Fekete-Szegö Problems for Quasi-Subordination Classes

Abstract

An analytic function 𝑓 is quasi-subordinate to an analytic function 𝑔, in the open unit disk if there exist analytic functions 𝜑 and 𝑤, with |𝜑(𝑧)|≤1, 𝑤(0)=0 and |𝑤(𝑧)|<1 such that 𝑓(𝑧)=𝜑(𝑧)𝑔(𝑤(𝑧)). Certain subclasses of analytic univalent functions associated with quasi-subordination are defined and the bounds for the Fekete-Szegö coefficient functional |𝑎₃−𝜇𝑎²₂| for functions belonging to these subclasses are derived.

1. Introduction and Motivation

Let 𝒜 be the class of analytic function 𝑓 in the open unit disk 𝔻={𝑧∶|𝑧|<1} normalized by 𝑓(0)=0 and 𝑓^′(0)=1 of the form 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛. For two analytic functions 𝑓 and 𝑔, the function 𝑓 is subordinate to 𝑔, written as follows: 𝑓(𝑧)≺𝑔(𝑧), (1.1) if there exists an analytic function 𝑤, with 𝑤(0)=0 and |𝑤(𝑧)|<1 such that 𝑓(𝑧)=𝑔(𝑤(𝑧)). In particular, if the function 𝑔 is univalent in 𝔻, then 𝑓(𝑧)≺𝑔(𝑧) is equivalent to 𝑓(0)=𝑔(0) and 𝑓(𝔻)⊂𝑔(𝔻). For brief survey on the concept of subordination, see [1].

Ma and Minda [2] introduced the following class 𝒮^∗𝜙=𝑓∈𝒜∶𝑧𝑓^′(𝑧)𝑓(𝑧)≺𝜙(𝑧), (1.2) where 𝜙 is an analytic function with positive real part in 𝔻, 𝜙(𝔻) is symmetric with respect to the real axis and starlike with respect to 𝜙(0)=1 and 𝜙^′(0)>0. A function 𝑓∈𝒮^∗(𝜙) is called Ma-Minda starlike (with respect to 𝜙). The class 𝒞(𝜙) is the class of functions 𝑓∈𝒜 for which 1+𝑧𝑓^′′(𝑧)/𝑓^′(𝑧)≺𝜙(𝑧). The class 𝒮^∗(𝜙) and 𝒞(𝜙) include several well-known subclasses of starlike and convex functions as special case.

In the year 1970, Robertson [3] introduced the concept of quasi-subordination. For two analytic functions 𝑓 and 𝑔, the function 𝑓 is quasi-subordinate to 𝑔, written as follows: 𝑓(𝑧)≺_𝑞𝑔(𝑧), (1.3) if there exist analytic functions 𝜑 and 𝑤, with |𝜑(𝑧)|≤1, 𝑤(0)=0 and |𝑤(𝑧)|<1 such that 𝑓(𝑧)=𝜑(𝑧)𝑔(𝑤(𝑧)). Observe that when 𝜑(𝑧)=1, then 𝑓(𝑧)=𝑔(𝑤(𝑧)), so that 𝑓(𝑧)≺𝑔(𝑧) in 𝔻. Also notice that if 𝑤(𝑧)=𝑧, then 𝑓(𝑧)=𝜑(𝑧)𝑔(𝑧) and it is said that 𝑓 is majorized by 𝑔 and written 𝑓(𝑧)≪𝑔(𝑧) in 𝔻. Hence it is obvious that quasi-subordination is a generalization of subordination as well as majorization. See [4–6] for works related to quasi-subordination.

Throughout this paper it is assumed that 𝜙 is analytic in 𝔻 with 𝜙(0)=1. Motivated by [2, 3], we define the following classes.

Definition 1.1. Let the class 𝒮^∗_𝑞(𝜙) consists of functions 𝑓∈𝒜 satisfying the quasi-subordination 𝑧𝑓^′(𝑧)𝑓(𝑧)−1≺_𝑞𝜙(𝑧)−1. (1.4)

Example 1.2. Since 𝑧𝑓^′(𝑧)𝑓(𝑧)−1=𝑧𝜙(𝑧)−1≺_𝑞𝜙(𝑧)−1, (1.5) the function 𝑓∶𝔻→ℂ defined by the following: 𝑓(𝑧)=𝑧exp−𝑧+^𝑧₀𝜙(𝜉)𝑑𝜉 (1.6) belongs to the class 𝒮^∗_𝑞(𝜙).

