# New Results for Some Generalizations of Starlike and Convex Functions

Authors: Ali Ebadian; Nafya Hameed Mohammed; Ebrahim Analouei Adegani; Teodor Bulboacă
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The purpose of the current paper is to investigate several various problems for the categoriesSTLs,SNe∗, and other related categories such as various new outcomes for the coefficients of f, together with majorization issues, the Hankel determinant, and the logarithmic coefficients with sharp inequalities and differential subordination implications.


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## Body

## 1. Introduction and Preliminaries

LetU≔z∈ℂ:z<1 denote the open unit disk in the complex plane ℂ. Let A be the category of functions f analytic in U that has the following representation(1)fz=z+∑n=2∞anzn,z∈U,and denoted by S the subclass of all functions of A which are univalent in U. Then, the logarithmic coefficients γn of f∈S are defined as the coefficients of the series expansion(2)logfzz=2∑n=1∞γnfzn,z∈U.These coefficients play an important role for various estimates in the theory of univalent functions (see for example [1–3] and [4], Chapter 2), and note that we will use the notation γn instead of γnf.Utilizing the principle of subordination, Ma and Minda [5] introduced the classes S∗φ and Cφ, where we make here the assumptions that the function φ is univalent in the unit disk U and satisfies φ0=1, with the power series expansion of the form(3)φz=1+L1z+L2z2+L3z3⋯,z∈U.They considered the abovementioned classes as follows:(4)S∗φ≔f∈A:zf′zfz≺φz,Cφ≔f∈A:1+zf″zf′z≺φz,where the symbol “≺” stands for the usual subordination. Some special subclasses of the class S∗φ and Cφ play a significant role in the Geometric Function Theory because of their geometric properties.For instance, the categoriesS∗φ and Cφ reduce to the categories S∗A,B and CA,B of the popular Janowski starlike and Janowski convex functions for φz=1+Az/1+Bz with −1≤B<A≤1, respectively. By replacing A=1−2α0≤α<1 and B=−1 in these function families, we get the classes S∗α and Cα of the starlike functions of orderα and convex functions of ordera, respectively. Especially, S∗≔S∗0 and C≔C0 are the family of starlike functions and of convex functions in the unit disk U, respectively.Further, forφz=1+z, we get the family SL∗ defined by Sok’ol and Stankiewicz [6], including functions f such that w=zf′z/fz stands in the region bounded by the right-half branch of the lemniscate of Bernoulli given by w2−1<1. Moreover, the properties of the classes Se∗≔S∗ez consisting of functions f∈A satisfying the condition logzf′z/fz<1,z∈U were considered by Mendiratta et al. in [7]. Raina and Sokół [8] studied the family Sc∗≔S∗h, where hz=z+1+z2 maps U onto the crescent-shaped region w∈ℂ:w2−1<2w,Rew>0.In addition, forφz=1−z−s0<s≤1, we obtain the categories SThpls and CVhpls studied by Kanas et al. [9], with the property that zf′z/fz and 1+zf″z/f′z lies in a domain bounded by a right branch of a hyperbola ρs=2cosϕ/s−s where ϕ<πs/2, respectively. Also, Goel and Kumar [10] introduced the classes SSG∗ and CSG for φz=2/1+e−z where this function maps U onto a domain ΔSG≔w∈ℂ:logw/2−w<1.Lately, Masih and Kanas [11] introduced and studied the categories STLs and CVLs by (5)STLs≔f∈A:zf′zfz≺Lsz≔1+sz2,0<s≤12,CVLs≔f∈A:1+zf″zf′z≺Lsz≔1+sz2,0<s≤12,and investigated some outcomes regarding the behavior of the functions of these classes. The function Ls maps the unit disk onto a domain bounded by the limacon u−12+v2−s42=4s2u−1+s2+v2. Further, LsU is symmetric respecting the