain ψ(p(z), p^J(z); z) = φ

.f (z)

, f ^J(z)

+ f (z) − 1

; z Σ

(41) z f (z) z

Hence (35) becomes

ψ(p(z), p^J(z); z) ∈ Ω (42)

From (40), we see that the admissibility condition for φ ∈ Φ∆,1[Ω, q] in Definition 5 is equivalent to the admissibility condition for ψ as given in Definition 2. Hence ψ ∈ Ψ∆[Ω, q], and by Theorem 1, we have p(z) ≺ q(z) or ^f(z) ≺ q(z). Q

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570

z ξ

ξ

We will denote the class Φ∆,1[h(∆), q] by Φ∆,1[h, q], where h is the conformal mapping of

∆ onto Ω ƒ= C.

Theorem 7. Let φ ∈ Φ∆,1[h, q]. If f ∈ H[∆] satisfies

Then

Our next result is an extension of Theorem 7 to the case where the behaviour of q(z) on

∂∆ is not known.

Theorem 8. Let Ω ⊂ C and q be univalent in ∆ with q ∈ Q(∆). Let φ ∈ Φ∆,1[h, qρ], for some ρ ∈ (0, 1), where qρ(z) = q(ρz). If f ∈ H[∆] satisfies (35), then (44) holds.

As a special case, when q(z) = z, we get the following corollary.

Corollary 3. Let Ω be a set in C and let φ : C²× ∆ → C that satisfy φ

.

ξ, ξ + k

; z Σ

∈/ Ω (45)

whenever z ∈ ∆, ξ ∈ R \ {0}, and k ≥ 0. If f ∈ H[∆] satisfies φ

.f (z)

, f ^J(z)

+ f (z) − 1

; z Σ

∈ Ω (46)

z f (z) z then

Im

.f (z)Σ

> 0 (z ∈ ∆) (47)

In the special case, when Ω = q(∆) = {w : Im(w) > 0}, Corollary 3 reduces to the following corollary.

Corollary 4. Let φ : C²× ∆ → C satisfies I m

. φ

.

ξ, ξ + k

; z ΣΣ

≤ 0 (48)

Whenever z ∈ ∆, ξ ∈ R \ {0}, and k ≥ 0. If f ∈ H[∆] satisfies I m

. φ

.f (z)

, f ^J(z)

+ f (z) − 1

; z ΣΣ

> 0 (49)

z f (z) z then

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571

Im z

.f (z)Σ

> 0 (z ∈ ∆) (50)

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[8] J. Stankiewicz and Z. Stankiewicz, On the classes of functions regular in a half-plane. II, Folia Scientiarum Universitatis Technicae Resoviensis, vol. 60, no. 9, pp. 111123, 1989.

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