We introduce a new subclass N_n(λ, α, β, A, B) of Non-Bazilevič functions of order α + iβ. Some subordination relations and inequality properties are discussed. The results obtained generalize the related work of some authors. In addition, some other new results are also obtained.

1. Introduction

Let A_n denote the class of the functions f of the form

(1)

which are analytic in the open unit disk

. Let f(z) and F(z) be analytic in

. Then we say that the function f(z) is subordinate to F(z) in

if there exists an analytic function w(z) in

such that |w(z)| ≤ 1 and f(z) = F(w(z)), denoted f≺F or f(z)≺F(z). If F(z) is univalent in

, then the subordination is equivalent to f(0) = F(0) and

Assume that 0 < α < 1, a function f(z) ∈ A_n, is in N(α) if and only if

(2)

The class N(α) was introduced by Obradović [1] recently. This class of functions was said to be of Non-Bazilevič type. To this date, this class was studied in a direction of finding necessary conditions over α that embeds this class into the class of univalent functions or its subclasses which is still an open problem.

Assume that 0 < α < 1,

, −1 ≤ B ≤ 1, A ≠ B, and

, we consider the following subclass of A_n:

(3)

where all the powers are principal values, and we apply this agreement to get the following definition.

Definition 1. Let N(λ, α, μ) denote the class of functions in A_n satisfying the inequality

(4)

where 0 < α < 1,

, 0 ≤ μ < 1, and

The classes N(λ, α, A, B) and N(λ, α, μ) were studied by Wang et al. [2].

In the present paper, similarly we define the following class of analytic functions.

Definition 2. Let N_n(λ, α, β, A, B) denote the class of functions in A_n satisfying the inequality

(5)

where

, α ≥ 0,

, −1 ≤ B ≤ 1, A ≠ B, and

. All the powers in (5) are principal values.

We say that the function f(z) in this class is Non-Bazilevič functions of type α + iβ.

Definition 3. Let f(z) ∈ N_n(λ, α, β, μ) if and only if f(z) ∈ A_n and it satisfies

(6)

where

, α ≥ 0,

, 0 ≤ μ < 1, and

In particular, if β = 0, it reduces to the class N(λ, α, A, B) studied in [2].

If β = 0, λ = −1, n = 1, A = 1, and B = −1, then the class N_n(λ, α, β, A, B) reduces to the class of non-Bazilevi functions. If β = 0, λ = −1, n = 1, A = 1 − 2μ, and B = −1, then the class N_n(λ, α, β, A, B) reduces to the class of non-Bazilevič functions of order μ (0 ≤ μ < 1). Tuneski and Darus studied the Fekete-Szegö problem of the class N(−1, α, 0,1 − 2μ, −1) [3]. Other works related to Bazilevič and non-Bazilevič can be found in ([4–9]).

In the present paper, we will discuss the subordination relations and inequality properties of the class N_n(λ, α, β, A, B). The results presented here generalize and improve some known results, and some other new results are obtained.

2. Some Lemmas

Lemma 4 (see [10].)Let F(z) = 1 + b_nzⁿ + b_n+1zⁿ⁺¹ + ⋯ be analytic in and h(z) be analytic and convex in , h(0) = 1. If

(7)

where c ≠ 0 and Rec ≥ 0, then

(8)

and

is the best dominant for the differential subordination (7).

Lemma 5 (see [11].)Let −1 ≤ B₁ ≤ B₂ < A₂ ≤ A₁ ≤ 1; then

(9)

Lemma 6 (see [12].)Let F(z) be analytic and convex in , f(z) ∈ A_n, g(z) ∈ A_n. If

(10)

then

(11)

Lemma 7 (see [13].)Let be analytic in and analytic and convex in . If f(z)≺g(z), then |a_k | ≤|b_k|, for k = 1,2, ….

Lemma 8. Let , α ≥ 0, , α + iβ ≠ 0, −1 ≤ B ≤ 1, A ≠ B, and . Then f(z) ∈ N_n(λ, α, β, A, B) if and only if

(12)

where

(13)

Proof. Let

(14)

Then, by taking the derivatives of both sides of (14) and through simple calculation, we have

(15)

since f(z) ∈ N_n(λ, α, β, A, B), we have

(16)

3. Main Results

Theorem 9. Let , α ≥ 0, , α + iβ ≠ 0, −1 ≤ B ≤ 1, A ≠ B, and . If f(z) ∈ N_n(λ, α, β, A, B), then

(17)

Proof. First let F(z) = (z/f(z))^α+iβ; then F(z) = 1 + b_nzⁿ + b_n+1zⁿ⁺¹ + ⋯ is analytic in . Now, suppose that f(z) ∈ N_n(λ, α, β, A, B); by Lemma 8, we know that

(18)

It is obvious that h(z) = (1 + Az)/(1 + Bz) is analytic and convex in

, h(0) = 1. Since α + iβ ≠ 0, α ≥ 0, λ ≠ 0, and

; therefore, it follows from Lemma 4 that

(19)

Corollary 10. Let , α ≥ 0, , α + iβ ≠ 0, and μ ≠ 1. If f(z) ∈ A_n satisfies

(20)

then

(21)

or equivalent to

(22)

Corollary 11. Let , α ≥ 0, , α + iβ ≠ 0, and ; then

(23)

Theorem 12. Let 0 ≤ λ₁ ≤ λ₂, α ≥ 0, , α + iβ ≠ 0, and −1 ≤ B₁ ≤ B₂ < A₂ ≤ A₁ ≤ 1; then

(24)

Proof. Suppose that f(z) ∈ N_n(λ₂, α, β, A₂, B₂) we have f(z) ∈ A_n, and

(25)

Since −1 ≤ B₁ ≤ B₂ < A₂ ≤ A₁ ≤ 1, therefore it follows from Lemma 5 that

(26)

that is f(z) ∈ N_n(λ₂, α, β, A₁, B₁). So Theorem 12 is proved when λ₁ = λ₂ ≥ 0.

