A Brief Study of Certain Class of Harmonic Functions of Bazilevič Type

Abstract

We define and investigate a new subclass of Bazilevič type harmonic univalent functions using a linear operator. We investigated the harmonic structures in terms of its coefficient conditions, extreme points, distortion bounds, convolution, and convex combination. So, also, we discussed the subordination properties for the functions in this class.

1. Introduction

Let 𝐴 denote the usual class of analytic functions of the form 𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘, (1) which are analytic in the unit disk 𝑈={𝑧∶|𝑧|<1} and normalized with 𝑓(0)=0 and 𝑓^′(0)−1=0. Also, we denote the subclass of 𝐴 consisting of analytic and univalent functions 𝑓(𝑧) in the unit disk 𝑈 by 𝑆.

Here, we recall some definitions and concepts of classes of analytic functions. Let 𝑓∈𝐴. Then, 𝑓∈𝑆^∗(𝜇) if and only if Re𝑧𝑓^′(𝑧)𝑓(𝑧)>𝜇,0≤𝜇<1,𝑧∈𝑈. (2) This class is called starlike class of analytic function.

Also, let 𝑓∈𝐴. Then, 𝑓∈𝐶(𝜇) if and only if Re1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)>𝜇,0≤𝜇<1,𝑧∈𝑈. (3) This class is called convex class of analytic function. The above two classes have been repeatedly investigated by various authors like [1–4] just to mention but few, as the literatures littered everywhere.

The theory of analytic functions has wide application in many physical problem: problems as in heat conduction, electrostatic potential and fluid flows, and theory of fractals constitute practical examples. The concern of this work is the study of a particular family of analytic functions defined in a given domain by certain geometric conditions which are useful in the above problems.

Let 𝛾∶𝐶²→𝐶, and let 𝜙 be univalent in 𝑈. If 𝑝 is analytic in 𝑈 and satisfies the differential subordination Φ(𝑝(𝑧),𝑧𝑝^′(𝑧))≺𝜙(𝑧), then 𝑝 is called a solution of the differential subordination. The univalent function 𝑞 is called a dominant of the solution of the differential subordination, 𝑝≺𝑞. If 𝑝 and Φ(𝑝(𝑧),𝑧𝑝^′(𝑧)) are univalent in 𝑈 and satisfy the differential superordination 𝜙(𝑧)≺Φ(𝑝(𝑧),𝑧𝑝^′(𝑧)), then 𝑝 is called a solution of the differential superordination. An analytic function 𝑞 is called subordinate of the solution of the differential superordination if 𝑞≺𝑝. For details (see [5–7]).

Sălăgean [8] introduced the following differential operator: 𝐷⁰𝑓(𝑧)=𝑧𝑓^′(𝑧),𝐷¹𝑓(𝑧)=𝐷𝑓(𝑧)=𝑧𝑓^′(𝑧),𝐷^𝑛𝑓(𝑧)=𝐷𝐷^𝑛−1𝑓(𝑧)=𝑧𝐷^𝑛−1𝑓(𝑧)^′. (4) From (1), we can write that 𝑓(𝑧)^𝛼=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘𝛼. (5) Using binomial expansion on (5), we have 𝑓(𝑧)^𝛼=𝑧^𝛼+𝛼𝑎₂𝑧^𝛼+1+𝛼𝑎₃+𝛼(𝛼−1)𝑎2!³²𝑧^𝛼+2+𝛼𝑎₄+𝛼(𝛼−1)2!2𝑎₂𝑎₃+𝛼(𝛼−1)(𝛼−2)𝑎3!³²𝑧^𝛼+3+⋯. (6) We then define the class of analytic functions of fractional power 𝐴_𝛼 as 𝑓(𝑧)^𝛼=𝑧^𝛼+^∞_𝑘=𝛼𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1}, (7) where𝛼>0 (is real, and it is principal determination only).

Thus, we obtain the differential operator 𝐷^𝑛𝑓(𝑧)^𝛼=𝛼^𝑛𝑧^𝛼+^∞_𝑘=𝛼(𝛼+𝑘−1)^𝑛𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1}. (8) Let us also define the function 𝜑_𝛼(𝑎,𝑐,𝑧) by 𝜑_𝛼(𝑎,𝑐;𝑧)=𝑧^𝛼+^∞_𝑘=2(𝑎)_𝑘−1𝑐_𝑘−1𝑧^{𝛼+𝑘−1},𝑧∈𝑈,𝛼∈𝑅⁺,𝑎∈𝑅,𝑐∈𝑅−{0,−1,−2,…}, (9) where (𝑎_𝑘) is the Pochhammer symbol defined by (𝑎)_𝑘=𝛾(𝑎+𝑘)𝛾(𝑎)=𝑎()⋯()1,𝑘=0,𝑎+1)(𝑎+2𝑎+𝑛−1𝑛∈𝑁. (10)

Corresponding to the function 𝜑_𝛼(𝑎,𝑐;𝑧), we defined a linear operator 𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼=𝜑_𝛼(𝑎,𝑐;𝑧)∗𝐷^𝑛𝑓(𝑧)^𝛼,𝑓(𝑧)^𝛼∈𝐴_𝛼. (11) Or equivalently 𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼=𝛼^𝑛𝑧^𝛼+^∞_𝑘=2(𝑎)_𝑘−1(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1}. (12)

Remark 1. For 𝑛=0,   𝛼=1 operator (11) reduces to Carlson-shaffer operator [3], and also for different values of 𝛼, it imposes the Saitoh operator [7] and recently the Mahzoon-Lotha [9]. for 𝑎=𝑐, 𝛼=1 poses the Salagean derivative operator.

