# Starlikeness of Analytic Functions with Subordinate Ratios

Authors: Rosihan M. Ali; Vaithiyanathan Ravichandran; Kanika Sharma
Journal: Journal of Mathematics
 (2021)

---

## Abstract

Leth be a nonvanishing analytic function in the open unit disc with h0=1. Consider the class consisting of normalized analytic functions f whose ratios fz/gz, gz/zpz, and pz are each subordinate to h for some analytic functions g and p. The radius of starlikeness of order α is obtained for this class when h is chosen to be either hz=1+z or hz=ez. Further, starlikeness radii are also obtained for each of these two classes, which include the radius of Janowski starlikeness, and the radius of parabolic starlikeness.


---

## Body

## 1. Two Subclasses of Normalized Analytic Functions

LetA denote the class of normalized analytic functions fz=z+∑k=2∞akzk in the unit disc D=z∈ℂ:z<1. A prominent subclass of A is the class S∗ consisting of functions f∈A such that fD is a starlike domain with respect to the origin. Geometrically, this means the linear segment joining the origin to every other point w∈fD lies entirely in fD. Every starlike function in A is necessarily univalent.Sincef′0 does not vanish, every function f∈A is locally univalent at z=0. Further, each function f∈A mirrors the identity mapping near the origin and thus, in particular, maps small circles z=r onto curves which bound starlike domains. If f∈A is also required to be univalent in D, then it is known that f maps the disc z<r onto a domain starlike with respect to the origin for every r≤r0≔tanhπ/4 (see [1], Corollary, p. 98). The constant r0 cannot be improved. Denoting by S the class of univalent functions f∈A, the number r0=tanhπ/4 is commonly referred to as the radius of starlikeness for the class S.Another informative description of the classS is its radius of convexity. Here, it is known that every f∈S maps the disc z<r onto a convex domain for every r≤r0≔2−3 ([1], Corollary, p. 44). Thus, the radius of convexity for S is r0=2−3.To formulate a radius description for other entities besides starlikeness and convexity, consider in general two familiesG and ℳ of A. The G-radius for the class ℳ, denoted by RGℳ, is the largest number R such that r−1frz∈G for every 0<r≤R and f∈ℳ. Thus, for example, an equivalent description of the radius of starlikeness for S is that the S∗-radius for the class S is RS∗S=tanhπ/4.In this paper, we seek to determine the radius of starlikeness and certain otherG-radius, for particular subclasses G of A. Several widely studied subclasses of A have simple geometric descriptions; these functions are often expressed as a ratio between two functions. Among the very early studies in this direction is the class of close-to-convex functions introduced by Kaplan [2] and Reade’s class [3] of close-to-starlike functions. Close-to-convex functions are necessarily univalent, but not so for close-to-starlike functions.In this paper, we examine two different subclasses of functions inA satisfying a certain subordination of ratios. Interestingly, these classes contain nonunivalent functions. An analytic function f is subordinate to an analytic function g, written f≺g, if(1)fz=gwz,z∈D,for some analytic self-map w in D with wz≤z. The function w is often referred to as a Schwarz function.Now, leth be a nonvanishing analytic function in D with h0=1. The classes treated in this paper consist of functions f∈A whose ratios fz/gz, gz/zpz, and pz are each subordinate to h for some analytic functions g and p:(2)fzgz≺hz,gzzpz≺hz,pz≺hz.Whenp is the constant one function, then the class contains functions f∈A satisfying the subordination of ratios(3)fzgz≺hz,gzz≺hz.Whenf∈A satisfies REfz/gz>0 and REgz/z>0, or their variants, these functions have earlier been studied, notably by MacGregor in [4–7] and Ratti in [8, 9]. For related investigations, see [10, 11] and several recent references therein. Under the present context, this amount to choosing hz=1+z/1−z or some other appropriate choices of h.In this paper, two specific choices of the functionh are made: hz=1+z and hz=ez.The classT1: this is the class given by(4)T1≔f∈A:fzgz≺1+z,gzzpz≺1+z,for someg∈A,pz≺1+z.This class is nonempty: letf1,g1,p1:D⟶ℂ be given by(5)f1z=z1+z3/2,g1z=z1+z,p1z=1+z.Then,f1z/g1z≺1+z and g1z/zp1z≺1+z, so that f1∈T1. The function f1 will be shown to play the role of an extremal function for the class T1. Since f1′ vanishes at z=−2/5, the function f1 is nonunivalent, and thus, the class T1 contains nonunivalent functions. Incidentally, f1 demonstrates the radius of univalence for T1 is at most 2/5. In Theorem 1, the radius of starlikeness for T1 is shown to be 2/5, whence T1 has radius of univalence 2/5.The following is a useful result in investigating the starlikeness of the classT1.Lemma 1.
