# Study of Generalizedq-Close-to-Convex Functions Related to Parabolic Domain

Authors: Khalida Inayat Noor; Alina Alb Lupas; Shujaat Ali Shah; Alanod M. Sibih; S. Abdel-Khalek
Journal: Journal of Function Spaces
 (2023)

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## Abstract

The concepts of bounded boundary rotation and parabolic domain are used to define certain new classes of analytic functions such asV1,qm,g, R1,qm,g, T1,qm,g, and Q1,qγm,g,a. The difference of coefficients, necessary conditions, distortion bounds, radius problem, and several other interesting properties of these newly defined classes are also studied.


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## Body

## 1. Introduction

LetA denote the class of functions f which are analytic in the open unit disc E=z:z<1 and are represented by the series expansion of the form
(1)fz=z+∑n=2∞anzn.Two functionsf and g, analytic in E, we say f is subordinate to g, written f≺g or fz≺gz, if there exists a Schwartz function wz analytic in E with w0=0, wz<1 such that fz=gwz. If the function g is univalent in E, then we have the following equivalence:
(2)fz≺gz⇔f0=g0,fE⊂gE.Forf,g∈A, f be given by (1) and gz=z+∑n=2∞bnzn, the convolution (Hadamard product) of f and g is defined as
(3)f∗gz=z+∑n=2∞anbnzn,z∈E.Kanas and Wisniowska [1, 2] defined the subclasses of A related to the conic domains Ωk, where
(4)Ωk=u+iv:u2>k2u−12+v2,u>0,k≥0.The domainΩ0 represents the right half plane, Ω1 is a domain related to parabola, Ωk, 0<k<1 represents hyperbola and an ellipse for k>1.We shall deal with the domainΩ1 and the function p1z given below as
(5)p1z=1+2π2log1+z1−z2plays the role of extremal function for Ω1.LetP denote the class of Caratheodory functions p, analytic in E with p0=1 and satisfying the property Repz>0z∈E.Using the concept ofq-calculus, Ismail et al. [3] defined a subclass Sq∗ of A consisting of functions called q-starlike in E and established that limq⟶1−Sq∗ is the well-known class S∗ of starlike functions with respect to origin. In [4], a firm foothold of the usage of the q-calculus in the context of geometric function theory was effectively established. For the study of various subclasses of A, the quantum (or q-) calculus has been used as an important tool (see [5–7]).Jackson [8, 9] first defined, with some applications, the q-derivative and q-integral operators. For more and potentially useful survey-cum-expository review article, refer to [10] for interested researchers and scholars. For convenience, we provide here some basics and details of q-calculus which we use in this study. Throughout this paper, we will assume that q satisfies the condition 0<q<1.
(i)
Letq∈0,1 and define the q-number(6)λq=1−qλ1−q,λ∈ℂ=1+q+q2+⋯+qn−1,λ=n∈ℕ=0,λ=0(ii)
Theq-derivative (or q-difference) Dq of f∈A is defined in a given subset of ℂ by(7)Dqfz=fqz−fzq−1z,z≠0,f′0,z=0,provided that f′0 exists.From (7), we note that