real axis, and Ls is starlike with respect to Ls0=1. Also, Ls′0>0 and ReLsz>0 in U (see for more details [11]); hence, Ls satisfies the conditions of the category of the Ma-Minda functions (see [5]). In addition, for n=1,2,⋯ the functions(6)Φs,nz≔zexp∫0zLstn−1tdt=z+2snzn+1+n+4s22n2z2n+1+⋯,z∈U,Ks,nz≔∫0zexp∫0xLstn−1tdtdx=z+2snn+1zn+1+n+4s22n22n+1z2n+1+⋯,z∈U,play as extremal functions for some problems for the classes STLs and CVLs, respectively.For instance, the quantityzf′z/fzor1+zf″z/f′z lies in a domain bounded by the lima¸con u−12+υ2−s42=4s2u−1+s2+υ2 in the category STLsorCVLs while zf′z/fzor1+zf″z/f′z lies in a domain bounded by a right branch of a hyperbola ρs=_sφ<πs/22cosϕ/s where in the category SThplsorCVhpls. Therefore, it is observed that these categories have different structures and geometric properties (see for more details [9, 11]).Recently, Wani and Swaminathan [12] investigated the new Ma-Minda-type function classes SNe∗ and CNe and obtained some characteristic properties of these classes defined by (7)SNe∗≕f∈A:zf′zfz≺φNez≔1+z−z33,CNe≕f∈A:1+zf″zf′z≺φNez≔1+z−z33.The functionφNe maps U onto the interior of the nephroid, a 2-cusped kidney-shaped curve,(8)u−12+υ2−493=4υ23.Further, forn=2,3,⋯, the functions (9)Ωnz≔zexp∫0zφNetn−1−1tdt=z+znn−1+z2n−12n−12+⋯,z∈U,Λnz≔∫0zexp∫0zφNetn−1−1tdtdx=z+znnn−1+z2n−122n−1n−12+⋯,z∈U, play the role of extremal functions for several problems for the categories SNe∗ and CNe, respectively.Finding the upper bound for coefficients has been one of the central topics of research in the Geometric Function Theory as it gives several properties of functions. In particular, the bound for the second coefficient gives growth and distortion theorems for the functions of the class S. In [13], Ebadian et al. studied some coefficient problems for the categories SThpls,CVhpls,SSG∗ and related categories like sharp bounds for initial coefficients, logarithmic coefficients, Hankel determinants, and Fekete-Szegö problems. Also, they investigated some geometric properties as applications of differential subordinations.According to the abovementioned issues, motivated essentially by the recent work [13], this paper is aimed at investigating some various problems for the categories STLs,SNe∗, and other related categories like various new outcomes for the coefficients of the power series expansions of the functions that belong to these classes, together with majorization issue, the Hankel determinant, and the logarithmic coefficients with sharp inequalities, and differential subordination implications.


## 2. Logarithmic Coefficients, Coefficient Estimates, and Majorization Issue

We obtain the estimates for the logarithmic coefficients, the first three coefficients, and majorization issue (see [14]) for the functions belonging to STLs,SNe∗ and similar classes. For this purpose, suppose Ω represents the set of all analytic functions ω in U, with ω0=0 and ωz<1 for z∈U, i.e., Ω is the set of Schwarz functions. Also, we recall the following lemmas:Lemma 1 (see [15], Theorem 1).
Let the functionf∈S∗φ. If φ is starlike with respect to 1, then the logarithmic coefficients of f satisfy the inequality:(10)γn≤L12,n∈ℕ≔1,2,3⋯.
The above inequality is sharp for anyn∈ℕ, for the function fn given by zfn′n/fnz=φzn.Lemma 2 (see [16], p. 172).
Assume thatω is a Schwarz function, such that ωz=∑n=1∞ωnzn,z∈U.
Then,(11)ωn≤1,ωn≤1−ω12,n=2,3,⋯Lemma 3 (see [15], Theorem 2).
Let the functionf∈Cφ. Then, the logarithmic coefficients of f satisfy the inequalities(12)γ1≤L14,(13)γ2≤L112,if4L2+L12≤4L14L2+L1248if4L2+L12≤4L1.
