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

Abstract

For 0≤𝜂<1, 0≤𝜆<1, −𝜋/2<𝛾<𝜋/2, 0≤𝛽≤𝛼, and 𝑚∈ℕ∪{0}, a new class 𝑆^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆) of analytic functions defined by means of the differential operator 𝐷^𝑚_𝛼,𝛽 is introduced. Our main object is to provide sharp upper bounds for Fekete-Szegö problem in 𝑆^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆). 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 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛, (1) 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 ℜ𝑒^𝑖𝛾𝑧𝑓^′(𝑧)𝑓(𝑧)>𝜆cos𝛾,𝑧∈𝒰, (2) 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 Špacek [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 𝑝_𝜆,𝛾(𝑧)=1+𝑒^−𝑖𝛾𝑒^−𝑖𝛾−2𝜆cos𝛾𝑧1−𝑧,𝑧∈𝒰. (3) The function 𝑝_𝜆,𝛾(𝑧) maps the open unit disk onto the half-plane 𝐻_𝜆,𝛾={𝑧∈ℂ∶ℜ(𝑒^𝑖𝛾𝑧)>𝜆cos𝛾}. If 𝑝_𝜆,𝛾(𝑧)=1+^∞_𝑛=1𝑝_𝑛𝑧^𝑛, (4) then it is easy to check that 𝑝_𝑛=2𝑒^−𝑖𝛾(1−𝜆)cos𝛾,∀𝑛≥1. (5)

For 𝑓∈𝒜 given by (1) and 𝑔∈𝒜 given by 𝑔(𝑧)=𝑧+^∞_𝑛=2𝑏_𝑛𝑧^𝑛, (6) the Hadamard product (or convolution), denoted by 𝑓∗𝑔, is defined by 𝑓∗𝑔(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑏_𝑛𝑧^𝑛,𝑧∈𝒰. (7)

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 𝑎₃−𝜇𝑎²₂≤3−4𝜇,if𝜇≤0,1+2exp−2𝜇1−𝜇,if0≤𝜇≤1,4𝜇−3,if𝜇≥1. (8) 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 |𝑎₃−𝜇𝑎²₂| 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 Raducanu and Orhan [13]:(i)𝐷⁰_𝛼,𝛽𝑓(𝑧)=𝑓(𝑧), (ii)𝐷¹_𝛼,𝛽𝑓(𝑧)=𝐷_𝛼,𝛽𝑓(𝑧)=𝛼𝛽𝑧²𝑓^′′(𝑧)+(𝛼−𝛽)𝑧𝑓^′(𝑧)+(1−𝛼+𝛽)𝑓(𝑧), (iii)𝐷^𝑚_𝛼,𝛽𝑓(𝑧)=𝐷_𝛼,𝛽(𝐷^𝑚−1_𝛼,𝛽𝑓(𝑧)),𝑧∈𝒰, where 0≤𝛽≤𝛼 and 𝑚∈ℕ₀={0,1,…}.

If the function 𝑓 is given by (1), then, from the definition of the operator 𝐷^𝑚_𝛼,𝛽𝑓, it is easy to observe that 𝐷^𝑚_𝛼,𝛽𝑓(𝑧)=𝑧+^∞_𝑛=2Φ_𝑛𝛼,𝛽,𝑚𝑎_𝑛𝑧^𝑛, (9) where Φ_𝑛𝛼,𝛽,𝑚=1+𝛼𝛽𝑛+𝛼−𝛽(𝑛−1)^𝑚,𝑛≥2. (10)

It should be remarked that the operator 𝐷^𝑚_𝛼,𝛽𝑓 generalizes other differential operators considered earlier. For 𝑓∈𝒜, we have(i)𝐷^𝑚_1,0𝑓(𝑧)=𝐷^𝑚𝑓(𝑧), the operator introduced by Salagean [14];(ii)𝐷^𝑚_𝛼,0𝑓(𝑧)=𝐷^𝑚_𝛼𝑓(𝑧), the operator studied by Al-Oboudi [15].