When λ₂ > λ₁ ≥ 0, then we can see from Corollary 11 that f(z) ∈ N_n(0, α, β, A₁, B₁); then

(27)

But

(28)

It is obvious that h₁(z) = (1 + A₁z)/(1 + B₁z) is analytic and convex in

. So we obtain from Lemma 6 and differential subordinations (26) and (27) that

(29)

that is, f(z) ∈ N_n(λ₁, α, β, A₁, B₁). Thus we have

(30)

Corollary 13. Let , and α + iβ ≠ 0; then

(31)

Theorem 14. Let , and . If f(z) ∈ N_n(λ, α, β, A, B), then

(32)

Proof. Suppose that f(z) ∈ N_n(λ, α, β, A, B); then from Theorem 9 we know that

(33)

Therefore, from the definition of the subordination, we have

(34)

Corollary 15. Let , and μ < 1. If f(z) ∈ N_n(λ, α, β, 1 − 2μ, −1), then

(35)

Corollary 16. Let , and μ > 1. If f(z) ∈ A_n; then

(36)

then

(37)

Corollary 17. Let , and −1 ≤ B < A ≤ 1. If f(z) ∈ N_n(λ, α, 0, A, B), then

(38)

and inequality (38) is sharp, with the extremal function defined by

(39)

Proof. Suppose that f(z) ∈ N_n(λ, α, 0, A, B); from Theorem 9 we know

(40)

Therefore, from the definition of the subordination and A > B, we have that

(41)

It is obvious that inequality (38) is sharp, with the extremal function given by (39).

Corollary 18. Let , and μ < 1. If f(z) ∈ N_n(λ, α, 0,1 − 2μ, −1), then

(42)

and inequality (42) is equivalent to

(43)

The inequality (42) is sharp, with the extremal function defined by

(44)

Corollary 19. Let , α ≥ 0, and −1 ≤ A < B ≤ 1. If f(z) ∈ N_n(λ, α, 0, A, B), then

(45)

and inequality (45) is sharp, with the extremal function given by (39).

Proof. Applying similar method as in Corollary 17, we get the result.

Corollary 20. Let , α ≥ 0, and μ > 1. If f(z) ∈ A_n satisfies

(46)

then

(47)

and inequality (47) is equivalent to

(48)

and inequality (47) is sharp, with the extremal function defined by equality (44).

If , then (see [2, 12]). So we have the following.

Corollary 21. Let , α ≥ 0, and − 1 ≤ B < A ≤ 1. If f(z) ∈ N_n(λ, α, 0, A, B), then

(49)

and inequality (49) is sharp, with the extremal function defined by equality (39).

Proof. From Theorem 9 we have

(50)

Since −1 ≤ A < B ≤ 1, we have

(51)

Thus, from inequality (38), we can get inequality (49). It is obvious that inequality (49) is sharp, with the extremal function defined by equality (39).

Corollary 22. Let , and −1 ≤ A < B ≤ 1. If f(z) ∈ N_n(λ, α, 0, A, B), then

(52)

and inequality (52) is sharp, with the extremal function defined by equality (39).

Proof. Applying similar method as in Corollary 21, we get the required result.

Remark 23. From Corollaries 21 and 22, we can generalize the corresponding results and some other special classes of analytic functions.

Corollary 24. Let , α ≥ 0, −1 ≤ B < A ≤ 1, and ; if , then one has

(53)

and inequality (53) is sharp, with the extremal function defined by equality (39).

Proof. Suppose that ; then we have

(54)

It follows from Lemma 7 that

(55)

Thus, we can get (53). Notice that

(56)

we obtain that the inequality (53) is sharp.

Remark 25. Setting λ = n = A = 1, and B = −1 in Corollary 24 we get the results obtained by [14].

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

Both authors read and approved the final paper.

Acknowledgments

The authors would like to acknowledge and appreciate the financial support received from Universiti Kebangsaan Malaysia under the Grant AP-2013-009. The authors also would like to thank the referees for the comments and suggestions to improve the paper.

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On Certain Class of Non-Bazilevič Functions of Order α + iβ Defined by a Differential Subordination

Abstract

1. Introduction

2. Some Lemmas

3. Main Results

Conflict of Interests

Authors’ Contribution

Acknowledgments

References

References

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On Certain Class of Non-Bazilevič Functions of Order α + iβ Defined by a Differential Subordination

Abstract

1. Introduction

2. Some Lemmas

3. Main Results

Conflict of Interests

Authors’ Contribution

Acknowledgments

References

References

Related

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