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

Authors: A. T. Oladipo; D. Breaz
Journal: ISRN Mathematical Analysis
 (2013)

---

## 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.


---

## Body

## 1. Introduction

LetA denote the usual class of analytic functions of the form
(1)f(z)=z+∑k=2∞‍akzk,
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. Letf∈A. Then, f∈S*(μ) if and only if
(2)Re{zf′(z)f(z)}>μ,(0≤μ<1,z∈U).
This class is called starlike class of analytic function.Also, letf∈A. Then, f∈C(μ) if and only if
(3)Re{1+zf′′(z)f′(z)}>μ,(0≤μ<1,z∈U).
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.Letγ: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:
(4)D0f(z)=zf′(z),D1f(z)=Df(z)=zf′(z),Dnf(z)=D(Dn-1f(z))=z(Dn-1f(z))′.
From (1), we can write that
(5)f(z)α=(z+∑k=2∞‍akzk)α.
Using binomial expansion on (5), we have
(6)f(z)α=zα+αa2zα+1+[αa3+α(α-1)2!a23]zα+2+[αa4+α(α-1)2!2a2a3+α(α-1)(α-2)3!a23]zα+3+⋯.
We then define the class of analytic functions of fractional power Aα as
(7)f(z)α=zα+∑k=α∞‍ak(α)zα+k-1,
whereα>0 (is real, and it is principal determination only).Thus, we obtain the differential operator(8)Dnf(z)α=αnzα+∑k=α∞‍(α+k-1)nak(α)zα+k-1.
Let us also define the function φα(a,c,z) by
(9)φα(a,c;z)=zα+∑k=2∞‍(a)k-1ck-1zα+k-1,(z∈U,α∈R+,a∈R,c∈R-{0,-1,-2,…}),
where (ak) is the Pochhammer symbol defined by
(10)(a)k=γ(a+k)γ(a)={1,k=0,a(a+1)(a+2)⋯(a+n-1)n∈N.Corresponding to the functionφα(a,c;z), we defined a linear operator
(11)Jnα(a,c)f(z)α=φα(a,c;z)*Dnf(z)α,f(z)α∈Aα.
Or equivalently
(12)Jnα(a,c)f(z)α=αnzα+∑k=2∞‍(a)k-1(c)k-1(α+k-1)nak(α)zα+k-1.Remark 1.
Forn=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, letBnα(γ) be the class of functions containing the operator (11) and satisfying the relation
(13)z(Jnα(a,c)f(z)α)′αJnα(a,c)f(z)α≺(1+γz1-z)αα>0,γ≠-1.
For a=c,  α=1, γ=1, and n=0, we obtain the well-known subclass
(14)f′(z)≺1+z1-z.
Also, for n=0, a=c, and γ=1, we have
(15)αzf′(z)f(z)·f(z)αzα≺(1+z1-z)α.
For n=0, α=1, and γ=1 we have the following subclass which contain Carlson-Shaffer operator:
(16)z(J(a,c)f(z))′z≺1+z1-z.
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
(17)f(z)={α1+ε2∫0z‍p(v)-iεV(1+(iαε/(1+ε2)))g(v)α/(1+ε2)dv}(1+iε)/α,
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 takeε=0, we have
(18)f(z)={α∫0z‍p(v)vg(v)αdv}1/α.
On differentiation, the expression (18) yields
(19)zf′(z)f(z)α-1g(z)α=p(z),z∈U.
Or equivalently
(20)Rezf′(z)f(z)α-1g(z)α>0,z∈U.
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 Uα(r)={εf(rexpiθ)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,
(21)Rezf′(z)g(z)>0,z∈U.
If we decide to choose g(z)=f(z) in (21), we have
(22)Rezf′(z)f(z)>0,z∈U,
which implies that f(z) is starlike. Furthermore, if we replace f(z) by zf′(z) in (22), we obtain
(23)Re{1+zf′′(z)f′(z)}>0,z∈U,
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
(24)Rezf′(z)f(z)α-1zα>0,z∈U.