In view of (9), 𝐷^𝑚_𝛼,𝛽𝑓(𝑧) can be written in terms of convolution as 𝐷^𝑚_𝛼,𝛽𝑓(𝑧)=𝑔_𝛼,𝛽∗𝑓(𝑧),𝑧∈𝒰, (11) where 𝑔_𝛼,𝛽(𝑧)=𝑧+^∞_𝑛=2Φ_𝑛𝛼,𝛽,𝑚𝑧^𝑛,𝑧∈𝒰. (12)

Define the function 𝑔⁽⁻¹⁾_𝛼,𝛽 such that 𝑔⁽⁻¹⁾_𝛼,𝛽(𝑧)∗𝑔_𝛼,𝛽(𝑧)=𝑧1−𝑧,𝑧∈𝒰. (13)

It is easy to observe that 𝑓(𝑧)=𝑔⁽⁻¹⁾_𝛼,𝛽(𝑧)∗𝐷^𝑚_𝛼,𝛽𝑓(𝑧). (14)

Making use of the differential operator 𝐷^𝑚_𝛼,𝛽𝑓, we define the following class of functions.

Definition 1. For 0≤𝜂<1, 0≤𝜆<1, and |𝛾|<𝜋/2, denote by 𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆) the class of functions 𝑓∈𝒜 which satisfy the condition ℜ𝑒^𝑖𝛾𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′1−𝜂𝐷^𝑚_𝛼,𝛽𝑓(𝑧)+𝜂𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′>𝜆cos𝛾,𝑧∈𝒰. (15)
The class 𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆) contains as particular cases the following classes of functions: 𝒮⁰_𝛼,𝛽0,𝛾,𝜆=𝒮^∗,𝒮𝛾,𝜆_0𝛼,𝛽0,𝛾,0=𝒮^∗𝛾,𝒮⁰_𝛼,𝛽(0,0,0)=𝒮^∗. (16) Also, the class 𝒮⁰_𝛼,𝛽(𝜂,𝛾,𝜆) consists of functions 𝑓∈𝒜 satisfying the inequality ℜ𝑒^𝑖𝛾𝑧𝑓^′(𝑧)1−𝜂𝑓(𝑧)+𝜂𝑧𝑓^′(𝑧)>𝜆cos𝛾,𝑧∈𝒰. (17) An analogous of the class 𝒮⁰_𝛼,𝛽(𝜂,𝛾,𝜆) 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 𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆). 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 𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆).

Theorem 2. Let 𝑓∈𝒜, and let 𝛿 be a real number with 0≤𝛿<1. If 𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′1−𝜂𝐷^𝑚_𝛼,𝛽𝑓(𝑧)+𝜂𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′−1≤1−𝛿,𝑧∈𝒰, (18) then 𝑓∈𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆) provided that 𝛾≤cos⁻¹1−𝛿1−𝜆. (19)

Proof. From (18), it follows that 𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′1−𝜂𝐷^𝑚_𝛼,𝛽𝑓(𝑧)+𝜂𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′=1+(1−𝛿)𝑤(𝑧), (20) where 𝑤(𝑧)∈ℬ. We have ℜ𝑒^𝑖𝛾𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′1−𝜂𝐷^𝑚_𝛼,𝛽𝑓(𝑧)+𝜂𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′=ℜ𝑒^𝑖𝛾(()𝑤(𝑧()ℜ𝑒1+1−𝛿))=cos𝛾+1−𝛿^𝑖𝛾𝑤(𝑧)≥cos𝛾−(1−𝛿)𝑒^𝑖𝛾𝑤(𝑧())>cos𝛾−1−𝛿≥𝜆cos𝛾, (21) provided that |𝛾|≤cos⁻¹((1−𝛿)/(1−𝜆)). Thus, the proof is completed.

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

Corollary 3. Let 𝑓∈𝒜. If 𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′1−𝜂𝐷^𝑚_𝛼,𝛽𝑓(𝑧)+𝜂𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′−1≤(1−𝜆)cos𝛾,𝑧∈𝒰 (22) then 𝑓∈𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆).

A sufficient condition for a function 𝑓∈𝒜 to be in the class 𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆), in terms of coefficients inequality, is obtained in the next theorem.

Theorem 4. If a function 𝑓∈𝒜 given by (1) satisfies the inequality 1−𝜂(^∞_𝑛=2𝑛−1)sec𝛾+(1−𝜆)1+𝜂(𝑛−1)×Φ_𝑛𝛼,𝛽,𝑚𝑎_n≤1−𝜆, (23) where 0≤𝜂<1, 0≤𝜆<1, |𝛾|<𝜋/2, and Φ_𝑛(𝛼,𝛽,𝑚) is defined by (10), then it belongs to the class 𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆).