Letpz≺1+z. Then, p satisfies the sharp inequalities(6)1−r≤pz≤1+r,z≤r,(7)zp′zpz≤r21−r,z≤r.Proof.
Ifpz≺1+z, then p2z=1+wz for some Schwarz function w. The well-known Schwarz lemma shows that wz≤z and(8)w′z≤1−wz21−z2.
Therefore,(9)pz2=1+wz≤1+wz≤1+z≤1+r,for z≤r, that is, pz≤1+r for z≤r. Similarly, pz≥1−r for z≤r.
Since2zp′z/pz=zw′z/1+wz, the inequality (8) readily shows(10)2zp′zpz≤zw′z1−wz≤z1+wz1−z2≤z1+z1−z2=z1−z≤r1−r,for z≤r. This proves (7). The inequalities are sharp for the function p:D⟶ℂ defined by pz=1+z.Forf∈T1, let p1z=fz/gz and p2z=gz/zpz. Then, fz=zpzp1zp2z and(11)zf′zfz−1≤zp′zpz+zp1′zp1z+zp2′zp2z.Sincep,p1,p2≺1+z, we deduce from (7) and (11) that(12)zf′zfz−1≤3r21−r,z≤r,for each function f∈T1. Sharp growth inequalities also follow from (6):(13)r1−r3/2≤fz≤r1+r3/2,for each f∈T1. Crude distortion inequalities can readily be obtained from (12) and the growth inequality; however, finding sharp estimates remain an open problem.The classT2: this class is defined by(14)T2≔f∈A:fzgz≺ez,gzzpz≺ez,for someg∈A,pz≺ez.Letf2,g2,p2:D⟶ℂ be given by(15)f2z=ze3z,g2z=ze2z,p2z=ez.Evidently,f2z/g2z≺ez, g2z/zp2z≺ez, so that f2∈T2, and the class T2 is nonempty. Similar to f1∈T1, the function f2 plays the role of an extremal function for the class T2. The Taylor series expansion for f2 is(16)f2z=z+3z2+9z32+9z42+27z58+⋯.Comparing the second coefficient, it is clear thatf2 is nonunivalent. Hence, the class T2 contains nonunivalent functions. The derivative f2′ vanishes at z=−1//3, which shows the radius of univalence for T2 is at most 1/3. From Theorem 1, the radius of starlikeness is shown to be 1/3, and so the radius of univalence for T2 is 1/3.Lemma 2.
Everypz≺ez satisfies the sharp inequalities(17)e−r≤pz≤er,z≤r,(18)zp′zpz≤r,z≤r≤2−1,1+r2241−r2,z=r≥2−1.Proof.
Letpz≺ez. Since pz=ewz for some Schwarz self-map w satisfying wz≤z, it follows that(19)e−z≤e−wz≤pz=eRewz≤ewz≤ez.
The inequalities become equality for the functionp:D⟶ℂ defined by pz=ez respectively at z=−r and z=r.
The functionw also satisfies the sharp inequality (see [1], Corollary, p. 199)(20)w′z≤1,r=z≤2−1,1+r224r1−r2,r≥2−1.
Fromzp′z/pz=zw′z, we conclude that(21)zp′zpz≤r,r=z≤2−1,1+r2241−r2,r≥2−1.
This inequality is sharp for the functionp:D⟶ℂ defined by pz=ez when r=z≤2−1. It is also sharp in the remaining interval for the function pz=ewz, where w is the extremal function for which equality holds in (20).Forf∈T2, let p1z=fz/gz and p2z=gz/zpz. Then, fz=zpzp1zp2z and(22)zf′zfz−1≤zp′zpz+zp1′zp1z+zp2′zp2z.Sincep,p1,p2≺ez, estimates (18) and (22) show that(23)zf′zfz−1≤3r,r=z≤2−1,31+r2241−r2,r≥2−1,for each function f∈T2. It also follows from (17) that(24)re−3r≤fz≤re3rholds for each function f∈T2 and that these estimates are sharp.In this paper, we shall adopt the commonly used notations for subclasses ofA. First, for 0≤α<1, let S∗α denote the class of starlike functions of order α consisting of functions f∈A satisfying the subordination(25)zf′zfz≺1+1−2αz1−z.Thus,(26)Rezf′zfz>α,z∈D.The caseα=0 corresponds to the classical functions whose image domains are starlike with respect to the origin. Various other starlike subclasses of A occurring in the literature can be expressed in terms of the subordination(27)zf′zfz≺φz,for suitable choices of the superordinate function φ. When φ:D⟶ℂ is chosen to be φz≔1+Az/1+Bz, −1≤B<A≤1, the subclass derived is denoted by S∗A,B. Functions f∈S∗A,B are known as Janowski starlike functions. When φz≔1+2/π2log1+z/1−z2, the subclass is denoted by Sp∗, and its functions are called parabolic starlike functions.In Section2 of this paper, the radius of starlikeness of order α, Janowski starlikeness, and parabolic starlikeness are found for the classes Ti, with i=1,2. Section 3 deals with the determination of the G-radius for the class Ti with i=1,2, for certain other subclasses G occurring in the literature. These classes are associated with particular choices of the superordinate function φ in (27). As mentioned earlier, the G-radius for a given class ℳ, denoted by RGℳ, is the largest number R such that r−1frz∈G for every 0<r≤R and f∈ℳ. It will become apparent in the forthcoming proofs that there are common features in the methodology of finding the G-radius for each of these subclasses.