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
(25)ReDnf(z)αzα>0,z∈U,
where the parameter α and the operator Dn are defined as earlier. He denoted this class of functions by Bn(α). It is easily seen that his generalization has extraneously included analytic functions satisfying
(26)Ref(z)αzα>0,z∈U,
which are largely nonunivalent in the unit disk. By proving the inclusion
(27)Bn+1(α)⊂Bn(α),
Abdulhalim was able to show that for all n∈N, each function of the class Bn(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 classBnα(γ,a,c).For this purpose, the following Lemmas will be necessary.Lemma 2 (see [21]).
Letβ≠0 be a complex number. Let q(z) (q(z)≠0) be a univalent function in U such that
(28)Re{1+zq′′(z)q′(z)-zq′(z)q(z)}>max{0,R(β-1βq(z))}.
If p, (p(z)≠0), z∈U satisfies the differential subordination
(29)(1-β)(p(z)-1)+βzp′(z)p(z)≺(1-β)(q(z)-1)+βzq′(z)q(z),
then p≺q and q is the best dominant.Lemma 3 (see [22]).
Letω 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
(30)z0ω′(z0)=mω(z0),
where m is a real number and m≥1.Lemma 4 (see [23]).
Iff∈A satisfies
(31)Re[1+zf′′(z)f′(z)]<(μ+1)2(μ-1),(2≤μ<3,z∈U),
then
(32)zf′(z)f(z)≺μ(1-z)μ-z,|zf′(z)f(z)-μμ+1|<μμ+1.Now, we begin our main results as the following.Theorem 5.
Letα>0 (α is real), n∈N0, and β≠0 be a complex number such that
(33)Re{1+z(1-γ+2γz)(1+γz)(1-z)}>max{0,R(β-1β)(1+γz1-z)α}.
If the subordination
(34)(1-β)[z(Jnα(a,c)f(z)α)′αnzα-1]+β(1-α+z(Jnα(a,c)f(z)α)′′(Jnα(a,c)f(z)α)′)≺(1-β)[(1+γz1-z)α-1]+αβ(1+γ)z(1+γz)(1-z)
holds, then fα∈Bnα(γ,a,c).Proof.
Suppose that(35)p(z)=z(Jnα(a,c)f(z)α)′αnzα,q(z)=(1+γz1-z)α.
Then, simple computations give
(36)Re{1+zq′′(z)q′(z)-zq′(z)q(z)}=Re{1+z(1-γ+2γz)(1+γz)(1-z)}>max{0,R(β-1β)(1+γz1-z)α}=max{0,R(β-1β)q(z)},(37)(1-β)(p(z)-1)+βzp′(z)p(z)=(1-β)[z(Jnα(a,c)f(z)α)′αnzα-1]+β(1-α+z(Jnα(a,c)f(z)α)′′(Jnα(a,c)f(z)α)′)≺(1-β)[(1+γz1-z)α-1]+αβ(1+γ)z(1+γz)(1-z)=(1-β)(q(z)-1)+βzq′(z)q(z).
Thus, in the view of Lemma 2, we have f(z)α∈Bnα(γ,a,c).Forα=1, γ=1, n=0. We have the following.Corollary 6.
Letβ≠0 be a complex number such that
(38)Re{1+2z21-z2}>max{0,Re(β-1β(1+z1-z))}.
If the subordination
(39)(1-β)[(J01(a,c)f(z))′-1]+β(z(J01(a,c)f(z))′′(J01(a,c)f(z))′)≺(1-β)[(1+z1-z)-1]+2βz1-z2
Holds, then f(z)α∈B01(1,a,c).Fora=c, α=1, γ=1, and n=0, we have the followingCorollary 7.
Letβ≠0 be a complex number such that
(40)Re{1+2z21-z2>max{0,R(β-1β(1+z1-z))}.
If the subordination
(41)(1-β)(f′(z)-1)+βzf′′(z)f′(z)≺(1-β)[1+z1-z-1]+2βz1-z2
holds, then f(z)α∈B01(1,1,1)≡B.Forγ=1,  a=c, we have the following.Corollary 8.
Letβ≠0 be a complex number such that
(42)Re{1+2z21-z2}>max{0,Re(β-1β(1+z1-z)α)}.