Proof. In virtue of Corollary 3, it suffices to show that the condition (22) is satisfied. We have 𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′1−𝜂𝐷^𝑚_𝛼,𝛽𝑓(𝑧)+𝜂𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′=∑−11−𝜂^∞𝑛=2(𝑛−1)Φ_𝑛𝛼,𝛽,𝑚𝑎_𝑛𝑧^𝑛−11+∑^∞_𝑛=21−𝜂+𝜂𝑛Φ_𝑛𝛼,𝛽,𝑚𝑎_𝑛𝑧^𝑛−1<∑1−𝜂^∞𝑛=2(𝑛−1)Φ_𝑛𝛼,𝛽,𝑚𝑎_𝑛1−∑^∞_𝑛=21−𝜂+𝜂𝑛Φ_𝑛𝛼,𝛽,𝑚𝑎_𝑛. (24)
The last expression is bounded previously by (1−𝜆)cos𝛾, if 1−𝜂(^∞_𝑛=2𝑛−1)Φ_𝑛𝛼,𝛽,𝑚𝑎_𝑛≤(1−𝜆)cos𝛾1−^∞_𝑛=21−𝜂+𝜂𝑛Φ_𝑛𝛼,𝛽,𝑚𝑎_𝑛, (25) which is equivalent to 1−𝜂(^∞_𝑛=2𝑛−1)sec𝛾+(1−𝜆)1+𝜂(𝑛−1)×Φ_𝑛𝛼,𝛽,𝑚𝑎_𝑛≤1−𝜆. (26)

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 𝒮⁰_𝛼,𝛽(𝜂,𝛾,𝜆),𝒮⁰_𝛼,𝛽(0,𝛾,𝜆)=𝒮^∗(𝛾,𝜆), and 𝒮⁰_𝛼,𝛽(0,𝛾,0)=𝒮^∗(𝛾), respectively.

Corollary 5. Let 𝑓∈𝒜. If 1−𝜂(^∞_𝑛=2𝑛−1)sec𝛾+(1−𝜆)1+𝜂(𝑛−1)𝑎_𝑛≤1−𝜆, (27) where 0≤𝜂<1, 0≤𝜆<1, and |𝛾|<𝜋/2, then 𝑓∈𝒮⁰_𝛼,𝛽(𝜂,𝛾,𝜆).

Corollary 6 (see [17]). Let 𝑓∈𝒜. If (𝑛−1)^∞_𝑛=2sec𝛾+1−𝜆𝑎_𝑛≤1−𝜆, (28) where 0≤𝜆<1, |𝛾|<𝜋/2, then 𝑓∈𝒮^∗(𝛾,𝜆).

Corollary 7 (see [18]). Let 𝑓∈𝒜. If 1+(^∞_𝑛=2𝑛−1)sec𝛾𝑎_𝑛≤1, (29) where |𝛾|<𝜋/2, then 𝑓∈𝒮^∗(𝛾).

A necessary and sufficient condition for a function to be in the class 𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆) can be given in terms of integral representation.

Theorem 8. A function 𝑓∈𝒜 is in the class 𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆) if and only if there exists 𝑤∈ℬ such that 𝑓(𝑧)=𝑔⁽⁻¹⁾_𝛼,𝛽(𝑧)∗𝑧exp^𝑧₀𝑝_𝜆,𝛾(𝑤(𝜁))−11−𝜂𝑝_𝜆,𝛾(𝑤(𝜁))𝑑𝜁𝜁,𝑧∈𝒰, (30) where 𝑝_𝜆,𝛾(𝑧) and 𝑔⁽⁻¹⁾_𝛼,𝛽(𝑧) are defined by (3) and (13), respectively.

Proof. In virtue of (15), 𝑓∈𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆) if and only if there exists 𝑤∈ℬ such that 𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′1−𝜂𝐷^𝑚_𝛼,𝛽𝑓(𝑧)+𝜂𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′=𝑝_𝜆,𝛾(𝑤(𝑧)). (31) From the last equality, we obtain 𝐷^𝑚_𝛼,𝛽𝑓(𝑧)=𝑧exp^𝑧₀𝑝_𝜆,𝛾(𝑤(𝜁))−11−𝜂𝑝_𝜆,𝛾(𝑤(𝜁))𝑑𝜁𝜁. (32) Making use of (14) and (32), we have 𝑓(𝑧)=𝑔⁽⁻¹⁾_𝛼,𝛽(𝑧)∗𝑧exp^𝑧₀𝑝_𝜆,𝛾(𝑤(𝜁))−11−𝜂𝑝_𝜆,𝛾(𝑤(𝜁))𝑑𝜁𝜁,𝑧∈𝒰, (33) and thus, the proof is completed.