(8)limq⟶1−Dqfz=limq⟶1−fqz−fzq−1z=f′z,for a differentiable function f in a given subset of ℂ.From (1) and (7), we observe that
(9)Dqfz=1+∑n=2∞nqanzn−1.By using a special case of general conic domainΩk, concept of q-calculus, and principle of subordination, we define as in [11] the class P1,q as follows.Letp be analytic in E with p0=1. Then, p∈P1,q if and ony if
(10)pz≺p1,qz=2p1z1+q+1−qp1z,and p1z is given by (5).Geometrically, the functionp1∈P1,q takes all values from the domain Ω1,q, which is defined as
(11)Ω1,q=w:1+qw2+q−1w−1<Re1+qw2+q−1w.It can easily be seen thatP1,q⊂Pα, with α=2/3+q, and p∈P1,q implies that Repz>2/3+q. Also, argpz<σπ/2, σ=2/πtan−11+q/2. Also, limq⟶1−P1,q⊂P1/2 and argpz<π/4.We generalize the classP1,q as follows.Definition 1.
An analytic functionp:pz=1+∑n=1∞cnzn is in the class P1,qm if and only if there exist p1, p2∈P1,q such that
(12)pz=m4+12p1z−m4−12p2z.It is clear thatP1,q2=P1,q.Using the classP1,qm, we now define the following classes of analytic function.Definition 2.
LetV1,qm,g denote the class of all functions f∈A, with g∈A such that g∗f′z≠0 in E and zg∗f′′/g∗f′∈P1,qmz∈E.
WithF=zf′, F∈R1,qm,g if and only if f∈V1,qm,g in E. For gz=z/1−z, F∈R1,qm.
Special cases:(i)
V1,qm,z/1−z=V1,qm⊂Vmα, α=2/3+q1/2<α<2/3 and Vm is well-known class of functions of bounded boundary rotation (see [12])(ii)
Let(13)gz=hqz=11−qlog1−qz1−z=∑n=1∞1−qn1−qznnbe convex univalent in E (see [13]).
We note that(14)zhq′z=z1−qz1+z=∑n=1∞nqzn,fz∗zhq′z=z+∑n=2∞nqanzn=zDqfz.
Thus, Definition2 can be written as
(15)zhq∗f′′hq∗f′=zhq′∗f′zhq′∗f=zDqf′Dqf∈P1,qm,that is, Dqf∈R1,qm in E.
(iii)
V1,q2,z/1−z=C1,q⊂C, where C is the class of convex functions in E. Similarly, R1,q2,z/1−z=R1,q is a subclass of the class S∗ of starlike functions in EDefinition 3.
Letf,g∈A with g∗fz≠0 in E. Then, f∈T1,qm,g, if there exists ψ∈V1,qm,g such that
(16)g∗f′g∗ψ′∈P1,q,z∈E.
Special cases:(i)
Choosegz=z/1−z. Then, T1,qm,z/1−z. Then, T1,qm,z/1−z reduces to a subclass of Tm consisting of generalized close-to-convex functions. The class Tm has been introduced and studied in [14], where T2=K is the class of close-to-convex functions (see [15])(ii)
Letgz=hqz, where hqz is given by (44). Then,(17)hq∗f′hq∗ψ′=zhq′∗fzhq′∗ψ=DqfDqψ∈P1,qm.
In this case, we say thatf∈T1,q∗m in E.Remark 4.
(i)
Forψ∈A, the class Cqofq-convex functions is defined as(18)DqzDqψDqψ∈P,z∈E.
Asq⟶1−, Cq⟶C and the class Sq∗ of q-starlike functions, first introduced and studied in [3]. Also, f∈Cq if and only if zDqf∈Sq∗, and it has been proved [3] that
(19)⋂0<q<1Cq=C,⋂0<q<1Sq∗=S∗.(ii)
Ifψ∈C, then zDqψ is starlike of order β=1−q/21+q for z∈E (see [13]).Definition 5.
Letf,g∈A with Fz=g∗fz≠0, FzF′z/z≠0 for z∈E. Then, f is said to belong to the class Q1,qγm,g,a if and only if there exists a function G∈T1,qm,g such that, for z∈E, Rea≥0, 0<γ≤1,
(20)zF′z+aFz=a+1zG′zγ.We note that(21)Q1,q1m,z1−z,0=T1,qmand Q1,q12,hq,0=K1,q⊂Kq, where Kq is the class of q-close-to-convex functions.