If the subordination
(43)(1-z)[z(Jnαf(z)α)′αnzα-1]+β(1-α+z(Jnαf(z)α)′′(Jnαf(z)α)′)≺(1-β)[(1+z1-z)α-1]+2αβz1-z2
holds, then fα∈Bnα(1,1,1).Forγ=1,  n=0,  a=c, we have the following.Corollary 9.
Letβ≠0 be a complex number such that
(44)Re{1+2z21-z2}>max{0,Re(β-1β(1+z1-z)α)}.
If the subordination
(45)(1-β)[αzf′(z)f(z)·f(z)αzα-1]+β(1-β+z(f(z)α)′′(f(z)α)′)≺(1-β)[(1+z1-z)α-1]+2αβz1-z2
holds, then fα∈B0α≡B(α).Theorem 10.
Let the functionsf(z)α take the form (7) and satisfy
(46)Re{1-α+z(Jnα(a,c)f(z)α)′′(Jnα(a,c)f(z)α)′}<α(γ+1)2(γ-1),z∈U,α>0,γ>1.
Then, f∈Bnα(γ,a,c).Proof.
Letω be defined by
(47)z(Jnα(a,c)f(z)α)′αnzα=(1+γω(z)1-ω(z))α,ω(z)≠1.
Then, ω is analytic in U, and since γ≠-1, then ω(0)=0. Also, it follows that
(48)Re{1-α+z(Jnα(a,c)f(z)α)′′(Jnα(a,c)f(z)α)′}=Re{α(γ+1)zω′(z)(1+γω(z))(1-ω(z))}<α(γ+1)2(γ-1),γ≠1.
Now, let us proceed to prove that |ω(z)|<1. Suppose that there exists a point z0∈U such that
(49)max|z|≤|z0||ω(z)|=|ω(z)|=1.
Then, using Lemma 3 and letting ω(z0)=eiθ and z0ω′(z0)=meiθ, m≥1 yields that
(50)Re{1-α+z(Jnα(a,c)f(z)α)′′(Jnα(a,c)f(z)α)′}=Re{α(γ+1)z0ω′(z0)(1+γω(z0))(1-ω(z0))}=Re{αmeiθ(γ+1)(1+γω(eiθ))(1-eiθ)}=αm(1+γ)2(γ-1)≥α(γ+1)2(γ-1),γ>1,α>0(αisreal).
Thus, we have
(51)Re{1-α+z(Jnα(a,c)f(z)α)′′(Jnα(a,c)f(z)α)′}≥α(γ+1)2(γ-1),z∈U,
which contradicts the hypothesis (46). Therefore, we conclude that |ω(z)|<1 for all z∈U and
(52)z(Jnα(a,c)f(z)α)′αnzα≺(1+γ(z)1-(z))α,z∈U,γ≠-1,α>0,
where (α is real). This completes the proof of the theorem.Lettingα=1, n=0, and a=c in Theorem 10, we have the following.Corollary 11.
Let the functionf take the form (1) and satisfy
(53)Re{zf′′(z)f′(z)}<γ+12(γ-1),z∈U,γ>1.
Then,
(54)f′(z)≺1+γz1-z.Corollary 12.
Let the functionf take the form (1) and satisfy
(55)Re{zf′′f′(z)}<γ+12(γ-1),(z∈U),γ>1.
Then,
(56)|f′(z)-γ2|<γ2,(z∈U).
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 functionf=u+iv defined in a simply connected domain D is said to be harmonic in D. In any simply connected domain, we can write
(57)f=h+g¯,
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
(58)|h′(z)|>|g′(z)|,(z∈U).
Denote by SH 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 functionsh and g as
(59)h(z)α=zα+∑k=2∞‍ak(α)zα+k-1,g(z)α=∑k=1∞‍bk(α)zα+k-1,|b1(α)|<1.
Therefore,
(60)f(z)α=h(z)α+g(z)α¯.
We define our linear operator as given in (11) such that
(61)Jnα(a,c)f(z)α=Jnα(a,c)h(z)α+(-1)nJnα(a,c)g(z)α→,
where
(62)Jnα(a,c)h(z)α=αnzα+∑k=2∞‍ak-1ck-1(α+k-1)nak(α)zα+k-1,Jnα(a,c)g(z)α=∑k=1∞‍ak-1ck-1(α+k-1)nbk(α)zα+k-1.