Now, let 𝐵^𝛼_𝑛(𝛾) be the class of functions containing the operator (11) and satisfying the relation 𝑧𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′𝛼𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼≺1+𝛾𝑧1−𝑧^𝛼𝛼>0,𝛾≠−1. (13) For 𝑎=𝑐, 𝛼=1, 𝛾=1, and 𝑛=0, we obtain the well-known subclass 𝑓^′(𝑧)≺1+𝑧.1−𝑧 (14) Also, for 𝑛=0, 𝑎=𝑐, and 𝛾=1, we have 𝛼𝑧𝑓^′(𝑧)𝑓(𝑧)⋅𝑓(𝑧)^𝛼𝑧^𝛼≺1+𝑧1−𝑧^𝛼. (15) For 𝑛=0, 𝛼=1, and 𝛾=1 we have the following subclass which contain Carlson-Shaffer operator: 𝑧𝐽(𝑎,𝑐)𝑓(𝑧)^′𝑧≺1+𝑧.1−𝑧 (16) The starting point in the study of functions defined in (13) is the discovery in 1995 by Russian Mathematician Bazilevič [10] of functions in 𝑈 defined by 𝑓(𝑧)=𝛼1+𝜀^2𝑧₀𝑝(𝑣)−𝑖𝜀𝑉1+𝑖𝛼𝜀/1+𝜀²𝑔(𝑣)^{𝛼/(1+𝜀2)}𝑑𝑣^{(1+𝑖𝜀)/𝛼}, (17) where 𝑝∈𝑃 and 𝑔∈𝑆^∗. The number 𝛼>0 and 𝜀 are real, and all powers are meant as principal determinant only. The family of functions in (17) became known as Bazilevič functions and is, in this work, denoted by 𝐵(𝛼,𝜀). Except that, Bazilevič showed that each function 𝑓∈𝐵(𝛼,𝜀) is univalent in 𝑈, very little is known regarding the family as a whole. However, with some simplifications, it may be possible to understand and investigate the family. Indeed, it is easy to verify that, with special choices of the parameters 𝛼 and 𝜀 and the function 𝑔(𝑧), the family 𝐵(𝛼,𝜀) cracks down to some well-known subclasses of univalent functions.

For instance, if we take 𝜀=0, we have 𝑓(𝑧)=𝛼^𝑧₀𝑝(𝑣)𝑣𝑔(𝑣)^𝛼𝑑𝑣^1/𝛼. (18) On differentiation, the expression (18) yields 𝑧𝑓^′(𝑧)𝑓(𝑧)^𝛼−1𝑔(𝑧)^𝛼=𝑝(𝑧),𝑧∈𝑈. (19) Or equivalently Re𝑧𝑓^′(𝑧)𝑓(𝑧)^𝛼−1𝑔(𝑧)^𝛼>0,𝑧∈𝑈. (20) The subclasses of Bazilevič functions satisfying (19) are called Bazilevič functions of type 𝛼 and are denoted by 𝐵(𝛼) (see [11]). In 1973, Noonan [12] gave a plausible description of functions of the class 𝐵(𝛼) as those functions in 𝑆 for which each 𝑟<1, and the tangent to the curve 𝑈_𝛼(𝑟)={𝜀𝑓(𝑟exp^𝑖𝜃)^𝛼,0≤𝜃<2𝜋} never turns back on itself as much as 𝜋 radian. If 𝛼=1, the class 𝐵(𝛼) reduces to the family of close-to-convex functions; that is, Re𝑧𝑓^′(𝑧)𝑔(𝑧)>0,𝑧∈𝑈. (21) If we decide to choose 𝑔(𝑧)=𝑓(𝑧) in (21), we have Re𝑧𝑓^′(𝑧)𝑓(𝑧)>0,𝑧∈𝑈, (22) which implies that 𝑓(𝑧) is starlike. Furthermore, if we replace 𝑓(𝑧) by 𝑧𝑓^′(𝑧) in (22), we obtain Re1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)>0,𝑧∈𝑈, (23) which shows that 𝑓(𝑧) is convex. Moreover, if 𝑔(𝑧)=𝑧 in (20), then we have the family 𝐵₁(𝛼) [11] of functions satisfying Re𝑧𝑓^′(𝑧)𝑓(𝑧)^𝛼−1𝑧^𝛼>0,𝑧∈𝑈. (24) The various subfamilies of Bazilevič functions are being studied repeatedly by many authors; the literatures in this direction littered everywhere (see Bernard's Bibliography of Schlich functions [13]).

In 1992, Abduhalim [14] introduced a generalization of functions satisfying (24) as 𝐷Re^𝑛𝑓(𝑧)^𝛼𝑧^𝛼>0,𝑧∈𝑈, (25) where the parameter 𝛼 and the operator 𝐷^𝑛 are defined as earlier. He denoted this class of functions by 𝐵_𝑛(𝛼). It is easily seen that his generalization has extraneously included analytic functions satisfying 𝑓(𝑧)Re^𝛼𝑧^𝛼>0,𝑧∈𝑈, (26) which are largely nonunivalent in the unit disk. By proving the inclusion 𝐵_𝑛+1(𝛼)⊂𝐵_𝑛(𝛼), (27) Abdulhalim was able to show that for all 𝑛∈𝑁, each function of the class 𝐵_𝑛(𝛼) is univalent in 𝑈.

Notable contributors like MacGregor, [15, 16], Noonan [12], Singh [11], Thomas [17], Tuan and Anh [18], Yamaguchi [19], and Opoola [20] had earlier considered various special cases of the parameters 𝑛 and 𝛼 of (25) and established many interesting properties of function in those particular cases.

In some general sense, it is possible to further improve work on the function defined by the geometric condition (25). Therefore, we intend to investigate this family from the viewpoints of subordination and harmonic univalent functions and determine coefficient inequalities, extreme points, distortion bounds, convolution, and convex combination.

2. Subordination Results

The objective of this section is to find the sufficient conditions of functions belonging to the class 𝐵^𝛼_𝑛(𝛾,𝑎,𝑐).

For this purpose, the following Lemmas will be necessary.

Lemma 2 (see [21]). Let 𝛽≠0 be a complex number. Let 𝑞(𝑧) (𝑞(𝑧)≠0) be a univalent function in 𝑈 such that Re1+𝑧𝑞^′′(𝑧)𝑞^′(𝑧)−𝑧𝑞^′(𝑧)𝑞(𝑧)>max0,𝑅𝛽−1𝛽𝑞(𝑧). (28) If 𝑝, (𝑝(𝑧)≠0), 𝑧∈𝑈 satisfies the differential subordination 1−𝛽𝑝(𝑧)−1+𝛽𝑧𝑝^′(𝑧)𝑝(𝑧)≺1−𝛽𝑞(𝑧)−1+𝛽𝑧𝑞^′(𝑧)𝑞(𝑧), (29) then 𝑝≺𝑞 and 𝑞 is the best dominant.