## 2. Preliminaries

Lemma 6 (see [16]).
Lethz=1+∑n=1∞cnzn be subordinate to Hz=1+∑n=1∞cnzn. IfHz is univalent in E and HE is convex, then
(22)cn≤C1,n≥1.Lemma 7 (see [17]).
Suppose thatβ,γ∈ℂ with β≠0, and Reβhz+γ>0, z∈E and q∈A such that qz≺hz, z∈E. If pz is analytic in E with p0=1, then
(23)pz+pzβqz+γ≺hz,z∈E,implies that pz≺hz.Lemma 8 (see [18]).
Leth∈P. Then, for z=reiθ(24)∫02πhreiθλdθ<cλ11−rλ−1,z∈E,λ>1,where cλ is a constant.Lemma 9 (see [19]).
Leth∈P. Then, for z=reiθ(25)12π∫02πhz2dθ≤1+3r21−r2.Lemma 10 (see [20]).
Letpz=1+∑n=1∞cnzn be analytic in E, δ≥0, Rec≥0, 0≤θ1<θ2≤2π. If
(26)∫θ1θ2Repz+δzp′zcδ+pzdθ>−π,then
(27)∫θ1θ2Repzdθ>−π.


## 3. The ClassV1,qm,g

Using the well-known results, for example, see [21, 22], the following results can easily be proved.Theorem 11.
Letf∈V1,qm,g. Then, with gz=z/1−z, m≥2,
(28)f′z=f1′z1−αm/4+1/2f2′z1−αm/4−1/2,α=23+q,and f1, f2∈V2=C.Whenq⟶1−, α⟶1/2, we obtain a known result. Also, for f∈V1,qm,hq, (28) can be written as
(29)f∗hq′=f1∗hq′1−αm/4+1/2f2∗hq′1−αm/4−1/2.That is,(30)Dqfz=Dqf11−αm/4+1/2Dqf21−αm/4−1/2,f1,f2∈C.Theorem 12.
Letf∈V1,qm,g and let F=f∗g. Then, for m≥2, z=reiθ, 0≤θ1<θ2≤2π, we have
(31)∫θ1θ2RezF′z′F′zdθ>−m2−11−απ,where α=2/3+q.We note that, withgz=z/1−z, f is univalent for m/2−11−α≤1, α∈1/2,2/3. See [23].Theorem 13.
Letf∈V1,qm,g, m/2−11−α≤1, 1/2<α=2/3+q<2/3. Then, FE, with F=f∗g, contains the Schlicht disc d:
(32)d=w:w<48+m2−α.Proof.
Let(33)pz=m4+12p1z−m4−12p2z,with pz=1+∑n=1∞cnzn and piz=∑n=1∞cn,izn.
Forp∈P1,qm, pi∈P1,q⊂Pα, α=2/3+q. Then, by Lemma 6, we have
(34)cn≤m22−α,α∈12,23foralln≥1.
From Theorem12, it follows that F:Fz=z+∑n=2∞Anzn is univalent in E.
Letw0w0≠0 be any complex number such that Fz≠w0 for z∈E. Then, the function
(35)F1z=w0Fzw0−Fz=z+A2+1w0z2+⋯is analytic and univalent in E.
Now,zF′z′/F′z∈P1,qm, and simple calculations yield that
(36)A2≤m42−α.
We use this bound forA2 and the well-known sharp bound for the second coefficient of univalent functions and have
(37)1w0−A2≤A2+1w0≤2;this gives us
(38)1w0≤2+A2≤2+m42−α=8+m2−α4.
The proof is complete.For the permissible values ofα and m, we obtain several interesting results as special cases of this result.Theorem 14.
Letf∈V1,qm,hq. Then,
(39)1+rm/2+11−rm/2−11−α1−β≤Dqf≤1−rm/2+11+rm/2−11−α1−β,where α=2/3+q, β=1−q/21+q, r⟶1.Proof.
