We introduce and study certain subclasses of analytic functions which are defined by differential subordination. Coefficient inequalities, some properties of neighborhoods, distortion and covering theorems, radius of starlikeness, and convexity for these subclasses are given.

1. Introduction

Let 𝒯(j) be the class of analytic functions f of the form

(1.1)

defined in the open unit disc 𝒰 = {z ∈ ℂ:|z | < 1}.

Let Ω be the class of functions ω analytic in 𝒰 such that ω(0) = 0, |ω(z)| < 1.

For any two functions f and g in 𝒯(j), f is said to be subordinate to g that is denoted f≺g, if there exists an analytic function ω ∈ Ω such that f(z) = g(ω(z)) [1].

Definition 1.1 (see [2].)For n ∈ ℕ and λ ≥ 0, the Al-Oboudi operator is defined as , , and .

For λ = 1, we get Sǎlǎgean differential operator [3].

Further, if

, then

(1.2)

For any function f ∈ 𝒯(j) and δ ≥ 0, the (j, δ)-neighborhood of f is defined as

(1.3)

In particular, for the identity function e(z) = z, we see that

(1.4)

The concept of neighborhoods was first introduced by Goodman [4] and then generalized by Ruscheweyh [5].

Definition 1.2. A function f ∈ 𝒯(j) is said to be in the class 𝒯_j(n, m, A, B, λ) if

(1.5)

where n ∈ ℕ₀, m ∈ ℕ, λ ≥ 1, and −1 ≤ B < A ≤ 1.

We observe that 𝒯_j(n, m, 1 − 2α, −1,1) ≡ 𝒯_j(n, m, α) [6], [7], the class of starlike functions of order α and 𝒯_j(1,1, 1 − 2α, −1,1) ≡ 𝒞_j(α) [7], the class of convex functions of order α.

2. Neighborhoods for the Class 𝒯_j(n, m, A, B, λ)

Theorem 2.1. A function f ∈ 𝒯(j) belongs to the class 𝒯_j(n, m, A, B, λ) if and only if

(2.1)

for n ∈ ℕ₀, m ∈ ℕ, λ ≥ 1, and −1 ≤ B < A ≤ 1.

Proof. Let f ∈ 𝒯_j(n, m, A, B, λ). Then,

(2.2)

Therefore,

(2.3)

Hence,

(2.4)

Thus,

(2.5)

Taking |z | = r, for sufficiently small r with 0 < r < 1, the denominator of (2.5) is positive and so it is positive for all r with 0 < r < 1, since ω(z) is analytic for |z | < 1. Then, inequality (2.5) yields

(2.6)

Equivalently,

(2.7)

and (2.1) follows upon letting r → 1.

Conversely, for |z| = r, 0 < r < 1, we have r^k < r. That is,

(2.8)

From (2.1), we have

(2.9)

This proves that

(2.10)

and hence f ∈ 𝒯_j(n, m, A, B, λ).

Theorem 2.2. If

(2.11)

then 𝒯_j(n, m, A, B, λ) ⊂ N_j,δ(e).

Proof. It follows from (2.1) that if f ∈ 𝒯_j(n, m, A, B, λ), then

(2.12)

which implies

(2.13)

Using (1.4), we get the result.

3. Neighborhoods for the Classes ℛ_j(n, A, B, λ) and 𝒫_j(n, A, B, λ)

Definition 3.1. A function f ∈ 𝒯(j) is said to be in the class ℛ_j(n, A, B, λ) if it satisfies

(3.1)

where −1 ≤ B < A ≤ 1, λ ≥ 1 and n ∈ 𝒩₀.

Definition 3.2. A function f ∈ 𝒯(j) is said to be in the class 𝒫_j(n, A, B, λ) if it satisfies

(3.2)

where −1 ≤ B < A ≤ 1, λ ≥ 1 and n ∈ 𝒩₀.

Lemma 3.3. A function f ∈ 𝒯(j) belongs to the class ℛ_j(n, A, B, λ) if and only if

(3.3)

Lemma 3.4. A function f ∈ 𝒯(j) belongs to the class 𝒫_j(n, A, B, λ) if and only if

(3.4)

Theorem 3.5. ℛ_j(n, A, B, λ) ⊂ 𝒩_j,δ(e), where

(3.5)

Proof. If f ∈ ℛ_j(n, A, B, λ), we have

(3.6)

which implies

(3.7)

Theorem 3.6. 𝒫_j(n, A, B, λ) ⊂ 𝒩_j,δ(e), where

(3.8)

Proof. If f ∈ 𝒫_j(n, A, B, λ), we have

(3.9)

which implies

(3.10)

Thus, in view of condition (1.4), we get the required result of Theorem 3.6.