Lemma 3 (see [22]). Let 𝜔 be analytic in 𝑈 with 𝜔(0)=0. If |𝜔(𝑧)| attains its maximum value on the circle |𝑧|<1 at a point 𝑧₀, then 𝑧₀𝜔^′𝑧₀=𝑚𝜔𝑧₀, (30) where 𝑚 is a real number and 𝑚≥1.

Lemma 4 (see [23]). If 𝑓∈𝐴 satisfies Re1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)<𝜇+12,,𝜇−12≤𝜇<3,𝑧∈𝑈 (31) then 𝑧𝑓^′(𝑧)𝑓(𝑧)≺𝜇(1−𝑧),𝜇−𝑧𝑧𝑓^′(𝑧)𝑓(𝑧)−𝜇<𝜇𝜇+1.𝜇+1 (32)

Now, we begin our main results as the following.

Theorem 5. Let 𝛼>0 (𝛼 is real), 𝑛∈𝑁₀, and 𝛽≠0 be a complex number such that 𝑧Re1+1−𝛾+2𝛾𝑧1+𝛾𝑧(1−𝑧)>max0,𝑅𝛽−1𝛽1+𝛾𝑧1−𝑧^𝛼. (33) If the subordination 𝑧𝐽1−𝛽^𝛼𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′𝛼^𝑛𝑧^𝛼𝑧𝐽−1+𝛽1−𝛼+^𝛼𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′′𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′≺1−𝛽1+𝛾𝑧1−𝑧^𝛼+𝑧−1𝛼𝛽1+𝛾1+𝛾𝑧(1−𝑧) (34) holds, then 𝑓^𝛼∈𝐵^𝛼_𝑛(𝛾,𝑎,𝑐).

Proof. Suppose that 𝑝(𝑧)=𝑧𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′𝛼^𝑛𝑧^𝛼,𝑞(𝑧)=1+𝛾𝑧1−𝑧^𝛼. (35) Then, simple computations give Re1+𝑧𝑞^′′(𝑧)𝑞^′(𝑧)−𝑧𝑞^′(𝑧)𝑞(𝑧)𝑧=Re1+1−𝛾+2𝛾𝑧1+𝛾𝑧(1−𝑧)>max0,𝑅𝛽−1𝛽1+𝛾𝑧1−𝑧^𝛼=max0,𝑅𝛽−1𝛽𝑞(𝑧,𝑝(𝑧))1−𝛽−1+𝛽𝑧𝑝^′(𝑧)𝑝(𝑧)=𝑧𝐽1−𝛽^𝛼𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′𝛼^𝑛𝑧^𝛼𝑧𝐽−1+𝛽1−𝛼+^𝛼𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′′𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′≺1−𝛽1+𝛾𝑧1−𝑧^𝛼+𝑧−1𝛼𝛽1+𝛾()=𝑞(𝑧)1+𝛾𝑧1−𝑧1−𝛽−1+𝛽𝑧𝑞^′(𝑧)𝑞(𝑧). (36)(37) Thus, in the view of Lemma 2, we have 𝑓(𝑧)^𝛼∈𝐵^𝛼_𝑛(𝛾,𝑎,𝑐).

For 𝛼=1, 𝛾=1, 𝑛=0. We have the following.

Corollary 6. Let 𝛽≠0 be a complex number such that Re1+2𝑧²1−𝑧²>max0,Re𝛽−1𝛽1+𝑧.1−𝑧 (38) If the subordination 1−𝛽𝐽¹₀(𝑎,𝑐)𝑓(𝑧)^′𝑧𝐽−1+𝛽¹⁰(𝑎,𝑐)𝑓(𝑧)^′′𝐽¹₀(𝑎,𝑐)𝑓(𝑧)^′≺1−𝛽1+𝑧+1−𝑧−12𝛽𝑧1−𝑧² (39) Holds, then 𝑓(𝑧)^𝛼∈𝐵¹₀(1,𝑎,𝑐).

For 𝑎=𝑐, 𝛼=1, 𝛾=1, and 𝑛=0, we have the following

Corollary 7. Let 𝛽≠0 be a complex number such that Re1+2𝑧²1−𝑧²>max0,𝑅𝛽−1𝛽1+𝑧.1−𝑧 (40) If the subordination 1−𝛽𝑓^′(𝑧)−1+𝛽𝑧𝑓^′′(𝑧)𝑓^′(𝑧)≺1−𝛽1+𝑧+1−𝑧−12𝛽𝑧1−𝑧² (41) holds, then 𝑓(𝑧)^𝛼∈𝐵¹₀(1,1,1)≡𝐵.

For 𝛾=1, 𝑎=𝑐, we have the following.

Corollary 8. Let 𝛽≠0 be a complex number such that Re1+2𝑧²1−𝑧²>max0,Re𝛽−1𝛽1+𝑧1−𝑧^𝛼. (42) If the subordination (𝑧𝐽1−𝑧)^𝛼𝑛𝑓(𝑧)^𝛼′𝛼^𝑛𝑧^𝛼𝑧𝐽−1+𝛽1−𝛼+^𝛼𝑛𝑓(𝑧)^𝛼′′𝐽^𝛼_𝑛𝑓(𝑧)^𝛼′≺1−𝛽1+𝑧1−𝑧^𝛼−1+2𝛼𝛽𝑧1−𝑧² (43) holds, then 𝑓^𝛼∈𝐵^𝛼_𝑛(1,1,1).

For 𝛾=1, 𝑛=0, 𝑎=𝑐, we have the following.

Corollary 9. Let 𝛽≠0 be a complex number such that Re1+2𝑧²1−𝑧²>max0,Re𝛽−1𝛽1+𝑧1−𝑧^𝛼. (44) If the subordination 1−𝛽𝛼𝑧𝑓^′(𝑧)𝑓(𝑧)⋅𝑓(𝑧)^𝛼𝑧^𝛼𝑧𝑓(𝑧)−1+𝛽1−𝛽+^𝛼′′𝑓(𝑧)^𝛼′≺1−𝛽1+𝑧1−𝑧^𝛼−1+2𝛼𝛽𝑧1−𝑧² (45) holds, then 𝑓^𝛼∈𝐵^𝛼₀≡𝐵(𝛼).