Definition 1.3. Let the class 𝒞_𝑞(𝜙) consists of functions 𝑓∈𝒜 satisfying the quasi-subordination 𝑧𝑓^′′(𝑧)𝑓^′(𝑧)≺_𝑞𝜙(𝑧)−1. (1.7)

Example 1.4. The function 𝑓∶𝔻→ℂ defined by the following: 𝑓(𝑧)=^𝑧₀exp−𝜁+^𝜁₀𝜙(𝜉)𝑑𝜉𝑑𝜁 (1.8) belongs to the class 𝒞_𝑞(𝜙).

The classes 𝒮^∗_𝑞(𝜙) and 𝒞_𝑞(𝜙) are analogous to the Ma-Minda starlike and convex classes defined in the form of quasi-subordination.

Definition 1.5. Let the class ℛ_𝑞(𝜙) consist of functions 𝑓∈𝒜 satisfying the quasi-subordination 𝑓^′(𝑧)−1≺_𝑞𝜙(𝑧)−1. (1.9)

Example 1.6. The function 𝑓∶𝔻→ℂ defined by the following: 𝑓(𝑧)𝑧=𝑧−²2+exp^𝑧₀𝜙(𝜉)𝑑𝜉 (1.10) belongs to the class ℛ_𝑞(𝜙).

It is known that a function 𝑓∈𝒜 with Re𝑓^′(𝑧)>0 in 𝔻 is univalent. The above class of functions defined in terms of the quasi-subordination is associated with the class of functions with positive real part.

Functions in the following classes, ℳ_𝑞(𝛼,𝜙) and ℒ_𝑞(𝛼,𝜙) are analogous to the 𝛼-convex functions of Miller et al. [7] and 𝛼-logarithmically convex functions introduced by Lewandowski et al. [8] (see also [9]), respectively.

Definition 1.7. Let the class ℳ_𝑞(𝛼,𝜙), (𝛼≥0) consist of functions 𝑓∈𝒜 satisfying the quasi-subordination (1−𝛼)𝑧𝑓^′(𝑧)𝑓(𝑧)+𝛼1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−1≺_𝑞𝜙(𝑧)−1. (1.11)

Example 1.8. The function 𝑓∶𝔻→ℂ defined by the following: (1−𝛼)𝑧𝑓^′(𝑧)𝑓(𝑧)+𝛼1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−1=𝑧𝜙(𝑧)−1 (1.12) belongs to the class ℳ_𝑞(𝜙).

Definition 1.9. Let the class ℒ_𝑞(𝛼,𝜙), (𝛼≥0) consist of functions 𝑓∈𝒜 satisfying the quasi-subordination 𝑧𝑓^′(𝑧)𝑓(𝑧)^𝛼1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)^1−𝛼−1≺_𝑞𝜙(𝑧)−1. (1.13)

Example 1.10. The function 𝑓∶𝔻→ℂ defined by the following: 𝑧𝑓^′(𝑧)𝑓(𝑧)^𝛼1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)^1−𝛼−1=𝑧𝜙(𝑧)−1 (1.14) belongs to the class ℒ_𝑞(𝜙).

It is well known (see [10]) that the 𝑛-th coefficient of a univalent function 𝑓∈𝒜 is bounded by 𝑛. The bounds for coefficient give information about various geometric properties of the function. Many authors have also investigated the bounds for the Fekete-Szegö coefficient for various classes [11–25]. In this paper, we obtain coefficient estimates for the functions in the above defined classes.

Let Ω be the class of analytic functions 𝑤, normalized by 𝑤(0)=0, and satisfying the condition |𝑤(𝑧)|<1. We need the following lemma to prove our results.

Lemma 1.11 (see [26]). If 𝑤∈Ω, then for any complex number 𝑡𝑤₂−𝑡𝑤²₁≤max{1;|𝑡|}. (1.15) The result is sharp for the functions 𝑤(𝑧)=𝑧² or 𝑤(𝑧)=𝑧.

2. Main Results

Although Theorems 2.1 and 2.4 are contained in the corresponding results for the classes ℳ_𝑞(𝛼,𝜙) and ℒ_𝑞(𝛼,𝜙), they are stated and proved separately here because of the importance of the classes.

Throughout, let 𝑓(𝑧)=𝑧+𝑎₂𝑧²+𝑎₃𝑧³+⋯,𝜙(𝑧)=1+𝐵₁𝑧+𝐵₂𝑧²+𝐵₃𝑧³+⋯,  𝜑(𝑧)=𝑐₀+𝑐₁𝑧+𝑐₂𝑧²+𝑐₃𝑧³+⋯,  𝐵₁∈ℝ and 𝐵₁>0.