The bounds (12) and (13) are sharp. Also, if L1, L2, and L3 are real values, then(14)γ3≤L124Kq1;q2,where q1=L1+4L2/L1/2,q2=L2+2L3/L1/2 and Kq1;q2 is given by (see [17, 18])(15)Kq1;q2=1ifq1,q2∈D1∪D2q2ifq1,q2∈D623q1+1q1+13q1+1+q2,ifq1,q2∈D8∪D9q23q12−4q12−4q2q12−43q2−1,ifq1,q2∈D10and the sets Dk, k=1,2,6,8,9,10 are stated as given below:(16)D1=q1,q2:q1≤12,q2≤1,D2=q1,q2:12≤q1≤2,427q1+12−q1+1≤q2≤1,D6=q1,q2:2≤q1≤4,q2≥112q12+8,D8=q1,q2:12≤q1≤2,−23q1+1≤q2≤427q1+13−q1+1,D9=q1,q2:q1≥2,−23q1+1≤q2≤2q1q1+1q12+2q1+4,D10=q1,q2:2≤q1≤4,2q1q1+1q12+2q1+4≤q2≤112q12+8.For two analytic functionsf and g in U, the function g is said to be majorized to f, written as gz≪fz, if there exists an analytic function ψ in U, with ψz≤1,z∈U, such that (see MacGregor [14]).(17)gz=ψzfz,z∈U.Cho et al. [19] studied the majorization issue for the category S∗φ of starlike functions as follows:Lemma 4 (see [19], Theorem 2).
Ifg∈A,f∈S∗φ with gz≪fz, then g′z≤f′z for all z in the disk∣z∣≤r1, where r1 is the smallest positive root of the equation(18)minz=rφz1−r2−2r=0,r∈0,1.Settingφ≕Ls and φ≕φNe in Lemma 1, since (19)logΦs,1zz=2∑n=1∞γnΦs,1zn=2sz+⋯,z∈U,logΩ2zz=2∑n=1∞γnΩ2zn=z+⋯,z∈U, we obtain the following two results:Theorem 5.
If the functionf∈STLs, then(20)γn≤s,n∈ℕ.
This inequality is sharp forn=1 for the function Φs,1.Theorem 6.
If the functionf∈SNe∗, then(21)γn≤12,n∈ℕ.
This inequality is sharp forn=1 for the function Ω2.Theorem 7.
If the functionf∈CVLs, then(22)γ1≤s2,γ2≤s6,γ3≤s12.
These bounds are sharp forf=Ks,n and n=1,2,3, respectively.Proof.
Applying Lemma3 with φ=Ls, the first two estimates follows immediately, and the results are sharp. To prove the above inequality |γ3|, by Lemma 3 we have(23)γ3≤L124Kq1;q2=s12Kq1;q2,where(24)q1=L1+4L2/L12=2s,andq2=L2+2L3/L12=s22.
First, regardingD1 it is clear that q1≤1/2 for s∈0,1/4, and the inequality q2≤1 holds for s∈0,1/2. Therefore, q1,q2∈D1 for s∈0,1/4.
Next, if we selectD2 then 1/2≤q1≤2 holds for s∈1/4,1/2. Also, the inequality 4/27q1+13−sq1+1≤q2 is equivalent to(25)Gs≔32s327+23s218−10s9−2327≤0,that holds for s∈0,1/2 (see Figure 1) and the fact that G1/2<0. Further, q2≤1 holds for s∈0,1/2. Therefore, regarding these results, we conclude q1,q2∈D2fors∈1/4,1/2.
From the abovementioned two results, using (23) and Lemma 3, we conclude that γ3≤s/24Kq1;q2=s/12.
Since(26)logKs,1zz=2∑n=1∞γnKs,1zn=sz+⋯,z∈U,logKs,2zz=2∑n=1∞γnKs,2zn=2s6z2+⋯,z∈U,logKs,3zz=2∑n=1∞γnKs,3zn=2s12z3+⋯,z∈U,it follows that the bounds are sharp for f=Ks,n and n=1,2,3, respectively.Theorem 8.
If the functionf∈CNe, then(27)γ1≤14,γ2≤112,γ3≤124.
These bounds are sharp forf=Λn+1 and n=1,2,3, respectively.Proof.
Applying Lemma3 for φ=φNe, we obtain the first two bounds, and these results are sharp. To find the upper bound for γ3, by Lemma 3, we have(28)γ3≤L124Kq1;q2=124Kq1;q2,where(29)q1=L1+4L2/L12=12,q2=L2+2L3/L12=−13.
Therefore,qq,q2∈D1 and from Lemma 3, we get(30)γ3≤124Kq1;q2=124.