Theorem 10. Let the functions 𝑓(𝑧)^𝛼 take the form (7) and satisfy 𝑧𝐽Re1−𝛼+^𝛼𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′′𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′<𝛼𝛾+12𝛾−1,𝑧∈𝑈,𝛼>0,𝛾>1. (46) Then, 𝑓∈𝐵^𝛼_𝑛(𝛾,𝑎,𝑐).

Proof. Let 𝜔 be defined by 𝑧𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′𝛼^𝑛𝑧^𝛼=1+𝛾𝜔(𝑧)1−𝜔(𝑧)^𝛼,𝜔(𝑧)≠1. (47) Then, 𝜔 is analytic in 𝑈, and since 𝛾≠−1, then 𝜔(0)=0. Also, it follows that 𝑧𝐽Re1−𝛼+^𝛼𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′′𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′𝛼=Re𝛾+1𝑧𝜔^′(𝑧)(𝑧)((𝑧<𝛼1+𝛾𝜔1−𝜔))𝛾+12𝛾−1,𝛾≠1. (48) Now, let us proceed to prove that |𝜔(𝑧)|<1. Suppose that there exists a point 𝑧₀∈𝑈 such that max_{|𝑧|≤𝑧0}|𝜔(𝑧)|=|𝜔(𝑧)|=1. (49) Then, using Lemma 3 and letting 𝜔(𝑧₀)=𝑒^𝑖𝜃 and 𝑧₀𝜔^′(𝑧₀)=𝑚𝑒^𝑖𝜃, 𝑚≥1 yields that 𝑧𝐽Re1−𝛼+^𝛼𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′′𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′𝛼𝑧=Re𝛾+1₀𝜔^′𝑧₀1+𝛾𝜔𝑧₀1−𝜔𝑧₀=Re𝛼𝑚𝑒^𝑖𝜃𝛾+11+𝛾𝜔𝑒^𝑖𝜃1−𝑒^𝑖𝜃=𝛼𝑚1+𝛾2≥𝛼𝛾−1𝛾+12().𝛾−1,𝛾>1,𝛼>0𝛼isreal (50) Thus, we have 𝑧𝐽Re1−𝛼+^𝛼𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′′𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′≥𝛼𝛾+12𝛾−1,𝑧∈𝑈, (51) which contradicts the hypothesis (46). Therefore, we conclude that |𝜔(𝑧)|<1 for all 𝑧∈𝑈 and 𝑧𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼′𝛼^𝑛𝑧^𝛼≺1+𝛾(𝑧)1−(𝑧)^𝛼,𝑧∈𝑈,𝛾≠−1,𝛼>0, (52) where (𝛼 is real). This completes the proof of the theorem.

Letting 𝛼=1, 𝑛=0, and 𝑎=𝑐 in Theorem 10, we have the following.

Corollary 11. Let the function 𝑓 take the form (1) and satisfy Re𝑧𝑓^′′(𝑧)𝑓^′(𝑧)<𝛾+12𝛾−1,𝑧∈𝑈,𝛾>1. (53) Then, 𝑓^′(𝑧)≺1+𝛾𝑧.1−𝑧 (54)

Corollary 12. Let the function 𝑓 take the form (1) and satisfy Re𝑧𝑓^′′𝑓^′(𝑧)<𝛾+12,()𝛾−1𝑧∈𝑈,𝛾>1. (55) Then, 𝑓^′(𝑧)−𝛾2<𝛾2,(𝑧∈𝑈). (56) With various special choices of the parameters involved, many existing and new subclasses of Bazilevič functions could be derived.

3. Harmonic Structure of Bazilevič Type

In this section, the authors wish to have a look into the Bazilevič type harmonic univalent functions.

A continuous complex-valued function 𝑓=𝑢+𝑖𝑣 defined in a simply connected domain 𝐷 is said to be harmonic in 𝐷. In any simply connected domain, we can write 𝑓=ℎ+𝑔, (57) where ℎ and 𝑔 are analytic in 𝐷. We call ℎ the analytic part and 𝑔 the coanalytic part of 𝐴. A necessary and sufficient condition for 𝑓 to be locally univalent and sense preserving in 𝐷 is that ℎ^′(𝑧)>𝑔^′(𝑧),(𝑧∈𝑈). (58) Denote by 𝑆_𝐻 the class of functions𝑓 of the form (57) that are harmonic univalent and sense-preserving in the disk 𝑈. The subclasses of harmonic functions have been studied by some authors for different purposes with different properties (see [24–26]). But unfortunately, it is becoming very difficult to see the literatures on Bazilevič-type harmonic univalent function, and this may be likely associated with the problem index 𝛼 always poses. This paper is designed to address this issue.

In this work, we may express the analytic functions ℎ and 𝑔 as ℎ(𝑧)^𝛼=𝑧^𝛼+^∞_𝑘=2𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1},𝑔(𝑧)^𝛼=^∞_𝑘=1𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−1},𝑏₁(𝛼)<1. (59) Therefore, 𝑓(𝑧)^𝛼=ℎ(𝑧)^𝛼+𝑔(𝑧)^𝛼. (60) We define our linear operator as given in (11) such that 𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼=𝐽^𝛼_𝑛(𝑎,𝑐)ℎ(𝑧)^𝛼+(−1)^𝑛𝐽^𝛼_𝑛(𝑎,𝑐)𝑔(𝑧)^𝛼, (61) where 𝐽^𝛼_𝑛(𝑎,𝑐)ℎ(𝑧)^𝛼=𝛼^𝑛𝑧^𝛼+^∞_𝑘=2𝑎_𝑘−1𝑐_𝑘−1(𝛼+𝑘−1)^𝑛𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1},𝐽^𝛼_𝑛(𝑎,𝑐)𝑔(𝑧)^𝛼=^∞_𝑘=1𝑎_𝑘−1𝑐_𝑘−1(𝛼+𝑘−1)^𝑛𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−1}. (62) We let 𝐵^𝛼_𝑛(𝛽,𝑎,𝑐) be the family of harmonic functions 𝑓 of the form (57) such that 𝐽Re^𝛼𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼𝛼^𝑛𝑧^𝛼>𝛽,0≤𝛽<1, (63) where 𝛼>0, (𝛼 is real), 𝑛∈𝑁₀, and 𝐽^𝛼_𝑛(𝑎,𝑐) is earlier defined in (11).