Theorem 2.1. If 𝑓∈𝒜 belongs to 𝒮^∗_𝑞(𝜙), then 𝑎₂≤𝐵₁,𝑎₃≤12𝐵₁+max𝐵₁,𝐵²₁+𝐵₂, (2.1) and, for any complex number 𝜇, 𝑎₃−𝜇𝑎²₂≤12𝐵₁+max𝐵₁,1−2𝜇𝐵²₁+𝐵₂. (2.2)

Proof. If 𝑓∈𝒮^∗_𝑞(𝜙), then there exist analytic functions 𝜑 and 𝑤, with |𝜑(𝑧)|≤1,  𝑤(0)=0 and |𝑤(𝑧)|<1 such that 𝑧𝑓^′(𝑧)𝑓(𝑧)−1=𝜑(𝑧)𝜙(𝑤(𝑧))−1. (2.3)
Since 𝑧𝑓^′(𝑧)𝑓(𝑧)−1=𝑎₂𝑧+−𝑎²₂+2𝑎₃𝑧²𝜙(𝑤(𝑧+⋯,))−1=𝐵₁𝑤₁𝑧+𝐵₁𝑤₂+𝐵₂𝑤²₁𝑧²𝜑(𝑧𝜙(𝑤(𝑧+⋯,)))−1=𝐵₁𝑐₀𝑤₁𝑧+𝐵₁𝑐₁𝑤₁+𝑐₀𝐵₁𝑤₂+𝐵₂𝑤²₁𝑧²+⋯, (2.4)(2.5) it follows from (2.3) that 𝑎₂=𝐵₁𝑐₀𝑤₁𝑎₃=12𝐵₁𝑐₁𝑤₁+𝐵₁𝑐₀𝑤₂+𝑐₀𝐵₂+𝐵²₁𝑐₀𝑤²₁. (2.6) Since 𝜑(𝑧) is analytic and bounded in 𝔻, we have [27, page 172] 𝑐_𝑛≤1−𝑐₀²≤1(𝑛>0). (2.7) By using this fact and the well-known inequality, |𝑤₁|≤1, we get 𝑎₂≤𝐵₁. (2.8) Further, 𝑎₃−𝜇𝑎²₂=12𝐵₁𝑐₁𝑤₁+𝑐₀𝐵₁𝑤₂+𝐵₂+𝐵²₁𝑐₀−2𝜇𝐵²₁𝑐₀𝑤²₁. (2.9) Then 𝑎₃−𝜇𝑎²₂≤12𝐵₁𝑐₁𝑤₁+𝐵₁𝑐₀𝑤₂−2𝜇𝐵₁𝑐₀−𝐵₁𝑐₀−𝐵₂𝐵₁𝑤²₁. (2.10) Again applying |𝑐_𝑛|≤1 and |𝑤₁|≤1, we have 𝑎₃−𝜇𝑎²₂≤𝐵₁21+𝑤₂−−1−2𝜇𝐵₁𝑐₀−𝐵₂𝐵₁𝑤²₁. (2.11) Applying Lemma 1.11 to 𝑤₂−−1−2𝜇𝐵₁𝑐₀−𝐵₂𝐵₁𝑤²₁ (2.12) yields 𝑎₃−𝜇𝑎²₂≤𝐵₁21+max1,−1−2𝜇𝐵₁𝑐₀−𝐵₂𝐵₁. (2.13) Observe that −1−2𝜇𝐵₁𝑐₀−𝐵₂𝐵₁≤𝐵₁𝑐₀+𝐵1−2𝜇²𝐵₁, (2.14) and hence we can conclude that 𝑎₃−𝜇𝑎²₂≤𝐵₁21+max1,𝐵₁+𝐵1−2𝜇²𝐵₁. (2.15) For 𝜇=0, the above will reduce to the estimate of |𝑎₃|.

Remark 2.2. For 𝜑(𝑧)≡1, Theorem 2.1 gives a particular case of the estimates in [13, Theorem 1] for 𝑝=1 and [14, Theorem 2.1] for 𝑘=1.

