Abstract and Applied Analysis
Volume 2013, Issue 1 415319
Research Article
Open Access

Coefficient Estimates and Other Properties for a Class of Spirallike Functions Associated with a Differential Operator

Halit Orhan, 

Corresponding Author

Halit Orhan

Department of Mathematics, Faculty of Science, Atatürk University, 25240 Erzurum, Turkey atauni.edu.tr

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Dorina Răducanu, 

Dorina Răducanu

Department of Mathematics, Faculty of Mathematics and Computer Science, Transylvania University of Braşov, Iuliu Maniu 50, 50091 Braşov, Romania unitbv.ro

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Murat Çağlar, 

Murat Çağlar

Department of Mathematics, Faculty of Science, Atatürk University, 25240 Erzurum, Turkey atauni.edu.tr

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Mustafa Bayram, 

Mustafa Bayram

Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University, Davutpasa, 34210 Istanbul, Turkey yildiz.edu.tr

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First published: 05 June 2013
https://doi.org/10.1155/2013/415319
Citations: 9
Academic Editor: Adem Kılıçman

Abstract

For 0 ≤ η < 1, 0 ≤ λ < 1, −π/2 < γ < π/2, 0 ≤ β ≤ α, and m ∈ ℕ ∪ {0}, a new class of analytic functions defined by means of the differential operator is introduced. Our main object is to provide sharp upper bounds for Fekete-Szegö problem in . We also find sufficient conditions for a function to be in this class. Some interesting consequences of our results are pointed out.

1. Introduction

Let 𝒜 denote the class of functions f of the form
()
which are analytic in the open unit disk 𝒰 = {z ∈ ℂ:|z | < 1}.

Let 𝒮 denote the subclass of 𝒜 consisting of functions that are univalent in 𝒰.

A function f ∈ 𝒜 is said to be in the class of γ-spirallike functions of order λ in 𝒰, denoted by 𝒮*(γ, λ), if
()
for 0 ≤ λ < 1 and some real γ with |γ | < π/2.

The class 𝒮*(γ, λ) was studied by Libera [1] and Keogh and Merkes [2].

Note that 𝒮*(γ, 0) is the class of spirallike functions introduced by Špaček [3], 𝒮*(0, λ) = 𝒮*(λ) is the class of starlike functions of order λ, and 𝒮*(0,0) = 𝒮* is the familiar class of starlike functions.

For the constants λ, γ with 0 ≤ λ < 1 and |γ| < π/2, denote
()
The function pλ,γ(z) maps the open unit disk onto the half-plane Hλ,γ = {z ∈ ℂ : ℜ(eiγz) > λcos   γ}. If
()
then it is easy to check that
()
For f ∈ 𝒜 given by (1) and g ∈ 𝒜 given by
()
the Hadamard product (or convolution), denoted by f*g, is defined by
()

Denote by ℬ the family of all analytic functions w(z) that satisfy the conditions w(0) = 0 and |w(z)| < 1, z ∈ 𝒰.

A function f ∈ 𝒜 is said to be subordinate to a function g ∈ 𝒜, written f≺g, if there exists a function w ∈ ℬ such that f(z) = g(w(z)), z ∈ 𝒰.

A classical theorem of Fekete and Szegö (see [4]) states that if f ∈ 𝒮 is given by (1), then
()
This inequality is sharp in the sense that for each μ there exists a function in 𝒮 such that the equality holds. Later Pfluger (see [5]) has considered the same problem but for complex values of μ. The problem of finding sharp upper bounds for the functional for different subclasses of 𝒜 is known as the Fekete-Szegö problem. Over the years, this problem has been investigated by many authors including [6–12].
For a function f ∈ 𝒜, we consider the following differential operator introduced by Răducanu and Orhan [13]:
  • (i)

    ,

  • (ii)

    ,

  • (iii)

    ,

where 0 ≤ β ≤ α and m ∈ ℕ0 = {0,1, …}.
If the function f is given by (1), then, from the definition of the operator , it is easy to observe that
()
where
()
It should be remarked that the operator generalizes other differential operators considered earlier. For f ∈ 𝒜, we have
  • (i)

    , the operator introduced by Sălăgean [14];

  • (ii)

    , the operator studied by Al-Oboudi [15].

In view of (9), can be written in terms of convolution as
()
where
()
Define the function such that
()
It is easy to observe that
()

Making use of the differential operator , we define the following class of functions.

Definition 1. For 0 ≤ η < 1, 0 ≤ λ < 1, and |γ| < π/2, denote by the class of functions f ∈ 𝒜 which satisfy the condition

()

The class contains as particular cases the following classes of functions:

()
Also, the class consists of functions f ∈ 𝒜 satisfying the inequality
()
An analogous of the class has been recently studied by Murugusundaramoorthy [16].

The main object of this paper is to obtain sharp upper bounds for the Fekete-Szegö problem for the class . We also find sufficient conditions for a function to be in this class.

2. Membership Characterizations

In this section, we obtain several sufficient conditions for a function f ∈ 𝒜 to be in the class .

Theorem 2. Let f ∈ 𝒜, and let δ be a real number with 0 ≤ δ < 1. If

()
then provided that
()

Proof. From (18), it follows that

()
where w(z) ∈ ℬ. We have
()
provided that |γ | ≤ cos −1((1 − δ)/(1 − λ)). Thus, the proof is completed.

If in Theorem 2 we take δ = 1 − (1 − λ)cos γ, we will obtain the following result.

Corollary 3. Let f ∈ 𝒜. If

()
then .

A sufficient condition for a function f ∈ 𝒜 to be in the class , in terms of coefficients inequality, is obtained in the next theorem.