## 2. Starlikeness of Orderα, Janowski Starlikeness, and Parabolic Starlikeness

The first result deals with theS∗α-radius (radius of starlikeness of order α) for the classes T1 and T2. This radius is shown to equal the Sα∗-radius, where Sα∗ is the subclass containing functions f∈A satisfying zf′z/fz−1<1−α. The latter condition also implies that Sα∗⊂S∗α.Theorem 1.
Let0≤α<1. The radii of starlikeness of order α for T1 and T2 are(i)
RS∗αT1=RSα∗T1=21−α/5−2α,(ii)
RS∗αT2=RSα∗T2=1−α/3.Proof.
(i)
The functionσr=2−5r/2−2r is a decreasing function on 0,1. Further, the number R1≔21−α/5−2α is the root of the equation σr=α. For f∈T1 and 0<r=z≤R1, the inequality (12) readily yields(28)Rezf′zfz≥1−3r21−r=2−5r2−2r=σr≥σR1=α,zf′zfz−1≤3r21−r=1−σr≤1−σR1=1−α.Atz=−R1, the function f1∈T1 given by f1z=z1+z3/2 yields(29)zf1′zf1z=2+5z2+2z=2−5R12−2R1=α.Thus,(30)Rezf1′zf1z=α,zf1′zf1z−1=1−α.This proves that theS∗α and Sα∗ radii for T1 are the same number R1.(ii)
Considerωr=1−3r,  0≤r<1. The number R2=1−α/3<1/3 is clearly the root of the equation ωr=α. Since ω is decreasing, then ωr≥ωR2=α for 0<r≤R2. It follows from (23) that for 0<r=z≤R2,(31)Rezf′zfz≥1−3r=ωr≥α,zf′zfz−1≤3r=1−ωr≤1−α.
Evaluating the functionf2z=ze3z at z=−R2 yields(32)zf2′zf2z=1−3R2=α.
Hence,(33)Rezf2′zf2z=α,zf2′zf2z−1=1−α.
This proves that theS∗α and Sα∗ radii for the class T2 are the same number R2.Next, we find theS∗A,B-radius (Janowski starlikeness) for T1 and T2. Recall that S∗A,B consists of analytic functions f∈A satisfying the subordination zf′z/fz≺1+Az/1+Bz, −1≤B<A≤1.Theorem 2.
(i)
Everyf∈T1 is Janowski starlike in the disc Dr=z:z<r for r≤2A−B/31+B+2A−B. If B<0, then RS∗A,BT1=2A−B/3+2A−5B.(ii)
The radius of Janowski starlikeness forT2 is RS∗A,BT2=A−B/31+B.Proof.
SinceS∗A,−1=S∗1−A/2, the results in the case B=−1 follow from Theorem 1. We now prove the results when −1<B<A≤1.(i)