Theorem 2.3. If 𝑓∈𝒜 satisfies 𝑧𝑓^′(𝑧)𝑓(𝑧)−1≪𝜙(𝑧)−1, (2.16) then the following inequalities hold: 𝑎₂≤𝐵₁,𝑎₃≤12𝐵₁+𝐵²₁+𝐵₂, (2.17) and, for any complex number 𝜇, 𝑎₃−𝜇𝑎²₂≤12𝐵₁+1−2𝜇𝐵²₁+𝐵₂. (2.18)

Proof. The result follows by taking 𝑤(𝑧)=𝑧 in the proof of Theorem 2.1.

Theorem 2.4. If 𝑓∈𝒜 belongs to 𝒞_𝑞(𝜙), then 𝑎₂≤𝐵₁2,𝑎₃≤16𝐵₁+max𝐵₁,𝐵²₁+𝐵₂, (2.19) and, for any complex number 𝜇, 𝑎₃−𝜇𝑎²₂≤16𝐵₁+max𝐵₁,31−2𝜇𝐵²₁+𝐵₂. (2.20)

Proof. Observe that when 𝑧𝑓^′∈𝒮^∗_𝑞, equality (2.3) becomes 𝑧𝑧𝑓^′(𝑧)^′𝑧𝑓^′(𝑧)−1=𝜑(𝑧)𝜙(𝑤(𝑧))−1, (2.21) or equally 𝑧𝑓^′′(𝑧)𝑓^′(𝑧)≺𝜙(𝑤(𝑧))−1, (2.22) and the converse can be verified easily. By the Alexander relation, that is 𝑓∈𝒞_𝑞 if and only if 𝑧𝑓^′∈𝒮^∗_𝑞, we can obtain the required estimates.

Theorem 2.5. If 𝑓∈𝒜 satisfies 𝑧𝑓^′′(𝑧)𝑓^′(𝑧)≪𝜙(𝑧)−1, (2.23) then the following inequalities hold: 𝑎₂≤𝐵₁2,𝑎₃≤16𝐵₁+𝐵²₁+𝐵₂, (2.24) and, for any complex number 𝜇, 𝑎₃−𝜇𝑎²₂≤16𝐵₁+31−2𝜇𝐵²₁+𝐵₂. (2.25)

Theorem 2.6. If 𝑓∈𝒜 belongs to ℛ_𝑞(𝜙), then 𝑎₂≤𝐵₁2,𝑎₃≤13𝐵₁+max𝐵₁,𝐵₂, (2.26) and, for any complex number 𝜇, 𝑎₃−𝜇𝑎²₂≤13𝐵₁+max𝐵₁,34𝜇𝐵²₁+𝐵₂. (2.27)

Proof. For 𝑓∈ℛ_𝑞(𝜙), we know that by Definition 1.5 there exist analytic functions 𝜑 and 𝑤, with |𝜑(𝑧)|≤1, 𝑤(0)=0 and |𝑤(𝑧)|<1 such that 𝑓^′(𝑧)−1=𝜑(𝑧)𝜙(𝑤(𝑧))−1. (2.28) Since 𝑓^′(𝑧)−1=2𝑎₂𝑧+3𝑎₃𝑧²+⋯, (2.29) it follows from (2.28) and (2.5) that 𝑎₂=12𝐵₁𝑐₀𝑤₁,𝑎₃=13𝐵₁𝑐₁𝑤₁+𝑐₀𝐵₁𝑤₂+𝐵₂𝑤²₁. (2.30) Following the same argument as in Theorem 2.1, where |𝑐₀|≤1 and |𝑐₁|≤1, we can deduce that 𝑎₂≤𝐵₁2,𝑎₃−𝜇𝑎²₂≤𝐵₁31+𝑤₂−3𝐵₁𝑐₀4𝐵𝜇−²𝐵₁𝑤²₁. (2.31) Applying Lemma 1.11, we get 𝑎₃−𝜇𝑎²₂≤𝐵₁31+max1,3𝐵₁𝑐₀4𝐵𝜇−²𝐵₁. (2.32) Since 3𝐵₁𝑐₀4𝐵𝜇−²𝐵₁≤3𝐵₁4𝜇𝑐₀+𝐵₂𝐵₁, (2.33) and |𝑐₀|≤1 we can conclude the hypothesis.