We let Bnα(β,a,c) be the family of harmonic functions f of the form (57) such that
(63)Re{Jnα(a,c)f(z)ααnzα}>β,0≤β<1,
where α>0, (α is real), n∈N0, and Jnα(a,c) is earlier defined in (11).Furthermore, let the subclassVBnαH(β,a,c) consist of harmonic functions
(64)fnα=hα+gnα¯,
so that hα and gnα are of the form
(65)h(z)α=zα+∑k=2∞‍|ak(α)|zα+k-1,g(z)α=(-1)n+∑k=1∞‍|bk(α)zα+k-1|,|b1(α)|<1.
The authors in this work wish to study the Bazilevič-type harmonic univalent functions defined by linear operator in which hα has positive coefficients. We claim that our results are quite new and not explored in the literatures.Assigning specific values ton, β, a, c, α in the subclass BnαH(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 inBnαH(β,a,c).Theorem 13.
Letfα=hα+gα¯, where fα and gα are as earlier defined if
(66)∑k=2∞‍(a)k-1(c)k-1(α+k-1α)n|ak(α)|+∑k=1∞‍(a)k-1(c)k-1(α+k-1α)n|bk(α)|≤1-β,
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∈BnαH(β,a,c).Proof.
Ifz1α≠z2α, then
(67)|f(z1)α-f(z2)αh(z1)α-h(z2)α|≥1-|g(z1)α-g(z2)αh(z1)α-h(z2)α|=1-|∑k=1∞‍bk(α)(z1α+k-1-z2α+k-1)(z1α-z2α)+∑k=2∞‍ak(α)(z1α+k-1-z2α+k-1)|>1-∑k=1∞‍(α+k-1)bk(α)α+∑k=2∞‍(α+k-1)ak(α)≥1-((∑k=1∞‍((a)k-1/(c)k-1)((α+k-1)/α)n×(1/(1-β))|bk(α)|∑k=1∞‍)×(1+∑k=2∞‍((a)k-1/(c)k-1)((α+k-1)/α)n×(1/(1-β))|ak(α)|∑k=2∞‍)-1)≥0,
which proves the univalence. Note that f is sense-preserving in U. This is because
(68)|h(z)α|≥α-∑k=2∞‍(α+k-1)|ak(α)||z|α+k-1>α-∑k=2∞‍(α+k-1)|ak(α)|≥αn-∑k=2∞‍(a)k-1(c)k-1(α+k-1)n1-β|ak(α)|≥∑k=1∞‍(a)k-1(c)k-1(α+k-1)n1-β|bk(α)|≥∑k=1∞‍(α+k-1)|bk(α)|>∑k=2∞‍(α+k-1)|bk(α)||zα+k-2|≥|g(z)α|.
By (63) and (64), we have
(69)Re{Jnα(a,c)f(z)ααnzn}=Re{Jnα(a,c)h(z)α+(-1)nJnα(a,c)g(z)α¯αnzn}>β.
Using the fact that Re(ω)>β if and only if |1-β+ω|≥|1+β-ω|, it suffices to show that
(70)|1-β+Jnα(a,c)f(z)ααnzn|-|1-β-Jnα(a,c)f(z)ααnzn|≥0.