For 0≤𝜃≤2𝜋, 0≤𝜏≤1, define the function Ψ(𝑧,𝜃,𝜏)=𝑔⁽⁻¹⁾_𝛼,𝛽(𝑧)∗𝑧exp^𝑧₀𝑝_𝜆,𝛾𝑒^𝑖𝜃𝜁(𝜁+𝜏)/(1+𝜏𝜁)−11−𝜂𝑝_𝜆,𝛾𝑒^𝑖𝜃𝜁(𝜁+𝜏)/(1+𝜏𝜁)𝑑𝜁𝜁, (34) where 𝑝_𝜆,𝛾(𝑧) and 𝑔⁽⁻¹⁾_𝛼,𝛽(𝑧) are defined by (3) and (13), respectively.

In virtue of Theorem 8, the function Ψ(𝑧,𝜃,𝜏) belongs to the class 𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆). 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 𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆), the following lemma is required (see, e.g., [19, page 108]).

Lemma 9. Let the function 𝑤∈ℬ be given by 𝑤(𝑧)=^∞_𝑛=1𝑤_𝑛𝑧^𝑛,𝑧∈𝒰. (35) Then 𝑤₁≤1,𝑤₂≤1−𝑤₁²,𝑤₂−𝑠𝑤²₁≤max{1,|𝑠|},𝑓𝑜𝑟𝑎𝑛𝑦𝑐𝑜𝑚𝑝𝑙𝑒𝑥𝑛𝑢𝑚𝑏𝑒𝑟𝑠. (36)(37)
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 |𝑎₃−𝜇𝑎²₂| with 𝜇 real parameter.