From Theorem11, for f∈V1,qm,hq, we have relation (30). Now, for fi∈C, it follows that zDqfiz=tiz is starlike of order β=1−q/21+q (see [13]). Also, it is known [24] that there exists Fi∈S∗ such that
(40)tizz=Fizz1−β,i=1,2.
Thus, we can write (30) as
(41)Dqfz=F1z/zm/4+1/2F2z/zm/4−1/21−α1−β,F1,F2∈S∗.
Now, using well-known distortion results forFi∈S∗, we obtain (39) as required.Theorem 15.
LetF=f∗g and f∈R1,q2,g, and let for γ,σ>0, F1 be defined by
(42)F1z=γ+1z−γ∫0ztγ−1Fσtdt1/σ,F1=f1∗g.Then,F1∈R1,q2,g in E.(i)
We note that (42) reduces to the generalized Bernardi operator for f∈R1,q2, when σ=1 and gz=z/1−z(ii)
By choosinggz=z/1−z2, we prove this result for f∈V1,q2Proof.
From (42), it follows that
(43)zγF1σz′=tγ−1Fσz,that is,
(44)F1σzγ+σHz=Fσz,Hz=zF1′zF1z.
With simple computations, we obtain from (44)
(45)Hz+zH′zσHz+γ=zF′zFz∈P1,q,z∈E.
Now, using Lemma7 along with (45), we have H∈P1,q and consequently F1∈R1,q2,g. This completes the proof.Theorem 16.
Letf:fz=z+∑n=2∞anzn∈V1,qm,g and F=f∗g with Fz=z+∑n=2∞Anzn. Then, for 1−αm+2>2−σ,
(46)An+1−An≤cm,qn1−αm/2+1+σ−3,where c is constant, α=2/3+q and σ=2/πtan−11+q/2.Proof.
Sincef∈V1,qm,g, it implies that F∈V1,qm⊂Vmα, α=2/3+q. We can write
(47)zF′z′=F′zhσz,h∈Pm,σ=2πtan−11+q2=F′zm4+12h1σz−m4−12h2σz,h1,h2∈P,=m4+12F′zh1σz−m4−12F′zh2σz.
Then,(48)z−ξF′z≤m4+12z−ξF′zh1σz−m4−12z−ξF′zh2σz.
First, we calculate, fori=1,2, by using Theorem 11 in the following integral:
(49)∫02πz−ξF′zhiσzdθ=∫02πz−ξs1z/zm/4+1/2s2z/zm/4−1/21−αhizσdθ,si∈S∗, ξ∈E and z=reiθ.
Let0<r<1. Then, by a result due to Golusin [25], there exists a ξ with ξ=r such that for z=r,
(50)z−ξs1′z≤2r21−r2.
We use (50) and distortion result for s2∈S∗ and apply Holder’s inequality to have
(51)∫02πz−ξF′zhiσzdθ≤2πcq2r21−r212π∫02πs1zμ−12/2−σdθ×12π∫02πhizσdθσ/2,where μ=1−αm/4+1/2.
Now, with subordination forhi∈P, starlike function s1 and Lemma 8 in (51), it follows that
(52)∫02πz−ξF′zhizσdθ≤cq11−r1−αm/2+1−1.
From (48), (49), and (52), it follows that
(53)n+12ξAn+1−n2An≤cm,q11−r1−αm/2+1+σ−1,r⟶1.
Thus, choosingr=1−1/nn⟶∞ in (53), we have
(54)An+1−An≤cm,qn1−αm/2+1+σ−3,where cq and cm,q are constants.As a special case, we note that, forf∈V1,qm,z/1−z, we have
(55)an+1−an≤o1·n1−αm/2+1+σ−3.Whenq⟶1−α=1/2, σ=1, and in this case, if we choose m=6, then
(56)an+1−an≤o1.