Furthermore, let the subclass 𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐) consist of harmonic functions 𝑓^𝛼_𝑛=ℎ^𝛼+𝑔^𝛼_𝑛, (64) so that ℎ^𝛼 and 𝑔^𝛼_𝑛 are of the form ℎ(𝑧)^𝛼=𝑧^𝛼+^∞_𝑘=2𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1},𝑔(𝑧)^𝛼=(−1)^𝑛+^∞_𝑘=1𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−1},𝑏₁(𝛼)<1. (65) The authors in this work wish to study the Bazilevič-type harmonic univalent functions defined by linear operator in which ℎ^𝛼 has positive coefficients. We claim that our results are quite new and not explored in the literatures.

Assigning specific values to 𝑛, 𝛽, 𝑎, 𝑐, 𝛼 in the subclass 𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐), we obtain the following subclasses which may be the expected results by using definition of earlier authors of subclasses of Bazilevič functions such as classes studied by Abduhalim [14], Yamaguchi [19], Macgregor [15], and Singh [11], just to mention but few.

We first prove a sufficient condition for the function in 𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐).

Theorem 13. Let 𝑓^𝛼=ℎ^𝛼+𝑔^𝛼, where 𝑓^𝛼 and 𝑔^𝛼 are as earlier defined if ^∞_𝑘=2(𝑎)_𝑘−1(𝑐)_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑎_𝑘(𝛼+_∞𝑘=1)(𝑎)_𝑘−1(𝑐)_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑏_𝑘(𝛼)≤1−𝛽, (66) where 𝑛∈𝑁₀, 0≤𝛽<1, 𝑎∈𝑅, 𝑐∈𝑅−{−1,−2,…} and 𝛼>0; is real then 𝑓(𝑧)^𝛼 is sense-preserving, harmonic univalent in 𝑈, and 𝑓∈𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐).

Proof. If 𝑧^𝛼₁≠𝑧^𝛼₂, then 𝑓𝑧₁^𝛼−𝑓𝑧₂^𝛼ℎ𝑧₁^𝛼−ℎ𝑧₂^𝛼𝑔𝑧≥1−^1𝛼−𝑔𝑧₂^𝛼ℎ𝑧₁^𝛼−ℎ𝑧₂^𝛼∑=1−^∞𝑘=1𝑏_𝑘(𝛼)𝑧₁^{𝛼+𝑘−1}−𝑧₂^{𝛼+𝑘−1}𝑧^𝛼₁−𝑧^𝛼₂+∑^∞_𝑘=2𝑎_𝑘(𝛼)𝑧₁^{𝛼+𝑘−1}−𝑧₂^{𝛼+𝑘−1}∑>1−^∞𝑘=1(𝛼+𝑘−1)𝑏_𝑘(𝛼)𝛼+∑^∞_𝑘=2(𝛼+𝑘−1)𝑎_𝑘(𝛼)≥1−^∞_𝑘=1(𝑎)_𝑘−1/(𝑐)_𝑘−1((𝛼+𝑘−1)/𝛼)^𝑛×1/1−𝛽𝑏_𝑘(𝛼×)1+^∞_𝑘=2(𝑎)_𝑘−1/(𝑐)_𝑘−1((𝛼+𝑘−1)/𝛼)^𝑛×1/1−𝛽𝑎_𝑘(𝛼)⁻¹≥0, (67) which proves the univalence. Note that 𝑓 is sense-preserving in 𝑈. This is because ℎ(𝑧)^𝛼≥𝛼−^∞_𝑘=2(𝛼+𝑘−1)𝑎_𝑘(𝛼)|𝑧|^{𝛼+𝑘−1}>𝛼−^∞_𝑘=2(𝛼+𝑘−1)𝑎_𝑘(𝛼)≥𝛼^𝑛−^∞_𝑘=2(𝑎)_𝑘−1(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛𝑎1−𝛽^𝑘(𝛼≥_∞𝑘=1)(𝑎)_𝑘−1(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛𝑏1−𝛽^𝑘(𝛼≥_∞𝑘=1)(𝛼+𝑘−1)𝑏_𝑘(𝛼>_∞𝑘=2)(𝛼+𝑘−1)𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−2}≥𝑔(𝑧)^𝛼. (68)
By (63) and (64), we have 𝐽Re^𝛼𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼𝛼^𝑛𝑧^𝑛𝐽=Re^𝛼𝑛(𝑎,𝑐)ℎ(𝑧)^𝛼+(−1)^𝑛𝐽^𝛼_𝑛(𝑎,𝑐)𝑔(𝑧)^𝛼𝛼^𝑛𝑧^𝑛>𝛽. (69) Using the fact that Re(𝜔)>𝛽 if and only if |1−𝛽+𝜔|≥|1+𝛽−𝜔|, it suffices to show that 𝐽1−𝛽+^𝛼𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼𝛼^𝑛𝑧^𝑛−𝐽1−𝛽−^𝛼𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼𝛼^𝑛𝑧^𝑛≥0. (70) That is, 1−𝛽𝛼^𝑛𝑧^𝑛+𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼−1−𝛽𝛼^𝑛𝑧^𝑛−𝐽^𝛼_𝑛(𝑎,𝑐)𝑓(𝑧)^𝛼𝛼≥0,1−𝛽^𝑛𝑧^𝛼+𝐽^𝛼_𝑛(𝑎,𝑐)ℎ(𝑧)^𝛼+(−1)^𝑛𝐽^𝛼_𝑛(𝑎,𝑐)𝑔(𝑧)^𝛼−1−𝛽𝛼^𝑛𝑧^𝛼−𝐽^𝛼_𝑛(𝑎,𝑐)ℎ(𝑧)^𝛼−(−1)^𝑛𝐽^𝛼_𝑛(𝑎,𝑐)𝑔(𝑧)^𝛼=𝛼^𝑛𝑧^𝛼−𝛽𝛼^𝑛𝑧^𝛼+𝛼^𝑛𝑧^𝛼+^∞_𝑘=2(𝑎)_𝑘−1(𝑐)_𝑘−2(𝛼+𝑘−1)^𝑛𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1}+(−1)^𝑛∞_𝑘=1(𝑎)_𝑘−1(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−1}−𝛼^𝑛𝑧^𝛼+𝛽𝛼^𝑛𝑧^𝛼−𝛼^𝑛𝑧^𝛼−^∞_𝑘=2(𝑎)_𝑘−1(𝑐)_𝑘−2(𝛼+𝑘−1)^𝑛𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1}−(−1)^𝑛∞_𝑘=1(𝑎)_𝑘−1(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−1},=𝛼^𝑛2−𝛽𝑧^𝛼+^∞_𝑘=2(𝑎)_𝑘−1(𝑐)_𝑘−2(𝛼+𝑘−1)^𝑛𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1}+(−1)^𝑛∞_𝑘=1(𝑎)_𝑘−1(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−1}−𝛽𝛼^𝑛𝑧^𝛼−^∞_𝑘=2(𝑎)_𝑘−1(𝑐)_𝑘−2(𝛼+𝑘−1)^𝑛𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1}−(−1)^𝑛∞_𝑘=1(𝑎)_𝑘−1(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−1}≥2𝛼^𝑛1−𝛽|𝑧|^𝛼+^∞_𝑘=22(𝑎)_𝑘−1(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛𝑎_𝑘(𝛼)|𝑧|^{𝛼+𝑘−1}−(−1)^𝑛∞_𝑘=12(𝑎)_𝑘−1(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛𝑏_𝑘(𝛼)|𝑧|^{𝛼+𝑘−1}×2𝛼^𝑛1−𝛽1+^∞_𝑘=21(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛×𝑎_𝑘(𝛼+())−1^𝑛∞_𝑘=11(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1×𝛼+𝑘−1𝛼^𝑛𝑏_𝑘(𝛼). (71) This last expression is nonnegative by (66), and so the proof is complete.