Since(31)logΛ2zz=2∑n=1∞γnΛ2zn=12z+⋯,z∈U,logΛ3zz=2∑n=1∞γnΛ3zn=16z2+⋯,z∈U,logΛ4zz=2∑n=1∞γnΛ4zn=112z3+⋯,z∈U,we conclude that the bounds are sharp for f=Λn+1 and n=1,2,3, respectively.Theorem 9.
If the functionf∈STLs has the power expansion series given by (1), then(32)a2≤2s,a3≤s,ifs∈0,25,5s22ifs∈25,12,and(33)a4≤2s3×1,ifs∈0,0.48481⋯,23q1+1q1+13q1+1+q21/2,ifs∈0.48481⋯,0.55177⋯,q23q12−4q12−4q2q12−43q2−11/2,ifs∈0.55177⋯.,0.55470=213,q2,ifs∈0.55470⋯.=213,12,where q1≕4s and q2≕7s2/2. The first two bounds and the first and last bound for a4 are sharp.Proof.
Iff∈STLs, by the concept of the subordination, there exists the function ωz=∑n=1∞ωnzn∈Ω such that(34)zf′zfz=φwz=1+L1w1z+L1w2+L2w12z2+L1w3+2w1w2L2+L3w13z3+⋯,z∈U,where φ=Ls. From the above equality, it follows that(35)a2=L1ω1,2a3−a22=L1ω2+L2ω12,3a4−3a2a3+a23=L1ω3+2ω1ω2L2+L3ω13.
For the estimate ofa2, applying Lemma 2 for the first of the above equalities, we get a2≤2s. Next, using again Lemma 2, for a3, it follows that(36)a3=L1w2+L2+L12w122≤L11−w12+L2+L12w122=L1+L2+L12−L1w122=L12,ifL2+L12≤L1,L2+L122,ifL2+L12≤L1=s,ifs∈0,25,5s22,ifs∈25,12.
Now, fora4, we obtain (see also [18])(37)a4≤L13w3+32L1+2L2L1w1w2+32L2+12L12+L3L1w13≤L13Kq1,q2,where(38)q1=32L1+2L2L1=4s,q2=32L2+12L12+L3L1=7s22.
First, it is clear thatq1≤1/2 for s∈0,1/8, and the inequality q2≤1 holds for s∈0,2/7.
Therefore, according to the definition ofD1, it follows that q1,q2∈D1 for s∈0,1/8.
Also, regarding the definition ofD2, we conclude that the inequality 1/2≤q1≤2 holds for s∈1/8,1/2. Also, the inequality 4/27q1+13−q1+1≤q2 is equivalent to(39)M1s≔256s327+65s218−20s9−2327≤0,which holds for s∈0,s0, where s0=0.48481⋯ (see Figure 2). Further, q2≤1holds for s∈0,2/7; therefore, considering these results, we conclude q1,q2∈D2 fors∈1/8,0.48481⋯.
Next, regarding the definition ofD6,2≤q1≤4 holds for s∈1/2,1, and the second inequality holds for s∈2/13=0.55470⋯,1/2. Therefore, q1,q2∈D6 for s∈2/13=0.55470⋯,1/2.
According to the definition ofD8,1/2≤q1≤2 holds for s∈1/8,1/2, and a simple computation shows that the first and second inequality from the right side hold for s∈0,1/2 and s∈0.48481⋯,1/2, respectively. Therefore, q1,q2∈D8 for s∈0.48481⋯,1/2.
Further, for the definition ofD9, it is clear that q1≥2 for s∈1/2,1/2. On other the hand, the second inequality from the right side holds for s∈0,1/2, and the first inequality from the right side is equivalent to(40)M2s≔56s4+28s3−18s2−8s≤0,holds for s∈0,0.55177⋯ (see Figure 3). Therefore, q1,q2∈D9 for s∈1/2,0.55177⋯.
Finally, according to the definition ofD10, from the above explanations, it follows that q1,q2∈D10 for s∈0.55177⋯,2/13=0.55470.
Hence, applying the mentioned outcomes from (37), we conclude that(41)a4≤L13Kq1,q2=2s3×1,ifs∈0,0.48481⋯,23q1+1q1+13q1+1+q21/2,ifs∈0,0.48481⋯,0.55177⋯,q23q12−4q12−4q2q12−43q2−11/2,ifs∈0.55177⋯,0.55470⋯=213,q2,ifs∈0.55470⋯=213,12.