Letf∈T1 and write w=zf′z/fz. Then, (12) shows that w−1≤3r/21−r for z≤r. For 0≤r≤R1≔2A−B/31+B+2A−B, then 3R1/21−R1=A−B/1+B.For0≤r≤R1, we first show that the disc(34)w:w−1≤3R121−R1=A−B1+Bis contained in the images of the unit disc under the mapping1+Az/1+Bz. As B≠−1, the image is the disc given by(35)w:w−1−AB1−B2<A−B1−B2.Silverman ([12], p. 50–51) has shown that the disc(36)w:w−c<d⊂w:w−a<b,if and only ifa−c≤b−d. With the choices c=1, d=A−B/1+B, a=1−AB/1−B2, and b=A−B/1−B2, then a−c=BA−B/1−B2=b−d. This proves that S∗A,B radius is at least R1.To prove sharpness, consider the functionf1∈T1 given by f1z=z1+z3/2. Evidently, zf1′z/f1z=2+5z/2+2z. For B<0, evaluating at z=−R1, then zf1′z/f1z=1+3z/2+2z=1−A−B/1+B=1−A/1−B. This shows that(37)zf1′zf1z−1−AB1−B2=1−A1−B−1−AB1−B2=A−B1−B2,proving sharpness in the caseB<0.(ii)
Letf∈T2 and w≔zf′z/fz. It follows from (23) that w−1≤3r for z≤r. For 0≤r≤R2≔A−B/31+B, we see that the disc w:w−1≤3R2=A−B/1+B is contained in the disc w:w−1−AB/1−B2<A−B/1−B2, as in the proof of (i). This proves that S∗A,B radius is at least R2. The result is sharp for the function f2∈T2 given by the function f2z=ze3z.The functionφPAR:D⟶ℂ given by(38)φPARz≔1+2π2log1+z1−z2,IMz≥0,maps D into the parabolic region(39)φPARD=w=u+iv:v2<2u−1=w:REw>w−1.The classCφPAR=f∈A:1+zf″z/f′z≺φPARz is the class of uniformly convex functions introduced by Goodman [13]. The corresponding class Sp∗≔S∗φPAR=f∈A:zf′z/fz≺φPARz introduced by Rønning [14] is known as the class of parabolic starlike functions. The class Sp∗ consists of functions f∈A satisfying(40)Rezf′zfz>zf′zfz−1,z∈D.Evidently, every parabolic starlike function is also starlike of order 1/2. The radius of parabolic starlikeness for the classesT1 and T2 is given in the next result.Corollary 1.
The radius of parabolic starlikeness forT1 and T2 is respectively equal to its radius of starlikeness of order 1/2. Thus,(i)
RSp∗T1=1/4,(ii)
RSp∗T2=1/6.Proof.
Shanmugam and Ravichandran ([15], p. 321) proved that(41)w:w−a<a−12⊆w:REw>w−1,for 1/2<a≤3/2. Choosing a=1, this implies that S1/2∗⊂Sp∗. Every parabolic starlike function is also starlike of order 1/2, whence the inclusion S1/2∗⊂Sp∗⊂S∗1/2. Therefore, for any class ℱ, readily RS1/2∗ℱ≤RSp∗ℱ≤RS∗1/2ℱ.
Whenℱ=Ti, i=1,2, Theorem 1 gives RS∗αTi=RSα∗Ti. This shows that RS1/2∗Ti=RSp∗Ti=RS∗1/2Ti. Since RS∗1/2T1=1/4 and RS∗1/2T2=1/6 from Theorem 1, it follows that RSp∗T1=1/4 and RSp∗T2=1/6.