The harmonic function 𝑓(𝑧)^𝛼=𝑧^𝛼+^∞_𝑘=2(𝑐)_𝑘−11−𝛽(𝑎)_𝑘−1𝛼𝛼+𝑘−1^𝑛𝑥_𝑘𝑧^{𝛼+𝑘−1}+^∞_𝑘=1(𝑐)_𝑘−11−𝛽(𝑎)_𝑘−1𝛼𝛼+𝑘−1^𝑛𝑦^{𝛼+𝑘−1}, (72) where 𝑛∈𝑁₀, 𝛼>0 (𝛼 is real), 0≤𝛽<1, 𝑎∈𝑅, 𝑐∈𝑅−{−1,0,1…} and ^∞_𝑘=2𝑥_𝑘+^∞_𝑘=1𝑦_𝑘=1, (73) shows that the coefficient bound given by (66) is sharp. The functions of the form (72) are in 𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐) because ^∞_𝑘=21(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑎_𝑘(𝛼+_∞𝑘=1)1(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑏_𝑘(𝛼=_∞𝑘=2)𝑥_𝑘+^∞_𝑘=1𝑦_𝑘=1. (74) In the following theorem, it is shown that the condition (66) is also necessary for functions 𝑓^𝛼_𝑛=ℎ^𝛼+𝑔^𝛼_𝑛, where ℎ^𝛼 and 𝑔^𝛼_𝑛 are as earlier defined.

Theorem 14. Let 𝑓^𝛼_𝑛=ℎ^𝛼+𝑔^𝛼_𝑛. Then, 𝑓_𝑛∈𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐) if and only if ^∞_𝑘=21(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑎_𝑘(𝛼+_∞𝑘=1)1(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑏_𝑘(𝛼)≤1. (75)

Proof. Since 𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐)⊂𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐), we only need to prove the “only if” part of the theorem. To this end, for functions 𝑓^𝛼_𝑛 of the form (64), we notice that the condition 𝐽Re^𝛼𝑛(𝑎,𝑐)𝑓_𝑛(𝑧)^𝛼𝛼^𝑛𝑧^𝛼>𝛽 (76) is equivalent to Re1−𝛽𝑧^𝛼−^∞_𝑘=2(𝑎)_𝑘−1(𝑐)_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1}−(−1)^∞2𝑛_𝑘=1(𝑎)_𝑘−1(𝑐)_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−1}𝑧^−𝛼≥0. (77) The above required condition (75) must hold for all values of 𝑧 in 𝑈. Upon clearing the values of 𝑧 on the positive real axis, where 0≤𝑧=𝑟<1, we must have 1−𝛽𝑧^𝛼−^∞_𝑘=2(𝑎)_𝑘−1(𝑐)_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑎_𝑘(𝛼)𝑟^𝑘−1−(−1)^𝑛∞_𝑘=1(𝑎)_𝑘−1(𝑐)_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑏_𝑘(𝛼)𝑟^𝑘−1≥0, (78) and the proof is complete.

Theorem 15. Let 𝑓^𝛼_𝑛=ℎ^𝛼+𝑔^𝛼, where ℎ^𝛼 and 𝑔^𝛼 are as given earlier. Then, 𝑓_𝑛∈𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐) if and only if 𝑓_𝑛(𝑧)^𝛼=^∞_𝑘=1𝑥_𝑘ℎ_𝑘(𝑧)+𝑦_𝑘𝑔_𝑛𝑘(𝑧), (79) where ℎ₁(𝑧)^𝛼=𝑧^𝛼, ℎ_𝑘(𝑧)^𝛼=𝑧^𝛼+((𝑐)_𝑘−1(1−𝛽)(𝛼)^𝑛/(𝑎)_𝑘−1(𝛼+𝑘−1)^𝑛)𝑧^{𝛼+𝑘−1} (𝑘=2,3,…), 𝑔_𝑚𝑘(𝑧)^𝛼=𝑧^𝛼+(−1)^𝑛𝛼^𝑛1−𝛽(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛(𝑎)_𝑘−1𝑧^{𝛼+𝑘−1}(𝑘=2,3,…), (80) where 𝑥_𝑘≥0, 𝑦_𝑘≥0, 𝑥₁=1−(∑^∞_𝑘=2𝑥_𝑘+∑^∞_𝑘=1𝑦_𝑘)≥0.

In particular, the extreme points of 𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐 are {𝑏_𝑘} and {𝑔_𝑛𝑘}).