Theorem 4. If a function f ∈ 𝒜 given by (1) satisfies the inequality

()
where 0 ≤ η < 1, 0 ≤ λ < 1, |γ| < π/2, and Φn(α, β, m) is defined by (10), then it belongs to the class .

Proof. In virtue of Corollary 3, it suffices to show that the condition (22) is satisfied. We have

()

The last expression is bounded previously by (1 − λ)cos  γ, if

()
which is equivalent to
()

For special values of m, η, γ, and λ, from Theorem 4, we can derive the following sufficient conditions for a function f ∈ 𝒜 to be in the classes , and , respectively.

Corollary 5. Let f ∈ 𝒜. If

()
where 0 ≤ η < 1, 0 ≤ λ < 1, and |γ | < π/2, then .

Corollary 6 (see [17].)Let f ∈ 𝒜. If

()
where 0 ≤ λ < 1, |γ | < π/2, then f ∈ 𝒮*(γ, λ).

Corollary 7 (see [18].)Let f ∈ 𝒜. If

()
where |γ | < π/2, then f ∈ 𝒮*(γ).

A necessary and sufficient condition for a function to be in the class can be given in terms of integral representation.

Theorem 8. A function f ∈ 𝒜 is in the class if and only if there exists w ∈ ℬ such that

()
where pλ,γ(z) and are defined by (3) and (13), respectively.

Proof. In virtue of (15), if and only if there exists w ∈ ℬ such that

()
From the last equality, we obtain
()
Making use of (14) and (32), we have
()
and thus, the proof is completed.

For 0 ≤ θ ≤ 2π, 0 ≤ τ ≤ 1, define the function
()
where pλ,γ(z) and are defined by (3) and (13), respectively.

In virtue of Theorem 8, the function Ψ(z, θ, τ) belongs to the class . Note that Ψ(z, 0,0) is an odd function.

3. The Fekete-Szegö Problem

In order to obtain sharp upper bounds for the Fekete-Szegö functional for the class , the following lemma is required (see, e.g., [19, page 108]).

Lemma 9. Let the function w ∈ ℬ be given by

()
Then
()
()

The functions w(z) = z and w(z) = z2, or one of their rotations, show that both inequalities (36) and (37) are sharp.

First we obtain sharp upper bounds for the Fekete-Szegö functional with μ real parameter.

Theorem 10. Let be given by (1), and let μ be a real number. Then

()
where
()
()
and Φ2(α, β, m), Φ3(α, β, m) are defined by (10) with n = 2 and n = 3, respectively.

All estimates are sharp.

Proof. Suppose that is given by (1). Then, from the definition of the class , there exist w ∈ ℬ, w(z) = w1z + w2z2 + w3z3 + ⋯ such that

()
Set pλ,γ(z) = 1 + p1z + p2z2 + p3z3 + ⋯. Equating the coefficients of z and z2 on both sides of (41), we obtain
()
From (5), we have p1 = p2 = 2e−iγ(1 − λ)cos   γ, and thus we obtain
()

It follows that

()

Making use of Lemma 9 (36), we have

()
or
()
where
()
Denote
()
where x = cos γ, y = |w1|, and (x, y) ∈ [0,1] × [0,1].

Simple calculation shows that the function F(x, y) does not have a local maximum at any interior point of the open rectangle (0,1)×(0,1). Thus, the maximum must be attained at a boundary point. Since F(x, 0) = 1, F(0, y) = 1, and F(1,1) = |1 + M|, it follows that the maximal value of F(x, y) may be F(0,0) = 1 or F(1,1) = |1 + M|.

Therefore, from (46), we obtain

()
where M is given by (47).

Consider first the case |1 + M | ≥ 1. If μ ≤ σ1, where σ1 is given by (39), then M ≥ 0, and from (49), we obtain

()
which is the first part of the inequality (38). If μ ≥ σ2, where σ2 is given by (40), then M ≤ −2, and it follows from (49) that
()
and this is the third part of (38).

Next, suppose that σ1 ≤ μ ≤ σ2. Then, |1 + M | ≤ 1, and thus, from (49), we obtain

()
which is the second part of the inequality (38).

In view of Lemma 9, the results are sharp for w(z) = z and w(z) = z2 or one of their rotations. From (41), we obtain that the extremal functions are Ψ(z, θ, 1) and Ψ(z, θ, 0) defined by (34) with τ = 1 and τ = 0.

Next, we consider the Fekete-Szegö problem for the class with μ complex parameter.

Theorem 11. Let be given by (1), and let μ be a complex number. Then,

()
The result is sharp.

Proof. Assume that . Making use of (43), we obtain

()
The inequality (53) follows as an application of Lemma 9 (37) with
()
The functions Ψ(z, θ, 1) and Ψ(z, θ, 0) defined by (34) with τ = 1 and τ = 0 show that the inequality (53) is sharp.

Our Theorems 10 and 11 include several various results for special values of m, η, γ, and λ. For example, taking m = η = γ = λ = 0, in Theorem 10, we obtain the Fekete-Szegö inequalities for the class 𝒮* (see [2, 11]). The special case m = η = λ = 0 leads to the Fekete-Szegö inequalities for the class 𝒮*(γ) (see [2]). The Fekete-Szegö inequalities for the class 𝒮*(γ, λ) (see [2]) are also included in Theorems 10 and 11.

Acknowledgments

The authors thank the referees for their valuable suggestions to improve the paper. The first and third authors’ research was supported by Atatürk University Rectorship under “The Scientific and Research Project of Atatürk University,” Project no. 2012/173.

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