Theorem 2.7. If 𝑓∈𝒜 satisfies 𝑓^′(𝑧)−1≪𝜙(𝑧)−1, (2.34) then the following inequalities hold: 𝑎₂≤𝐵₁2,𝑎₃≤13𝐵₁+𝐵₂, (2.35) and, for any complex number 𝜇, 𝑎₃−𝜇𝑎²₂≤13𝐵₁+34𝜇𝐵²₁+𝐵₂. (2.36)

Let the class ℛ^𝜌_𝑞(𝜙) consist of functions 𝑓∈𝒜 satisfying the quasi-subordination 1𝜌𝑓^′(𝑧)−1≺_𝑞𝜙(𝑧)−1, (2.37) where 𝜌∈ℂ⧵{0}. The following corollary gives the results for 𝑓∈ℛ^𝜌_𝑞(𝜙).

Corollary 2.8. Let 𝜌∈ℂ⧵{0}. If 𝑓∈𝒜 belongs to ℛ^𝜌_𝑞(𝜙), then 𝑎₂≤𝜌2𝐵₁,𝑎₃≤𝜌3𝐵₁+max𝐵₁,𝐵₂, (2.38) and, for any complex number 𝜇, 𝑎₃−𝜇𝑎²₂≤𝜌3𝐵₁+max𝐵₁,34𝜇𝜌𝐵²₁+𝐵₂. (2.39)

Remark 2.9. (1) For 𝜑(𝑧)≡1, Corollary 2.8 gives a particular case of the estimates in [13, Theorem 3] for 𝑝=1 and [14, Theorem 2.3] for 𝑘=1.

(2) For 𝜑(𝑧)≡1 and 𝜙(𝑧)=(1+𝐴𝑧)/(1+𝐵𝑧), (−1≤𝐵<𝐴≤1), Corollary 2.8 reduces to the results in [19, Theorem 4].

Theorem 2.10. Let 𝛼≥0. If 𝑓∈𝒜 belongs to ℳ_𝑞(𝛼,𝜙), then 𝑎₂≤𝐵₁,𝑎1+𝛼₃≤12()𝐵1+2𝛼¹+max𝐵₁,1+3𝛼(1+𝛼)²𝐵²₁+𝐵₂, (2.40) and, for any complex number 𝜇, 𝑎₃−𝜇𝑎²₂≤12()𝐵1+2𝛼¹+max𝐵₁,2𝜇(1+2𝛼)−(1+3𝛼)(1+𝛼)²𝐵²₁+𝐵₂. (2.41)

Proof. If 𝑓∈ℳ_𝑞(𝛼,𝜙), for 𝛼≥0 then there are analytic functions 𝜑 and 𝑤, with |𝜑(𝑧)|≤1, 𝑤(0)=0 and |𝑤(𝑧)|<1 such that (1−𝛼)𝑧𝑓^′(𝑧)𝑓(𝑧)+𝛼1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−1=𝜑(𝑧)𝜙(𝑤(𝑧))−1. (2.42) A computation shows that (1−𝛼)𝑧𝑓^′(𝑧)𝑓(𝑧)=(1−𝛼)+(1−𝛼)𝑎₂𝑧+(1−𝛼)−𝑎²₂+2𝑎₃𝑧²𝛼+⋯,1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)=𝛼+2𝛼𝑎₂𝑧+2𝛼−2𝑎²₂+3𝑎₃𝑧²+⋯. (2.43) Hence from (2.43), we have (1−𝛼)𝑧𝑓^′(𝑧)𝑓(𝑧)+𝛼1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−1=(1+𝛼)𝑎₂𝑧+−(1+3𝛼)𝑎²₂+2(1+2𝛼)𝑎₃𝑧²+⋯, (2.44) It then follows from relation (2.42) and (2.5) that 𝑎₂=𝐵₁𝑐₀𝑤₁,𝑎1+𝛼₃=12()𝐵1+2𝛼¹𝑐₁𝑤₁+𝐵₁𝑐₀𝑤₂+𝐵₂𝑐₀+1+3𝛼(1+𝛼)²𝐵²₁𝑐²₀𝑤²₁. (2.45) We can then conclude the proof by proceeding similarly as previous theorems.

Remark 2.11. (1) When 𝛼=0, Theorem 2.10 reduces to Theorem 2.1.
(2) When 𝛼=1, Theorem 2.10 reduces to Theorem 2.4.
(3) For 𝜑(𝑧)≡1, Theorem 2.10 gives a particular case of the estimates in [14, Theorem 2.9] for 𝑘=1.