4. Neighborhood of the Class

Definition 4.1. A function f ∈ 𝒯(j) is said to be in the class if it satisfies

(4.1)

for −1 ≤ B < A ≤ 1, −1 ≤ D < C ≤ 1, λ ≥ 1 and g ∈ 𝒯_j(n, m, C, D, λ).

Theorem 4.2. For g ∈ 𝒯_j(n, m, C, D, λ), one has and

(4.2)

where

(4.3)

Proof. Let f ∈ 𝒩_j,δ(g) for g ∈ 𝒯_j(n, m, C, D, λ). Then,

(4.4)

Consider

(4.5)

This implies that

5. Distortion and Covering Theorems

Theorem 5.1. If f ∈ 𝒯_j(n, m, A, B, λ), then

(5.1)

with equality for

(5.2)

Proof. In view of Theorem 2.1, we have

(5.3)

Hence,

(5.4)

This completes the proof.

Theorem 5.2. Any function f ∈ 𝒯_j(n, m, A, B, λ) maps the disk |z | < 1 onto a domain that contains the disk

(5.5)

Proof. The proof follows upon letting r → 1 in Theorem 5.1.

Theorem 5.3. If f ∈ 𝒯_j(n, m, A, B, λ), then

(5.6)

with equality for

(5.7)

Proof. We have

(5.8)

In view of Theorem 2.1,

(5.9)

Thus,

(5.10)

On the other hand,

(5.11)

This completes the proof.

6. Radii of Starlikeness and Convexity

In this section, we find the radius of starlikeness of order ρ and the radius of convexity of order ρ for functions in the class 𝒯_j(n, m, A, B, λ).

Theorem 6.1. If f ∈ 𝒯_j(n, m, A, B, λ), then f is starlike of order ρ, (0 ≤ ρ < 1) in |z | < r₁(n, m, A, B, λ, ρ), where

(6.1)

Proof. It is sufficient to show that |z(f^′(z)/f(z)) − 1| ≤ 1 − ρ (0 ≤ ρ < 1) for |z | < r₁(n, m, A, B, λ, ρ).

We have

(6.2)

Thus, |z(f^′(z)/f(z)) − 1| ≤ 1 − ρ if

(6.3)

Hence, by Theorem 2.1, (6.3) will be true if

(6.4)

or if

(6.5)

This completes the proof.

Theorem 6.2. If f ∈ 𝒯_j(n, m, A, B, λ), then f is convex of order ρ, (0 ≤ ρ < 1) in |z | < r₂(n, m, A, B, λ, ρ), where

(6.6)

Proof. It is sufficient to show that |z(f^′′(z)/f^′(z))| ≤ 1 − ρ (0 ≤ ρ < 1) for |z | < r₁(n, m, A, B, λ, ρ).

We have

(6.7)

Thus, |z(f^′′(z)/f^′(z))| ≤ 1 − ρ if

(6.8)

Hence, by Theorem 2.1, (6.8) will be true if

(6.9)

or if

(6.10)

This completes the proof.

Acknowledgment

The author wish to thank the referee for his valuable suggestions.

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All articles

On Certain Subclasses of Analytic Functions Defined by Differential Subordination

Abstract

1. Introduction

2. Neighborhoods for the Class 𝒯_j(n, m, A, B, λ)

3. Neighborhoods for the Classes ℛ_j(n, A, B, λ) and 𝒫_j(n, A, B, λ)

4. Neighborhood of the Class

5. Distortion and Covering Theorems

6. Radii of Starlikeness and Convexity

Acknowledgment

References

References

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On Certain Subclasses of Analytic Functions Defined by Differential Subordination

Abstract

1. Introduction

2. Neighborhoods for the Class 𝒯j(n, m, A, B, λ)

3. Neighborhoods for the Classes ℛj(n, A, B, λ) and 𝒫j(n, A, B, λ)

4. Neighborhood of the Class

5. Distortion and Covering Theorems

6. Radii of Starlikeness and Convexity

Acknowledgment

References

References

Related

Information

2. Neighborhoods for the Class 𝒯_j(n, m, A, B, λ)

3. Neighborhoods for the Classes ℛ_j(n, A, B, λ) and 𝒫_j(n, A, B, λ)