Theorem 10. Let 𝑓∈𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆) be given by (1), and let 𝜇 be a real number. Then 𝑎₃−𝜇𝑎²₂≤(1−𝜆)cos𝛾1−𝜂²Φ₃×4()Φ𝛼,𝛽,𝑚𝜂+3−2𝜆1+𝜂−𝜇1−𝜆₃𝛼,𝛽,𝑚Φ²₂𝛼,𝛽,𝑚,if𝜇≤𝜎₁,(1−𝜆)cos𝛾1−𝜂Φ₃𝛼,𝛽,𝑚,if𝜎₁≤𝜇≤𝜎₂,(1−𝜆)cos𝛾1−𝜂²Φ₃×𝜇4()Φ𝛼,𝛽,𝑚1−𝜆₃𝛼,𝛽,𝑚Φ²₂𝛼,𝛽,𝑚+2𝜆1+𝜂−𝜂−3,if𝜇≥𝜎₂, (38) where 𝜎₁=Φ1+𝜂²²𝛼,𝛽,𝑚2Φ₃,𝜎𝛼,𝛽,𝑚₂=2−𝜆1+𝜂Φ1−𝜆²²𝛼,𝛽,𝑚2Φ₃,𝛼,𝛽,𝑚 (39)(40) 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 𝑓∈𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆) is given by (1). Then, from the definition of the class 𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆), there exist 𝑤∈ℬ, 𝑤(𝑧)=𝑤₁𝑧+𝑤₂𝑧²+𝑤₃𝑧³+⋯ such that 𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′1−𝜂𝐷^𝑚_𝛼,𝛽𝑓(𝑧)+𝜂𝑧𝐷^𝑚_𝛼,𝛽𝑓(𝑧)^′=𝑝_𝜆,𝛾(𝑤(𝑧)),𝑧∈𝒰. (41) Set 𝑝_𝜆,𝛾(𝑧)=1+𝑝₁𝑧+𝑝₂𝑧²+𝑝₃𝑧³+⋯. Equating the coefficients of 𝑧 and 𝑧² on both sides of (41), we obtain 𝑎₂=𝑝₁𝑤₁1−𝜂Φ₂,𝑎𝛼,𝛽,𝑚₃=121−𝜂Φ₃𝛼,𝛽,𝑚1+𝜂𝑝1−𝜂²¹+𝑝₂𝑤²₁+𝑝₁𝑤₂. (42) From (5), we have 𝑝₁=𝑝₂=2𝑒^−𝑖𝛾(1−𝜆)cos𝛾, and thus we obtain 𝑎₂=2𝑒^−𝑖𝛾(1−𝜆)cos𝛾1−𝜂Φ₂𝑤𝛼,𝛽,𝑚¹,𝑎₃=𝑒^−𝑖𝛾(1−𝜆)cos𝛾1−𝜂Φ₃×𝛼,𝛽,𝑚2𝑒^−𝑖𝛾(1−𝜆)cos𝛾1+𝜂𝑤1−𝜂+1²₁+𝑤₂. (43)
It follows that 𝑎₃−𝜇𝑎²₂≤(1−𝜆)cos𝛾1−𝜂Φ₃×𝛼,𝛽,𝑚2𝑒^−𝑖𝛾(1−𝜆)cos𝛾1−𝜂1+𝜂−𝜇2Φ₃𝛼,𝛽,𝑚Φ²₂×𝑤𝛼,𝛽,𝑚+1₁₂+𝑤₂. (44)
Making use of Lemma 9 (36), we have 𝑎₃−𝜇𝑎²₂≤(1−𝜆)cos𝛾1−𝜂Φ₃×𝛼,𝛽,𝑚1+2𝑒^−𝑖𝛾(1−𝜆)cos𝛾×1−𝜂1+𝜂−𝜇2Φ₃𝛼,𝛽,𝑚Φ²₂×𝑤𝛼,𝛽,𝑚+1−1₁₂ (45) or 𝑎₃−𝜇𝑎²₂≤(1−𝜆)cos𝛾1−𝜂Φ₃×𝛼,𝛽,𝑚1+1+𝑀(2+𝑀)cos²𝛾−1𝑤₁², (46) where 2()𝑀=1−𝜆1−𝜂1+𝜂−𝜇2Φ₃𝛼,𝛽,𝑚Φ²₂.𝛼,𝛽,𝑚 (47) Denote 𝐹𝑥,𝑦=1+1+𝑀(2+𝑀)𝑥²−1𝑦², (48) where 𝑥=cos𝛾, 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 𝑎₃−𝜇𝑎²₂≤(1−𝜆)cos𝛾1−𝜂Φ₃{|,𝛼,𝛽,𝑚max1,1+𝑀|} (49) 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 𝑎₃−𝜇𝑎²₂≤(1−𝜆)cos𝛾1−𝜂²Φ₃×4()Φ𝛼,𝛽,𝑚𝜂+3−2𝜆1+𝜂−𝜇1−𝜆₃𝛼,𝛽,𝑚Φ²₂,𝛼,𝛽,𝑚 (50) 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 𝑎₃−𝜇𝑎²₂≤(1−𝜆)cos𝛾1−𝜂²Φ₃×𝜇4()Φ𝛼,𝛽,𝑚1−𝜆₃𝛼,𝛽,𝑚Φ²₂,𝛼,𝛽,𝑚+2𝜆1+𝜂−𝜂−3 (51) and this is the third part of (38).
Next, suppose that 𝜎₁≤𝜇≤𝜎₂. Then, |1+𝑀|≤1, and thus, from (49), we obtain 𝑎₃−𝜇𝑎²₂≤(1−𝜆)cos𝛾1−𝜂Φ₃,𝛼,𝛽,𝑚 (52) 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 𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆) with 𝜇 complex parameter.

Theorem 11. Let 𝑓∈𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆) be given by (1), and let 𝜇 be a complex number. Then, 𝑎₃−𝜇𝑎²₂≤(1−𝜆)cos𝛾1−𝜂Φ₃2()𝛼,𝛽,𝑚×max1,1−𝜆cos𝛾×𝜇1−𝜂2Φ₃𝛼,𝛽,𝑚Φ²₂𝛼,𝛽,𝑚−1−𝜂−𝑒^𝑖𝛾. (53) The result is sharp.

Proof. Assume that 𝑓∈𝒮^𝑚_𝛼,𝛽(𝜂,𝛾,𝜆). Making use of (43), we obtain 𝑎₃−𝜇𝑎²₂≤(1−𝜆)cos𝛾1−𝜂Φ₃×𝑤𝛼,𝛽,𝑚₂−2𝑒^−𝑖𝛾(1−𝜆)cos𝛾×𝜇1−𝜂2Φ₃𝛼,𝛽,𝑚Φ²₂𝑤𝛼,𝛽,𝑚−1−𝜂−1²₁. (54) The inequality (53) follows as an application of Lemma 9 (37) with 𝑠=2𝑒^−𝑖𝛾(1−𝜆)cos𝛾𝜇1−𝜂2Φ₃𝛼,𝛽,𝑚Φ²₂𝛼,𝛽,𝑚−1−𝜂−1. (55) 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.