## 3. Further Radius of Starlikeness

In this section, we find theG-radius for the class Ti with i=1,2, for certain other widely studied subclasses G. These are associated with particular choices of the superordinate function φ in (27).Denote bySexp∗≔S∗ez the class associated with φz≔ez in (27). This class was introduced by Mendiratta et al. [16], and it consists of functions f∈A satisfying the condition logzf′z/fz<1. The following result gives the radius of exponential starlikeness for the classes T1 and T2.Corollary 2.
TheSexp∗-radius for the class T1 is(42)RSexp∗T1=2−2e2−5e≈0.296475,while that of T2 is(43)RSexp∗T2=e−13e.Proof.
Mendiratta et al. ([16], Lemma 2.2) proved that(44)w:w−a<a−1e⊆w:logw<1,for e−1≤a≤e+e−1/2, and this inclusion with a=1 gives S1/e∗⊂Sexp∗. It was also shown in ([16], Theorem 2.1 (i)) that Sexp∗⊂S∗1/e. Therefore, S1/e∗⊂Sexp∗⊂S∗1/e, which, as a consequence of Theorem 1, established the result.Corollary3 investigates the radius of cardioid starlikeness for each class T1 and T2. The class SC∗≔S∗φCAR, where φCARz=1+4z/3+2z2/3 in (27), was introduced and studied in [17]. Descriptively, f∈SC∗ provided zf′z/fz lies in the region bounded by the cardioid ΩC≔w=u+iv:9u2+9v2−18u+52−169u2+9v2−6u+1=0.Corollary 3.
The following are theSC∗-radius for the classes T1 and T2:(i)
RSC∗T1=4/13,(ii)
RSC∗T2=2/9.Proof.
Sharma et al. [17] proved that w:w−a<a−1/3⊆ΩC for 1/3<a≤5/3, and this inclusion with a=1 gives S1/3∗⊂SC∗. Thus, RS1/3∗Ti≤RSC∗Ti for i=1,2. To complete the proof, we demonstrate RSC∗Ti≤RS1/3∗Ti for i=1,2.(i)
Evaluating the functionf1z=z1+z3/2 at z=−R=−RS1/3∗T1=−4/13 gives(45)zf1′zf1z=2+5z2+2z=2−5R2−2R=13=φCAR−1.Thus,RSC∗T1≤4/13.(ii)
Similarly, atz=−R=−RS1/3∗T2=−2/9, the function f2z=ze3z yields(46)zf2′zf2z=1+3z=1−3R=13=φCAR−1.
This proves thatRSC∗T2≤2/9.In 2019, Cho et al. [18] studied the class Ssin∗≔S∗1+sinz consisting of functions f∈A satisfying the condition zf′z/fz≺1+sinz. We find the Ssin∗-radius for the classes T1 and T2.Corollary 4.
The following are theSsin∗-radius for each class T1 and T2:(i)
RSsin∗T1=2sin1/3+2sin1≈0.35938,(ii)
RSsin∗T2=sin1/3.Proof.
It was proved in [18] that w:w−a<sin1−a−1⊆qD for a−1≤sin1, where qz≔1+sinz. For a=1, this implies that S1−sin1∗⊂Ssin∗. Thus, RS1−sin1∗Ti≤RSsin∗Ti for i=1,2. The proof is completed by demonstrating RSsin∗Ti≤RS1−sin1∗Ti for i=1,2.(i)
Evaluating the functionf1z=z1+z3/2 at z=−R=−RS1−sin1∗T1=−2sin1/3+2sin1 gives(47)zf1′zf1z=2+5z2+2z=2−5R2−2R=1−sin1=q−1.Thus,RSsin∗T1≤22sin1/3+2sin1.(ii)
Similarly, atz=±R=±RS1−sin1∗T2=±sin1/3, the function f2z=ze3z yields(48)zf2′zf2z=1+3z=1±3R=1±sin1=q±1.
This proves thatRSsin∗T2≤sin1/3.Consider next the classS©∗≔S∗z+1+z2 introduced by Raina and Sokół in [19]. Functions f∈S©∗ provided zf′z/fz lies in the region bounded by the lune Ωl≔w:w2−1<2w. The result below gives the radius of lune starlikeness for each class T1 and T2.Corollary 5.