The extremal function fora2 is f=Φs,1, and for a3 in the first and second inequalities is given by f=Φs,2 and f=Φs,1, respectively. Also, the extremal function for a4 in the first and fourth above inequalities is given by f=Φs,3 and f=Φs,1, respectively.Figure 1
The plot ofGs for s∈0,1/2.Figure 2
The plot ofM1s for s∈0,1/2.Figure 3
The plot ofM2(s) for s∈0,1/2.For a functionf∈A, we have f∈CVLs if and only if zf′z∈STLs. Setting gz≔zf′z, we have gz=z+∑n=2∞bnzn where bn=nan for n≥2. Hence, we can get similarly the following theorem:Theorem 10.
If the functionf∈CVLs has the power expansion series given by (1), then(42)a2≤s,a3≤s3,ifs∈0,25,5s26,ifs∈25,12,a4≤2s12×1,ifs∈0,0.48481⋯,23q1+1q1+13q1+1+q21/2,ifs∈0.48481⋯,0.55177⋯,q23q12−4q12−4q2q12−43q2−11/2,ifs∈0.55177⋯,0.55470⋯=213,q2,ifs∈0.55470⋯=213,12,where q1≕4s and q2≕7s2/2. The extremal function for ∣a2∣ is f=Ks,1, and for ∣a3∣ in the first and second inequalities is given by f=Ks,2 and f=Ks,1, respectively. Also, the extremal function for ∣a4∣ in the first and fourth above inequalities is given by f=Ks,3 and f=Ks,1, respectively.Settingφ≕φNe in the proof of Theorem 9, hence, L1=1, L2=0, and L3=−1/3, and we get similarly the following result:Theorem 11.
If the functionf∈SNe∗ has the power expansion series given by (1), then(43)a2≤1,a3≤12,a4≤118.
The inequalities are sharp forf=Ω2.For the above choice ofφ, the analogue of Theorem 10 is the next one.Theorem 12.
If the functionf∈CNe has the power expansion series given by (1), then(44)a2≤12,a3≤16,a4≤172.
The inequalities are sharp forf=Λ2.Our next result deals with a majorization problem for the functions of the classSTLs:Theorem 13.
Letg∈A and f∈STLs, with gz≪fz. Then, for all z in the disk z≤r2, we have g′z≤f′z, where r2 is the smallest positive root of the equation(45)1−s21−r2−2r=0,r∈0,1.Proof.
The result follows from Lemma4 using the fact that min minz=rLsz=Ls−r that can be found in [11], Lemma 2.


## 3. Second Hankel Determinant Problem

In this section, we investigate the problem of coefficients for the second Hankel determinant problem (see [20–24]) for the classes STLs,SNe∗ and similar classes. For this purpose, we need some parts of Theorems 1 and 2 of [25], as follows:Lemma 14 (see [25], Theorem 1).
Let the functionf∈S∗φ. If L1, L2, and L3 satisfy the conditions(46)L2≤L1,4L1L3−L14−3L22−3L12≤0,then the second Hankel determinant satisfies(47)a2a4−a32≤L124.Lemma 15 (see [25], Theorem 2).
Let the functionf∈Cφ.(i)
IfL1, L2, and L3 satisfy the conditions(48)L12+4L2−2L1≤0,6L1L3+L12L2−L14−4L22−4L12≤0,then the second Hankel determinant satisfies(49)a2a4−a32≤L1236.(ii)
IfL1, L2, and L3 satisfy the conditions(50)L12+4L2−2L1>0,26L1L3+L12L2−L14−4L22−L13−4L1L2−6L12≤0,then the second Hankel determinant satisfies(51)a2a4−a32≤L12576166L1L3+L12L2−L14−4L22−12L13−48L1L2−36L12−L14−8L12L2−16L226L1L3+L12L2−L14−4L22−L13−4L1L2−2L12.By choosingφ≕Ls and φ≕φNe in Lemmas 14 and 15, then we obtain the following outcomes:Theorem 16.
If the functionf∈STLs, then the second Hankel determinant satisfies the inequality(52)a2a4−a32≤s2.