## 4. The ClassT1,qm,g

We discuss here some basic properties off∈T1,qm,g.Theorem 17.
Letf∈T1,qm,g. Then, there exist f1, f2∈T1,q2,g if and only if
(57)f′z=f1′zm/4+1/2f2′zm/4−1/2,z∈E.Proof.
From Definition3, we can write
(58)f′z=G′zhz,G∈V1,qm,g,h∈P1,q=g1′zm/4+1/2g2′zm/4−1/2hz,g1,g2∈V1,q2,g=g1′zhzm/4+1/2g2′zhzm/4−1/2=f1′zm/4+1/2f2′zm/4−1/2,f1,f2∈T1,q2,g.
The converse case follows easily, and the proof is complete.SinceP1,q⊂Pα, α=2/3+q, we note that V1,qm,z/1−z⊂Vmα, and for the class Vmα, we refer to [22]. From Definition 3, we have
(59)T1,qm,z1−z⊂Tm23+q.We now prove the following inclusion result.Theorem 18.
For0<α1<α2, with αi=2/3+q,
(60)Tmα1⊂Tmα2.Proof.
Letf∈Tmα1. Then, there exists G∈Vmα1 such that
(61)f′zG′z=1−αhz+α1,h∈P.
SinceVmα1⊂Vmα2, and Pα1⊂Pα2, for α1<α2, it follows that f∈Tmα2.We now discuss a geometrical property for the classT1,qm,g and investigate the behaviour of the inclination of the tangent at a point w=freiθ to the image of the circle Cr=z:z=r, 0≤r<1, and θ is any number of interval 0,2π under the mapping of f∈T1,qm,g.LetF=f∗g, G=ψ∗g. Let
(62)ϕθ=π2+θ+argF′reiθ,and for θ2>θ1, θ1,θ2∈0,2π,
(63)ϕθ2−ϕθ1=θ2+argF′reiθ2−θ1−argF′reiθ1.Now, since(64)θ+argF′reiθ=θ+Re−ilnF′reiθ,we have
(65)∂∂θθ+argF′reiθ=Re1+reiθF′′reiθF′′reiθ.Therefore,(66)∫θ1θ2∂∂θθ+argF′reiθdθ=∫θ1θ2Re1+reiθF′′reiθF′′reiθdθ.On the other hand,(67)∫θ1θ2∂∂θθ+argF′reiθdθ=θ2+argF′reiθ2−θ1−argF′reiθ1=ϕθ2−ϕθ1.So, the integral on the left hand side of (67) characterizes the increment of the angle of the tangent to the curve Γr between the points wθ2 and wθ1 for θ2>θ1.We now prove the following.Theorem 19.
Letf∈T1,qm,g and F=f∗g. Then, for 0≤r<1, θ2>θ1, θ1,θ2∈0,2π, we have
(68)∫θ1θ2Re1+reiθF′′reiθF′′reiθdθ>1−αm2−1+σπ,where σ=2/πtan−11+q/2 and α=2/3+q.Proof.
From Definition3, we can write
(69)F′z=G′zhσz,G∈V1,qm,g,h∈P,where σ=2/πtan−11+q/2 and α=2/3+q.
Since, forh∈P, it is well known that
(70)hz−1+r21−r2≤2r1−r2.
Thus, the values ofhz are contained in the circle whose diameter is the line segment from 1−r/1+r to 1+r/1−r. The circle is centered at the point 1+r2/1−r2 and has the radius 2r/1−r2. So, arghz attains its maximum at points where a ray from the origin is tangent to the circle, that is, when
(71)arghz=±sin−12r1−r2.
Thus, it follows that(72)maxh∈P∫θ1θ2Rereiθh′reiθhreiθdθ≤2sin−12r1−r2=π−cos−12r1−r2.
Now, from (69), (72), and Theorem 12, it follows that, for z=reiθ, θ1<θ2,
(73)∫θ1θ2Re1+reiθF′′reiθF′′reiθdθ>1−αm2−1+σπ.Remark 20.
In [23], the class Kβ1 of functions f∈A if and only if, for θ1<θ2, z=reiθ, β1≥0,
(74)∫θ1θ2Re1+zf′′reiθf′′reiθdθ>−β1π.For0≤β1≤1,f is close-to-convex and hence univalent. For β>1, f need not to be finitely valent. We note that T1,qm,g⊂K1−αm/2−1+σ. It can easily be seen that T1,qm,g consists of univalent functions for 1−αm/2−1+σ≤1. Also, it can easily be seen that T1,qm,g forms a subset of a linear-invariant family of order 1−αm/2−1+σ−1. For this, we refer to [26].Theorem 21.
Letf∈A and zf′z′/G′z∈P1,q, G=zG1′ and G1∈V1,qm. Let h be defined by
(75)h′z=zf′z′1+zG1′′z/G1′z.Then,h∈T1,qm in E.Proof.
As(76)G′z=zG1′z′=G1′z1+zG1′′zG1′z.
Therefore,(77)zf′z′G′z=zf′z′G1′z1+zG1′′z/G1′z=h′zG1′z∈P1,q.
This provesh∈T1,qm in E.Theorem 22.
Letf:fz=z+∑n=2∞anzn belong to the class T1,qm. Then, for m>2+2−σ/1−α, α=2/3+q, σ=2/πtan−11+q/2,
(78)an=o1·n1−αm/2+1+σ−2,where o1 is a constant.Proof.
Forf∈T1,qm, we can write
(79)f′z=G′zhσz,G∈Vmα,σ=2πtan−11+q2=G1′1−αhσz,G1∈Vm,α=23+q.
It is known [27] that, for all m>2, G1∈Vm, we can write
(80)zG1′z=szpm/2−1z,s∈S∗,p∈P.
Thus, by using Cauchy theorem and using (79) and (80), we have
(81)nan≤12πrn∫02πsz1−αpz1−αm/2−1hzσdθ≤1rnr1−r21−α12π∫02πpz1−αm/2−12/2−σdθ2−σ/212π∫02πhz2dθσ/2≤cm,q11−r1−αm/2−1+σ−1,where we have used Holder’s inequality and Lemma 9. Setting r=1−1/n, n⟶∞ in (81), we get
(82)an=o1·n1−αm/2−1+σ−2.As a special case, whenq⟶1−, f∈T1,qm=Tm, α=1/2, σ=1/2, and in this case,
(83)an=o1·nm/4−1n⟶∞.We now study a radius problem forf∈T1,qm,g.Theorem 23.
Letf∈T1,qm,g, and let F=f∗g. Then, f∈V1,q2,g for z<rm=m−m2−4/2.Proof.
Letψ∈V1,qm,g and G=ψ∗g. Then, by Definition 2,
(84)zG′z′G′z=Hz∈P1,qm.
Using (12), it can be easily shown that H∈P1,q for z<rm=m−m2−4/2. Now, for f∈T1,qm,g, we can write
(85)F′zG′z=pz,p∈P1,q,G∈V1,qm,g,and since G∈V1,q2,g in z<rm, the required result follows at once.Special cases:(i)
Letgz=z/1−z and q⟶1−. Then, f∈Tm, and in this case, f is convex for z<rm(ii)
Letgz=hqz, where hqz is given by (13). Then,(86)DqzDqψzDqψz∈P1,qforz<rm,which implies ψ is q-convex in z<rm. Consequently, f is q-close-to-convex in z<rm.