The following are theS©∗-radius for each class T1 and T2:(i)
RS©∗T1=22−2/22−7≈0.280847,(ii)
RS©∗T2=2−2/3.Proof.
It was shown by Gandhi and Ravichandran ([20], Lemma 2.1) that w:w−a<1−2−a⊆Ωl for 2−1<a≤2+1. Choosing a=1, the inclusion gives S2−1∗⊂S©∗. Thus, RS2−1∗Ti≤RS©∗Ti for i=1,2. We complete the proof by demonstrating RS©∗Ti≤RS2−1∗Ti for i=1,2.(i)
Evaluating the functionf1z=z1+z3/2 at z=−R=−RS2−1∗T1=−22−2/22−7 gives(49)zf1′zf1z2−1=2+5z2+2z2−1=2−5R2−2R2−1=22−1=2zf1′zf1z.Thus,RS©∗T1≤22−2/22−7.(ii)
Similarly, atz=−R=−RS2−1∗T2=−2−2/3, the function f2z=ze3z yields(50)zf2′zf2z2−1=1+3z2−1=1−3R2−1=22−1=2zf2′zf2z.
This proves thatRS©∗T2≤2−2/3.As a further example, consider next the classSR∗≔S∗ηz, where ηz=1+zk+z2/k2−kz, k=2+1. This class associated with a rational function was introduced and studied by Kumar and Ravichandran in [21].Corollary 6.
The following are theSR∗-radius for the classes T1 and T2:(i)
RSR∗T1=2−3+22/42−9≈0.102642,(ii)
RSR∗T2=3−22/3.Proof.
It was shown in [21] that w:w−a<a−22−1⊆ηD for 22−1<a≤2. This inclusion with a=1 gives S22−1∗⊂SR∗. Thus, RS22−1∗Ti≤RSR∗Ti for i=1,2. We next show that RSR∗Ti≤RS22−1∗Ti for i=1,2.(i)
Atz=−R=−RS22−1∗T1=−2−3+22/42−9, the function f1z=z1+z3/2 yields(51)zf1′zf1z=2−5R2−2R=22−1=η−1.Thus,RSR∗T1≤2−3+22/42−9.(ii)
Evaluatingf2z=ze3z at z=−R=−RS22−1∗T2=−3−22/3 gives(52)zf2′zf2z=1−3R=22−1=η−1.
Thus,RSR∗T2≤3−22/3.The classSNe∗≔S∗ψz, where ψz=1+z−z3/3, was introduced and studied by Wani and Swaminathan in [22]. Geometrically, f∈SNe∗ provided zf′z/fz lies in the region bounded by the nephroid: a 2-cusped kidney-shaped curve ΩNe≔w=u+iv:u−12+v2−4/93−4v2/3=0.Corollary 7.
The following are theSNe∗-radius for the classes T1 and T2:(i)
RSNe∗T1=4/13,(ii)
RSNe∗T2=2/9.Proof.
It was shown in [22] that w:w−a<a−1/3⊆ΩNe for 1/3<a≤1. This inclusion with a=1 gives S1/3∗⊂SNe∗. This shows that RS1/3∗Ti≤RSNe∗Ti for i=1,2. We next show that RSNe∗Ti≤RS1/3∗Ti for i=1,2.(i)
Evaluating the functionf1z=z1+z3/2 at z=−R=−RS1/3∗T1=−4/13 results in(53)zf1′zf1z=2−5R2−2R=13=ψ−1.Thus,RSNe∗T1≤4/13.(ii)
Similarly, evaluatingf2z=ze3z at z=−R=−RS1/3∗T2=−2/9 yields(54)zf2′zf2z=1−3R=13=ψ−1.
This proves thatRSNe∗T2≤2/9.Finally, we consider the classSSG∗≔S∗2/1+e−z introduced by Goel and Kumar in [23]. Here, 2/1+e−z is the modified sigmoid function that maps D onto the region ΩSG≔w=u+iv:logw/2−w<1. Thus, f∈SSG∗ provided the function zf′z/fz maps D onto the region lying inside the domain ΩSG.Corollary 8.
TheSSG∗-radius for the class T1 is(55)RSSG∗T1=2e−21+5e≈0.23552,while that of T2 is(56)RSSG∗T2=e−131+e.Proof.
The inclusionw:w−a<e−1/e+1−a−1⊆ΩSG holds for 2/1+e<a<2e/1+e (see [23]). At a=1, the set inclusion shows that S2/e+1∗⊂SSG∗. It was also shown in [23] that SSG∗⊂S∗α for 0≤α≤2/e+1. The desired result is now an immediate consequence of Theorem 1.



---

*Source: 8373209-2021-10-29.xml*