The inequality is sharp forf=Φs,2.Theorem 17.
If the functionf∈SNe∗, then the second Hankel determinant satisfies(53)a2a4−a32≤14.
The inequality is sharp forf=Ω3.Theorem 18.
If the functionf∈CVLs, then the second Hankel determinant satisfies the inequality(54)a2a4−a32≤a29,ifs∈0,12,s2244s2−4s−32s2−2s−1,ifs∈12,12.
The first inequality is sharp forf=Ks,2.Proof.
From Lemma15 (i), a simple computation shows that the first and second inequalities of the assumption hold for s∈0,1/2 and s∈0,1/2, respectively. Also, regarding Lemma 15 (ii), the first and second (which is equivalent to 8s24s2−2s−3≤0) inequalities of the assumption hold for s∈1/2,1/2 and s∈0,1/2, respectively; hence, we obtain the required result.Theorem 19.
If the functionf∈CNe, then the second Hankel determinant satisfies(55)a2a4−a32≤136.
The inequality is sharp forf=Λ3.Proof.
The result follows immediately from Lemma15 (i).


## 4. Differential Subordinations

The principle of differential subordination has important usages in the theory of analytic functions (for details see [26]). The significant importance of the Briot-Bouquet differential subordination inspired many authors to study these types of subordinations, and recently, many generalizations and extensions of the Briot-Bouquet differential subordination have been obtained; for example, see [27–30]. The following lemma will be a useful tool to get the main results:Lemma 20 (see [26], Theorem 3.4h, p. 132).
Letq be analytic in U and let ψ and θ be analytic in a domain D containing qU, with ψw6=0 when w∈qU. Set Qz≔zq′zψqz and hz≔θqz+Qz. Suppose that:(i)
Eitherh is convex, or Q is starlike univalent in U(ii)
Rezh′z/Qz>0 for z∈U
Ifp is analytic in U, with p0=q0,pU⊂D(56)θpz+zp′zψpz≺θqz+zq′zψqz,then pz≺qz, and q is the best dominant of (56).We will obtain sufficient conditions for certain subordinations involving the different mentioned functions and the functionLs.Theorem 21.
Letp be an analytic function in U, with p0=1, such that(57)1+βzp′z≺Lsz,with β>0, and let jsz≔2sz+1/2s2z2.(i)
For−1<A≤1 if β≥−2/1+Ajs−1, then pz≺1+Az/1−z;(ii)
Ifβ≥max−j1/2−1/2−2;−j1/21/2=1.98743⋯, then pz≺z+1+z2;(iii)
Ifβ≥max−j1/2−1;−j1/21/2−1=4.01776⋯,z then pz≺1+z(iv)
Ifβ≥maxe/1−e−j1/2−1;j1/21/e−1=1.84175⋯, then pz≺ezProof.
(i)
The differential equation(58)1+βzq′z=Lsz,has an analytic solution q:U⟶ℂ, with q0=1, defined by(59)qz=1+1β∫0zLst−1tdt=1+1β2sz+12s2z2=1+1βjsz.
In order to prove our result, we will use Lemma20. Thus, let define the functions θw=1 and ψw=β, for w∈ℂ. These functions are analytic in the domain D≔ℂ containing qU and ψw≠0 for all w∈qU. Let Q,h:U⟶ℂ be defined by(60)Qz≔zq′zψqz=βzq′z=2sz+s2z2=Lsz−1,hz≔θqz+Qz=1+βzq′z.
SinceLsz−1 is starlike with respect to 0, hence, Q is a starlike function in U, and on the other hand(61)Rezh′zQz=RezQ′zQz>0,z∈U.
Thus, according to Lemma 20(62)1+βzp′z=θpz+zp′zψpz≺θqz+zq′zψqz=1+βzq′z,implies pz≺qz.
The required subordinationpz≺1+Az/1−z holds if qz≺1+Az/1−z. For the function Jz≔1+Az/1−z, with −1<A≤1, we have(63)JU=w∈ℂ:Rew>1−A2.