That is,
(71)|(1-β)αnzn+Jnα(a,c)f(z)α|-|(1-β)αnzn-Jnα(a,c)f(z)α|≥0,|(1-β)αnzα+Jnα(a,c)h(z)α+(-1)nJnα(a,c)g(z)α¯|-|(1-β)αnzα-Jnα(a,c)h(z)α-(-1)nJnα(a,c)g(z)α¯|=|∑k=1∞‍(a)k-1(c)k-1αnzα-βαnzα+αnzα+∑k=2∞‍(a)k-1(c)k-2(α+k-1)nak(α)zα+k-1+(-1)n∑k=1∞‍(a)k-1(c)k-1(α+k-1)nbk(α)zα+k-1¯|-|-∑k=2∞‍(a)k-1(c)k-2(α+k-1)nak(α)zα+k-1αnzα+βαnzα-αnzα-∑k=2∞‍(a)k-1(c)k-2(α+k-1)nak(α)zα+k-1-(-1)n∑k=1∞‍(a)k-1(c)k-1(α+k-1)nbk(α)zα+k-1¯|,=|αn(2-β)zα+∑k=2∞‍(a)k-1(c)k-2(α+k-1)nak(α)zα+k-1+(-1)n∑k=1∞‍(a)k-1(c)k-1(α+k-1)nbk(α)zα+k-1|-|βαnzα-∑k=2∞‍(a)k-1(c)k-2(α+k-1)nak(α)zα+k-1-(-1)n∑k=1∞‍(a)k-1(c)k-1(α+k-1)nbk(α)zα+k-1|≥2αn(1-β)|z|α+∑k=2∞‍2(a)k-1(c)k-1(α+k-1)n|ak(α)||z|α+k-1-(-1)n∑k=1∞‍2(a)k-1(c)k-1(α+k-1)n|bk(α)||z|α+k-1×2αn(1-β)[1+∑k=2∞‍11-β(a)k-1(c)k-1(α+k-1α)n×|ak(α)|+(-1)n∑k=1∞‍11-β(a)k-1(c)k-1×(α+k-1α)n|bk(α)|].
This last expression is nonnegative by (66), and so the proof is complete.The harmonic function(72)f(z)α=zα+∑k=2∞‍(c)k-1(1-β)(a)k-1(αα+k-1)nxkzα+k-1+∑k=1∞‍(c)k-1(1-β)(a)k-1(αα+k-1)nyα+k-1¯,
where n∈N0, α>0 (α is real), 0≤β<1, a∈R, c∈R-{-1,0,1…} and
(73)∑k=2∞‍|xk|+∑k=1∞‍|yk|=1,
shows that the coefficient bound given by (66) is sharp. The functions of the form (72) are in BnαH(β,a,c) because
(74)∑k=2∞‍11-β(a)k-1(c)k-1(α+k-1α)n|ak(α)|+∑k=1∞‍11-β(a)k-1(c)k-1(α+k-1α)n|bk(α)|=∑k=2∞‍|xk|+∑k=1∞‍|yk|=1.
In the following theorem, it is shown that the condition (66) is also necessary for functions fnα=hα+gnα, where hα and gnα are as earlier defined.Theorem 14.
Letfnα=hα+gnα¯. Then, fn∈VBnαH(β,a,c) if and only if
(75)∑k=2∞‍11-β(a)k-1(c)k-1(α+k-1α)n|ak(α)|+∑k=1∞‍11-β(a)k-1(c)k-1(α+k-1α)n|bk(α)|≤1.Proof.
SinceVBnαH(β,a,c)⊂BnαH(β,a,c), we only need to prove the “only if” part of the theorem. To this end, for functions fnα of the form (64), we notice that the condition
(76)Re{Jnα(a,c)fn(z)ααnzα}>β
is equivalent to
(77)Re{((1-β)zα-∑k=2∞‍(a)k-1(c)k-1(α+k-1α)nak(α)zα+k-1-(-1)2n∑k=1∞‍(a)k-1(c)k-1(α+k-1α)nbk(α)zα+k-1)(z-α)}≥0.
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
(78)(1-β)zα-∑k=2∞‍(a)k-1(c)k-1(α+k-1α)nak(α)rk-1-(-1)n∑k=1∞‍(a)k-1(c)k-1(α+k-1α)nbk(α)rk-1≥0,
and the proof is complete.Theorem 15.
Letfnα=hα+gα¯, where hα and gα are as given earlier. Then, fn∈VBnαH(β,a,c) if and only if
(79)fn(z)α=∑k=1∞‍(xkhk(z)+ykgnk(z)),
where h1(z)α=zα, hk(z)α=zα+((c)k-1(1-β)(α)n/(a)k-1(α+k-1)n)zα+k-1 (k=2,3,…),
(80)gmk(z)α=zα+(-1)nαn(1-β)(c)k-1(α+k-1)n(a)k-1zα+k-1¯(k=2,3,…),
where xk≥0, yk≥0, x1=1-(∑k=2∞‍xk+∑k=1∞‍yk)≥0.In particular, the extreme points ofVBnαH(β,a,c are {bk} and {gnk}).Proof.