Proof. For functions 𝑓^𝛼=ℎ^𝛼+𝑔^𝛼, where ℎ^𝛼 and 𝑔^𝛼 are as earlier defined, we have 𝑓_𝑛(𝑧)^𝛼=^∞_𝑘=1𝑥_𝑘ℎ_𝑘(𝑧)+𝑦_𝑘𝑔_𝑚𝑘(𝑧=_∞𝑘=1)𝑥_𝑘+𝑦_𝑘𝑧^𝛼+^∞_𝑘=2𝛼^𝑛1−𝛽(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛(𝑎)_𝑘−1𝑥_𝑘𝑧^{𝛼+𝑘−1}+(−1)^𝑛∞_𝑘=1𝛼^𝑛1−𝛽(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛(𝑎)_𝑘−1𝑦_𝑘𝑧^{𝛼+𝑘−1}. (81) Then, ^∞_𝑘=2(𝑎)_𝑘−1(𝑐)_𝑘−1𝛼+𝑘−1𝛼^𝑛1𝑎1−𝛽^𝑘(𝛼+_∞𝑘=1)(𝑎)_𝑘−1(𝑐)_𝑘−1𝛼+𝑘−1𝛼^𝑛1𝑏1−𝛽^𝑘(𝛼=_∞𝑘=2)𝑥_𝑘+^∞_𝑘=2𝑥_𝑘=1−𝑥₁≤1, (82) and so 𝑓^𝛼_𝑛∈𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐).
Conversely, suppose that 𝑓^𝛼_𝑛∈𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐). Setting 𝑥_𝑘=1(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑎_𝑘(𝛼),0≤𝑥_𝑘(),𝑦≤1𝑘=2,3,…_𝑘=1(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑏_𝑘(𝛼),0≤𝑦_𝑘≤1(𝑘=1,2,3,…), (83) and 𝑥₁=1−∑^∞_𝑘=2𝑥_𝑘−∑^∞_𝑘=1𝑦_𝑘, therefore, 𝑓^𝛼_𝑛 can be written as 𝑓_𝑛(𝑧)^𝛼=𝑧^𝛼+^∞_𝑘=2𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1}+(−1)^𝑛∞_𝑘=1𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−1}=𝑧^𝛼+^∞_𝑘=2𝛼^𝑛1−𝛽(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛(𝑎)_𝑘−1𝑥_𝑘𝑧^{𝛼+𝑘−1}+(−1)^𝑛∞_𝑘=1𝛼^𝑛1−𝛽(𝑐)_𝑘−1(𝛼+𝑘−1)^𝑛(𝑎)_𝑘−1𝑦_𝑘𝑧^{𝛼+𝑘−1}=^∞_𝑘=2ℎ_𝑘(𝑧)^𝛼𝑥_𝑘+^∞_𝑘=1𝑔_𝑛𝑘(𝑧)^𝛼𝑦_𝑘+𝑧^𝛼1−^∞_𝑘=2𝑥_𝑘−^∞_𝑘=1𝑦_𝑘=^∞_𝑘=1ℎ_𝑘(𝑧)^𝛼𝑥_𝑘+𝑔_𝑛𝑘(𝑧)^𝛼𝑦_𝑘, (84) as required.

Our next result is on distortion bounds for the functions in the class 𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐).

Theorem 16. Let 𝑓^𝛼_𝑛∈𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐). Then, for |𝑧|=𝑟<1, one has 𝑓_𝑛(𝑧)^𝛼≤1+𝑏₁(𝛼)𝑟^𝛼+𝛼𝛼+1^𝑛𝑐1−𝛽𝑎−1+𝛽𝑎𝑏₁(𝛼)𝑟^𝛼+1,𝑓_𝑛(𝑧)^𝛼≥1−𝑏₁(𝛼)𝑟^𝛼−𝛼𝛼+1^𝑛𝑐1−𝛽𝑎−1+𝛽𝑎𝑏₁(𝛼)𝑟^𝛼+1. (85)

Proof. We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted.
Let 𝑓^𝛼_𝑛∈𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐). Taking the absolute value of 𝑓^𝛼_𝑛, we obtain 𝑓_𝑛(𝑧)^𝛼=𝑧^𝛼+^∞_𝑘=2𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1}+(−1)^𝑛∞_𝑘=1𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−1}≤1+𝑏₁(𝛼)𝑟^𝛼+^∞_𝑘=2𝑎_𝑘(𝛼)+𝑏_𝑘(𝛼)𝑟^{𝛼+𝑘−1}≤1+𝑏₁(𝛼)𝑟^𝛼+𝑟^∞𝛼+1_𝑘=2𝑎_𝑘(𝛼)+𝑏_𝑘(𝛼≤𝑏)1+₁(𝛼)𝑟^𝛼+𝛼^𝑛1−𝛽𝑐(𝛼+1)^𝑛𝑎×^∞_𝑘=211−𝛽𝛼+1𝛼^𝑛𝑎𝑐𝑎_𝑘(𝛼+𝑎)𝑐1+𝛽𝛼+1𝛼^𝑛𝑏_𝑘(𝛼)𝑟^𝛼+1≤1+𝑏₁(𝛼)𝑟^𝛼+𝛼^𝑛1−𝛽(𝑐)(𝛼+1)^𝑛(𝑎)×^∞_𝑘=21(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑎_𝑘(𝛼+1)(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑏_𝑘(𝛼)𝑟^𝛼+1≤1+𝑏₁𝛼𝑟^𝛼𝛼𝛼+1^𝑛×𝑐1−𝛽𝑎−𝑐1+𝛽𝑎𝑏₁(𝛼)𝑟^𝛼+1, (86) for 𝑏₁(𝛼)<1. This shows that the bound given in Theorem 16 is sharp.

The following covering results follow from the left-hand inequality in Theorem 18.

Corollary 17. If function 𝑓^𝛼_𝑛=ℎ^𝛼+𝑔^𝛼, where ℎ^𝛼 and 𝑔^𝛼 are as given earlier in 𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐), then |𝜔|≤𝛼𝜔∶𝛼+1^𝑛𝛼+1𝛼^𝑛−𝛼𝛼+1^𝑛𝑐𝑎−𝛼1−𝛽𝛼+1^𝑛𝛼+1𝛼^𝑛−𝛼𝛼+1^𝑛𝑐𝑎1−𝛽𝑏₁(𝛼)⊂𝑓_𝑛(𝑈). (87) For harmonic functions 𝑓_𝑛(𝑧)^𝛼=𝑧^𝛼+^∞_𝑘=2𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1}+(−1)^𝑛∞_𝑘=1𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−1},𝐹_𝑛(𝑧)^𝛼=𝑧^𝛼+^∞_𝑘=2𝐴_𝑘(𝛼)𝑧^{𝛼+𝑘−1}+(−1)^𝑛∞_𝑘=1𝐵_𝑘(𝛼)𝑧^{𝛼+𝑘−1}. (88) The convolution of 𝑓^𝛼_𝑛 and 𝐹^𝛼_𝑛 is given by 𝑓^𝛼_𝑛∗𝐹^𝛼_𝑛(𝑧)=𝑓_𝑛(𝑧)^𝛼∗𝐹^𝛼_𝑛(𝑧)^𝛼=𝑧^𝛼+𝑎_𝑘(𝛼)𝐴_𝑘(𝛼)𝑧^{𝛼+𝑘−1}+(−1)^𝑛∞_𝑘=1𝑏_𝑘(𝛼)𝐵_𝑘(𝛼)𝑧^{𝛼+𝑘−1}. (89) Using this definition, one shows that the class 𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐) is closed under convolution.