Theorem 2.12. Let 𝛼≥0. If 𝑓∈𝒜 satisfies (1−𝛼)𝑧𝑓^′(𝑧)𝑓(𝑧)+𝛼1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−1≪𝜙(𝑧)−1, (2.46) then the following inequalities hold: 𝑎₂≤𝐵₁,𝑎1+𝛼₃≤12()𝐵1+2𝛼¹+1+3𝛼(1+𝛼)²𝐵²₁+𝐵₂, (2.47) and, for any complex number 𝜇, 𝑎₃−𝜇𝑎²₂≤12()𝐵1+2𝛼¹+2𝜇(1+2𝛼)−(1+3𝛼)(1+𝛼)²𝐵²₁+𝐵₂. (2.48)

Theorem 2.13. Let 𝛼≥0 and 𝛽=1−𝛼. If 𝑓∈𝒜 belongs to ℒ_𝑞(𝛼,𝜙), then 𝑎₂≤𝐵₁,𝑎𝛼+2𝛽₃≤12𝐵𝛼+3𝛽¹+max𝐵₁,𝛼+2𝛽²−3𝛼+4𝛽2𝛼+2𝛽²𝐵²₁+𝐵₂, (2.49) and, for any complex number 𝜇, 𝑎₃−𝜇𝑎²₂≤12𝐵𝛼+3𝛽¹+max𝐵₁,𝛼+2𝛽²−3𝛼+4𝛽−4𝜇𝛼+3𝛽2𝛼+2𝛽²𝐵²₁+𝐵₂. (2.50)

Proof. If 𝑓∈ℒ_𝑞(𝛼,𝜙), for 𝛼≥0 and 𝛽=1−𝛼 then there are analytic functions 𝜑 and 𝑤, with |𝜑(𝑧)|≤1, 𝑤(0)=0 and |𝑤(𝑧)|<1 such that 𝑧𝑓^′(𝑧)𝑓(𝑧)^𝛼1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)^𝛽−1=𝜑(𝑧)𝜙(𝑤(𝑧))−1. (2.51) A computation shows that 𝑧𝑓^′(𝑧)𝑓(𝑧)^𝛼=1+𝛼𝑎₂1𝑧+2𝛼²−3𝛼𝑎²₂+4𝛼𝑎₃𝑧²+⋯,1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)^𝛽=1+2𝛽𝑎₂𝑧+2𝛽²−3𝛽𝑎²₂+6𝛽𝑎₃𝑧²+⋯. (2.52) Thus (2.52) give 𝑧𝑓^′(𝑧)𝑓(𝑧)^𝛼1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)^𝛽=𝑎−1𝛼+2𝛽₂1𝑧+2𝛼+2𝛽²−3𝛼+4𝛽𝑎²₂+4𝛼+3𝛽𝑎₃𝑧²+⋯, (2.53) By using the above equation and (2.5) in (2.51) we have 𝑎₂=𝐵₁𝑐₀𝑤₁𝑎𝛼+2𝛽₃=𝐵₁2𝐵𝛼+3𝛽¹𝑐₁𝑤₁+𝐵₁𝑐₀𝑤₂+𝐵₂𝑐₀−𝛼+2𝛽²−3𝛼+4𝛽2𝛼+2𝛽²𝐵²₁𝑐²₀𝑤²₁. (2.54) We can proceed similarly as previous theorems and proof the hypothesis.

Remark 2.14. (1) When 𝛼=0, Theorem 2.13 reduces to Theorem 2.4.
(2) When 𝛼=1, Theorem 2.13 reduces to Theorem 2.1.
(3) For 𝜑(𝑧)≡1, Theorem 2.13 gives a particular case of the estimates in [14, Theorem 2.7] for 𝑘=1.

Theorem 2.15. Let 𝛼≥0 and 𝛽=1−𝛼. If 𝑓∈𝒜 satisfies 𝑧𝑓^′(𝑧)𝑓(𝑧)^𝛼1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)^1−𝛼−1≪𝜙(𝑧)−1, (2.55) then the following inequalities hold: 𝑎₂≤𝐵₁,𝑎𝛼+2𝛽₃≤12𝐵𝛼+3𝛽¹+𝛼+2𝛽²−3𝛼+4𝛽2𝛼+2𝛽²𝐵²₁+𝐵₂, (2.56) and, for any complex number 𝜇, 𝑎₃−𝜇𝑎²₂≤12𝐵𝛼+3𝛽¹+𝛼+2𝛽²−3𝛼+4𝛽−4𝜇𝛼+3𝛽2𝛼+2𝛽²𝐵²₁+𝐵₂. (2.57)

Acknowledgment

The work presented here was supported in part by research Grant LRGS/TD/2011/UKM/ICT/03/02. The authors are thankful to the referees for their useful comments.