For functionsfα=hα+gα¯, where hα and gα are as earlier defined, we have
(81)fn(z)α=∑k=1∞‍(xkhk(z)+ykgmk(z))=∑k=1∞‍(xk+yk)zα+∑k=2∞‍αn(1-β)(c)k-1(α+k-1)n(a)k-1xkzα+k-1+(-1)n∑k=1∞‍αn(1-β)(c)k-1(α+k-1)n(a)k-1ykzα+k-1.
Then,
(82)∑k=2∞‍(a)k-1(c)k-1(α+k-1α)n11-β|ak(α)|+∑k=1∞‍(a)k-1(c)k-1(α+k-1α)n11-β|bk(α)|=∑k=2∞‍xk+∑k=2∞‍xk=1-x1≤1,
and so fnα∈BnαH(β,a,c).
Conversely, suppose thatfnα∈BnαH(β,a,c). Setting
(83)xk=11-β(a)k-1(c)k-1(α+k-1α)n|ak(α)|,0≤xk≤1(k=2,3,…),yk=11-β(a)k-1(c)k-1(α+k-1α)n|bk(α)|,0≤yk≤1(k=1,2,3,…),
and x1=1-∑k=2∞‍xk-∑k=1∞‍yk, therefore, fnα can be written as
(84)fn(z)α=zα+∑k=2∞‍|ak(α)|zα+k-1+(-1)n∑k=1∞‍|bk(α)|zα+k-1¯=zα+∑k=2∞‍αn(1-β)(c)k-1(α+k-1)n(a)k-1xkzα+k-1+(-1)n∑k=1∞‍αn(1-β)(c)k-1(α+k-1)n(a)k-1ykzα+k-1¯=∑k=2∞‍hk(z)αxk+∑k=1∞‍gnk(z)αyk+zα(1-∑k=2∞‍xk-∑k=1∞‍yk)=∑k=1∞‍(hk(z)αxk+gnk(z)αyk),
as required.Our next result is on distortion bounds for the functions in the classVBnαH(β,a,c).Theorem 16.
Letfnα∈VBnαH(β,a,c). Then, for |z|=r<1, one has
(85)|fn(z)α|≤(1+|b1(α)|)rα+(αα+1)n(c(1-β)a-1+βa|b1(α)|)rα+1,|fn(z)α|≥(1-|b1(α)|)rα-(αα+1)n(c(1-β)a-1+βa|b1(α)|)rα+1.Proof.
We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted.
Letfnα∈VBnαH(β,a,c). Taking the absolute value of fnα, we obtain
(86)|fn(z)α|=|zα+∑k=2∞‍ak(α)zα+k-1+(-1)n∑k=1∞‍bk(α)zα+k-1¯|≤(1+|b1(α)|)rα+∑k=2∞‍(|ak(α)|+|bk(α)|)rα+k-1≤(1+|b1(α)|)rα+rα+1∑k=2∞‍(|ak(α)|+|bk(α)|)≤(1+|b1(α)|)rα+αn(1-β)c(α+1)na×[∑k=2∞‍11-β(α+1α)nac|ak(α)|+ac(1+β)(α+1α)n|bk(α)|]rα+1≤(1+|b1(α)|)rα+αn(1-β)(c)(α+1)n(a)×[∑k=2∞‍11-β(a)k-1(c)k-1(α+k-1α)n|ak(α)|+11-β(a)k-1(c)k-1(α+k-1α)n|bk(α)|]rα+1≤(1+|b1α|)rα(αα+1)n×[c(1-β)a-c(1+β)a|b1(α)|]rα+1,
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 Theorem18.Corollary 17.
If functionfnα=hα+gα¯, where hα and gα are as given earlier in VBnαH(β,a,c), then
(87){ω:|ω|≤(αα+1)n(α+1α)n-(αα+1)nca(1-β)-(αα+1)n(α+1α)n-(αα+1)nca(1-β)|b1(α)|(αα+1)n(α+1α)n}⊂fn(U).
For harmonic functions
(88)fn(z)α=zα+∑k=2∞‍|ak(α)|zα+k-1+(-1)n∑k=1∞‍|bk(α)|zα+k-1¯,Fn(z)α=zα+∑k=2∞‍|Ak(α)|zα+k-1+(-1)n∑k=1∞‍|Bk(α)|zα+k-1¯.