Theorem 18. For 0≤𝜆≤𝛽<1, let 𝑓^𝛼_𝑛∈𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐), and let 𝐹^𝛼_𝑛∈𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐). Then, 𝑓^𝛼_𝑛∗𝐹^𝛼_𝑛∈𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐)⊆𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐).

Proof. Let the functions 𝑓_𝑛(𝑧)^𝛼=𝑧^𝛼+^∞_𝑘=2𝑎_𝑘(𝛼)𝑧^{𝛼+𝑘−1}+(−1)^𝑛∞_𝑘=1𝑏_𝑘(𝛼)𝑧^{𝛼+𝑘−1}bein𝑉𝐵^𝛼_𝑛𝐻𝛽,𝑎,𝑐 (90) and 𝐹_𝑛(𝑧)^𝛼=𝑧^𝛼+∑^∞_𝑘=2|𝐴_𝑘(𝛼)|𝑧^{𝛼+𝑘−1}+(−1)^𝑛∑^∞_𝑘=1|𝐵_𝑘(𝛼)|𝑧^{𝛼+𝑘−1} be in 𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐). Then, the convolution 𝑓^𝛼_𝑛∗𝐹^𝛼_𝑛 is given by (89).
We wish to show that the coefficients of 𝑓^𝛼_𝑛∗𝐹^𝛼_𝑛 satisfy the required condition given in Theorem 14. For 𝐹^𝛼_𝑛∈𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐), 𝐴_𝑘(𝛼)≤1, and |𝐵_𝑘(𝛼)|≤1. Now, for the convolution function (𝑓^𝛼_𝑛∗𝐹^𝛼_𝑛)(𝑧), we obtain ^∞_𝑘=21(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑎_𝑘(𝛼)𝐴_𝑘(𝛼+_∞𝑘=1)1(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑏_𝑘(𝛼)𝐵_𝑘(𝛼≤_∞𝑘=2)1(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑎_𝑘(𝛼+_∞𝑘=1)1(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑏_𝑘(𝛼)≤1. (91) Therefore (𝑓^𝛼_𝑛∗𝐹^𝛼_𝑛)(𝑧)∈𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐)⊆𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐).
Next, we show that the class 𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐) is closed under convex combinations of its members.
Let functions 𝑓_𝑛𝑗(𝑧)^𝛼 be defined, for 𝑗=1,2,…, by 𝑓_𝑛𝑗(𝑧)^𝛼=𝑧^𝛼+^∞_𝑘=2𝑎_𝑘,𝑗(𝛼)𝑧^{𝛼+𝑘−1}+(−1)^𝑛∞_𝑘=1𝑏_𝑘,𝑗(𝛼)𝑧^{𝛼+𝑘−1}. (92)

Theorem 19. Let the functions 𝑓_𝑛𝑗(𝑧)^𝛼 defined by (92) be in the class 𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐) for every 𝑗=1,2,…,𝑚. Then, the functions 𝑡_𝑗(𝑧)^𝛼 defined by 𝑡_𝑗(𝑧)^𝛼=^𝑚_𝑗=1𝛾_𝑗𝑓_𝑛𝑗(𝑧)0≤𝛾_𝑗≤1 (93) are also in the class 𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐), where ∑^𝑚_𝑗=1𝛾_𝑗=1.

Proof. According to the definition of 𝑡^𝛼, we can write 𝑡^𝛼=𝑧^𝛼+^∞_𝑘=2^𝑚_𝑗=1𝛾_𝑗𝑎_𝑘,𝑗(𝛼)𝑧^{𝛼+𝑘−1}+(−1)^𝑛∞_𝑘=1^𝑚_𝑗=1𝛾_𝑗𝑎_𝑘,𝑗(𝛼)𝑧^{𝛼+𝑘−1}. (94) Furthermore, since 𝑓_𝑛𝑗(𝑧)^𝛼 are in 𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐) for every (𝑗=1,2,…,), then by (66), we have ^∞_𝑘=21(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑚_𝑗=1𝛾_𝑗𝑎_𝑘,𝑗(𝛼+_∞𝑘=1)1(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑚_𝑗=1𝛾_𝑗𝑏_𝑘,𝑗(𝛼=_𝑚𝑗=1)𝛾_𝑗^∞_𝑘=21(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑎_𝑘,𝑗(𝛼+_∞𝑘=1)1(𝑎)1−𝛽(𝑐)^𝑘−1_𝑘−1𝛼+𝑘−1𝛼^𝑛𝑏_𝑘,𝑗(𝛼≤_𝑚𝑗=1)𝛾_𝑗=1. (95) Hence, the theorem is proved.

Corollary 20. The class 𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐) is closed under convex linear combination.

Proof. Let the functions 𝑓_𝑛𝑗(𝑧)^𝛼 (𝑗=1,2) defined by (92) be in the class 𝑀𝑉𝐵^𝛼_𝑛𝐻(𝛽,𝑎,𝑐). Then, the function Ψ(𝑧)^𝛼 defined by Ψ(𝑧)^𝛼=𝜇𝑓_𝑛1(𝑧)^𝛼+1−𝜇𝑓_𝑛2(𝑧)^𝛼0≤𝜇≤1 (96) is in the class 𝑀𝑉𝐵^𝛼_𝑛𝐻(𝜆,𝑎,𝑐). Also, by taking 𝑚=2, 𝑡₁=𝜇 and 𝑡₂=(1−𝜇) in Theorem 19, we have the above corollary.

With various special choices of the parameters involved, both the existing classes and new one could be derived.