SinceJ is univalent in U and q0=J0, the subordination qz≺Jz holds whenever qU⊂JU. On the other hand, the domain qU is symmetric with respect to the real axis because q has real coefficients. If jsz=2sz+1/2s2z2, then zjs′z=Lsz−1=Qz, Q0=0, and Q is starlike with respect to Q0=0. It follows that(64)Re1+zjs″zjs′z=RezQ′zQz>0,z∈U,that is js is convex in U, and therefore, qis convex in U. Since qU is convex, qU is symmetric with respect to the real axis, q−1<q1, so it follows that the inclusion qU⊂JU holds if(65)q−1=1+1βjs−1≥1−A2,that is β≥−2/1+Ajs−1. Thus, the subordination qz≺1+Az/1−z holds whenever β≥−2/1+Ajs−1.(ii)
Similarly to the proof of Theorem21 (i), it is sufficient to establish that(66)qz=1+1βjsz≺hz≔z+1+z2,for some values of the parameters s and β. For this work, first, we will prove(67)js1z≺js2z,if0<s1<s2≤12.
Let denote byϕζ≔2ζ+1/2ζ2, and Ur≔z∈ℂ:z<r, 0<r≤1. Since the function js is convex in U, it follows that it is univalent in U for all 0<s≤1/2, and js10=js20. Therefore, to prove the subordination (67), we need to show that js1U⊂js2U.
It is easy to see that(68)js1U=js1z:z∈U=ϕs1z:z∈U=ϕζ:ζ∈Us1=ϕUs1⊂ϕUs2=ϕζ:ζ∈Us2=ϕs2z:z∈U=js1z:z∈U=js2U,therefore, the subordination (67) holds. Thus, from (67), it follows that(69)jsz≺j1/2z,0<s≤12,which implies that (66) holds for all 0<s≤1/2 if(70)q~z≔1+1βj1/2z≺hz=z+1+z2.
Sinceh is univalent in U (see [8]) and q~0=h0, it is enough to establish q~U⊂hU. Using the MAPLE™ software, from Figure 4, it follows that this subordination holds provided 2−1≤q~−1<q~1≤2+1, and these inequalities are equivalent to(71)β≥β1=−j1/2−12−2,β=β2≥−j1/212.
Thus, the subordinationqz≺z+1+z2 holds for all 0<s≤1/2 whenever β≥maxβ1;β2=1.98743⋯.(iii)
Regarding the same proof of Theorem21 (i), it is sufficient to establish that q~z≔1+1/βj1/2z≺1+z. Using again the MAPLE™ software, since the function Hz≔1+z is univalent in U, from Figure 5, it follows that this subordination holds provided 0≤q~−1≤q~1≤2. Hence, it is sufficient to assume that(72)β≥β3=−j1/2−1,β=β4≥j1/212−1.
Thus, the subordinationqz≺1+z holds for all 0<s≤1/2 whenever β≥maxβ3;β4=4.01776⋯.(iv)
Similarly to the proof of Theorem21 (i) and the previous parts of this theorem, using the MAPLE™ software, since the function Tz≔ez is univalent in U, from Figure 6, it follows that the subordination qz≺ez holds provided(73)β≥β5=e1−ej1/2−1,β≥β6=j1/21e−1.
Thus, the subordinationqz≺ez holds for all 0<s≤1/2 whenever β≥maxβ5;β6=1.84175⋯.Figure 4
The image ofz=1 under qzs=1/2,β=1.98743⋯ and hz.Figure 5
The image of∣z∣=1 under qzs=1/2;β=4.01776⋯ and 1+z.Figure 6
The image of∣z∣=1 under qzs=1/2;β=1.84175⋯ and ez.Settingpz≔zf′z/fz in the above theorem, we obtain the next special cases:Corollary 22.
Suppose that the functionf∈A satisfies(74)1+βzf′zfz1−zf′zfz+zf″zf′z≺Lsz.(i)
For−1<A≤1, if β≥−2/1+Ajs−1, then f∈S∗A,−1(ii)
Ifβ≥max−j1/2−1/2−2;j1/21/2=1.98743⋯, then f∈Sc∗(iii)
Ifβ≥max−j1/2−1;j1/21/2−1=4.01776⋯, then f∈SL∗(iv)
Ifβ≥maxe/1−ej1/2−1;j1/21/e−1=1.84175⋯, then f∈Se∗



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*Source: 7428648-2020-09-25.xml*