## 5. The ClassQ1,qγm,g,a

Using Definition5, we shall discuss the class Q1,qγm,g,a in this section.Theorem 24 (integral representation).
A functionf∈Q1,qγm,g,a if and only if there exists some function F∈Q1,qγm,g,∞ such that
(87)fz=1+aza∫0zta−1Ftdt.Proof.
From Definition5, it follows that F∈Q1,qγm,g,∞ if and only if we can write
(88)Fz=zG′zγ,G∈T1,qm,g.
Thus,(89)zf′z+afz=a+1Fz,F∈Q1,qγm,g,∞,and from this observation, the proof is immediate.Theorem 25.
Let0<γ1<γ2≤1. Then,
(90)Q1,qγ1m,g,a⊂Q1,qγ2m,g,a.Proof.
Letf∈Q1,qγ1m,g,a. Then, with F=f∗g, G=ψ∗g, we have
(91)zF′z+aFz=a+1·zG′zγ1,G∈T1,qm.
We can write (91) as
(92)zF′z+aFz=a+1zJ′zγ2,where J′z=G′zγ1/γ2.
We now show thatJ∈T1,qm.
Consider(93)J′z=G′zγ1/γ2,G∈T1,qm,γ1γ2<1.
Let(94)G1′z=g1′zγ1/γ2,g1∈V1,qm.
Then,(95)zG1′z′G1′z=1−γ1γ2+γ1γ2zg1′z′g1′z.
This impliesG1∈V1,qm. Now,
(96)J′zG1′z=G′zg1′zγ1/γ2=hzγ1/γ2,h∈P1,q.
Sinceγ1/γ2<1, hzγ1/γ2∈P1,q in E. Thus, from (92), it follows that F∈Q1,qγ2m,a.Theorem 26.
Letf∈Q1,qγm,g,a, Rea≥0, 0<γ≤1. Then, for 0≤θ1<θ2≤2π, z=reiθ, we have
(97)∫θ1θ2Repz+zp′za+pzdθ>−γ1−αm2−1+σπ,where pz=zF′z/Fz, F=f∗g.Proof.
SinceF∈Q1,qγm,a, we can write
(98)zF′z+aFz=a+1zG′zγ,G∈T1,q,Rea≥0.
That is(99)Fzpz+a=a+1G′zγ.
Logarithmic differentiation of (99) and using Theorem 19, the required result immediately follows.We note the following special cases.(i)
Letγ1−αm/2−1+σ≤1. Then, for z=reiθ, θ1<θ2, using Lemma 10, with δ=1 and cδ=a, we get(100)∫θ1θ2Repzdθ>−π(ii)
If we choosegz=z/1−z2 in (i), then from (100) and a result due to Kaplan [15], it follows that F is close-to-convex and hence univalent in ETheorem 27.
Letf∈Q1,qγm,z/1−z2,∞. Then, f is starlike of order λ=1−γ1−α, α=2/3+q, for z<r∗, where
(101)r∗=2m+2+m2+4m.Proof.
By Definition5, we have
(102)fz=zG′zγ,G∈T1,qm.
SinceT1,qm⊂Tmα, α=2/3+q, it is known [28] that there exists G1∈Tm such that
(103)G′z=G1′z1−α.
Using (103) in (102), with logarithmic differentiation and simple calculations, we obtain (see [14])
(104)11−λRezf′zfz−λ=RezG1′z′G1z,λ=1−γ1−α≥r2−m+2r+11−r2.
The right-hand side is positive forz<r∗, where r∗ is given by (101).Ifq⟶1− and γ=1, then f is starlike of order 1/2 in z<r∗.



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*Source: 2608060-2023-02-04.xml*