The convolution of fnα and Fnα is given by
(89)(fnα*Fnα)(z)=fn(z)α*Fnα(z)α=zα+|ak(α)||Ak(α)|zα+k-1+(-1)n∑k=1∞‍|bk(α)||Bk(α)|zα+k-1¯.
Using this definition, one shows that the class VBnαH(β,a,c) is closed under convolution.Theorem 18.
For0≤λ≤β<1, let fnα∈VBnαH(β,a,c), and let Fnα∈VBnαH(β,a,c). Then, fnα*Fnα∈VBnαH(β,a,c)⊆VBnαH(β,a,c).Proof.
Let the functions(90)fn(z)α=zα+∑k=2∞‍|ak(α)|zα+k-1+(-1)n∑k=1∞‍|bk(α)|zα+k-1¯beinVBnαH(β,a,c)
and Fn(z)α=zα+∑k=2∞‍|Ak(α)|zα+k-1+(-1)n∑k=1∞‍|Bk(α)|zα+k-1¯ be in VBnαH(β,a,c). Then, the convolution fnα*Fnα is given by (89).
We wish to show that the coefficients offnα*Fnα satisfy the required condition given in Theorem 14. For Fnα∈VBnαH(β,a,c), Ak(α)≤1, and |Bk(α)|≤1. Now, for the convolution function (fnα*Fnα)(z), we obtain
(91)∑k=2∞‍11-β(a)k-1(c)k-1(α+k-1α)n|ak(α)||Ak(α)|+∑k=1∞‍11-β(a)k-1(c)k-1(α+k-1α)n|bk(α)||Bk(α)|≤∑k=2∞‍11-β(a)k-1(c)k-1(α+k-1α)n|ak(α)|+∑k=1∞‍11-β(a)k-1(c)k-1(α+k-1α)n|bk(α)|≤1.
Therefore (fnα*Fnα)(z)∈VBnαH(β,a,c)⊆VBnαH(β,a,c).
Next, we show that the classVBnαH(β,a,c) is closed under convex combinations of its members.
Let functionsfnj(z)α be defined, for j=1,2,…, by
(92)fnj(z)α=zα+∑k=2∞‍|ak,j(α)|zα+k-1+(-1)n∑k=1∞‍|bk,j(α)|zα+k-1.Theorem 19.
Let the functionsfnj(z)α defined by (92) be in the class VBnαH(β,a,c) for every j=1,2,…,m. Then, the functions tj(z)α defined by
(93)tj(z)α=∑j=1m‍γjfnj(z)(0≤γj≤1)
are also in the class VBnαH(β,a,c), where ∑j=1m‍γj=1.Proof.
According to the definition oftα, we can write
(94)tα=zα+∑k=2∞‍(∑j=1m‍γjak,j(α))zα+k-1+(-1)n∑k=1∞‍(∑j=1m‍γjak,j(α))zα+k-1.
Furthermore, since fnj(z)α are in VBnαH(β,a,c) for every (j=1,2,…,), then by (66), we have
(95)∑k=2∞‍11-β(a)k-1(c)k-1(α+k-1α)n(∑j=1m‍γj|ak,j(α)|)+∑k=1∞‍11-β(a)k-1(c)k-1(α+k-1α)n(∑j=1m‍γj|bk,j(α)|)=∑j=1m‍γj(∑k=2∞‍11-β(a)k-1(c)k-1(α+k-1α)n|ak,j(α)|+∑k=1∞‍11-β(a)k-1(c)k-1(α+k-1α)n|bk,j(α)|)≤∑j=1m‍γj=1.
Hence, the theorem is proved.Corollary 20.
The classVBnαH(β,a,c) is closed under convex linear combination.Proof.
Let the functionsfnj(z)α (j=1,2) defined by (92) be in the class MVBnαH(β,a,c). Then, the function Ψ(z)α defined by
(96)Ψ(z)α=μfn1(z)α+(1-μ)fn2(z)α(0≤μ≤1)
is in the class MVBnαH(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.



---

*Source: 179856-2013-09-09.xml*