# Fourth Hankel Determinant for the Set of Star-Like Functions

Authors: Muhammad Arif; Lubna Rani; Mohsan Raza; Pawel Zaprawa
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we derive a bound of the fourth Hankel determinant for the class of star-like functions. We also consider this problem for 2-fold and3-fold symmetric star-like functions. In this case, we obtain sharp results.


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## Body

## 1. Introduction and Definitions

LetA denote the class of all analytic functions f of the form(1)fz=z+∑n=2∞anzn,z∈D,which are analytic in the open unit disc D=z∈ℂ:z<1. Let S denote the subclass of all univalent functions in A. Also, let S∗,C, and K denote the classes of star-like, convex, and close-to-convex functions, respectively, which are defined as follows:(2)S∗=f∈S:Rezf′zfz>0,z∈D,C=f∈S:Re1+zf″zf′z>0,z∈D,Kg=f∈S:Rezf′zgz>0,forgz∈S∗,z∈D.It is easy to see that whengz=z, the class K reduces to the class ℛ of bounded turning functions. Let P denote the class of all analytic functions p of the form(3)pz=1+∑n=1∞cnzn,z∈D,whose real parts are positive in D.The problem of finding coefficient bounds plays an important role in exploring the geometry of a complex-valued function. In particular, the second coefficient provides information about the growth and distortion theorems for functions in class S. Similarly, the Hankel determinants are very useful in the investigations of the singularities and power series with integral coefficients.The Hankel determinantHq,nfq,n∈ℕ=1,2,… for a function fof form (1) was defined by Pommerenke [1, 2] (see also [3, 4]) as(4)Hq,nf≔anan+1…an+q−1an+1an+2…an+q⋮⋮…⋮an+q−1an+q…an+2q−2,for fixed integer q and n, and the growth of Hq,nf has been studied for different subclasses of univalent functions. The sharp bounds of H2,2f for S∗, C, and ℛ were investigated by Janteng et al. [5–7]. They proved that(5)H2,2f≤1,forf∈S∗,18,forf∈C,49,forf∈ℛ,for the class of Bazilevič functions, and the exact estimate of H2,2f was obtained by Krishna et al. [8]. On the other hand, the sharp bound of this determinant for the class of close-to-convex functions is unknown (see [9]). For more results on H2,2f, see [10–16].The estimation ofH3,1f is much more difficult to obtain as compared to H2,2f. In the first paper on H3,1f, published in 2010, Babalola [17] obtained the upper bound of H3,1f for S∗, C, and ℛ. Later on, some authors [18–24] derived the bounds of H3,1f for different subclasses of analytic and univalent functions. In 2016, Zaprawa [25] improved the results of Babalola [17] and showed that(6)H3,1f≤1,forf∈S∗,49540,forf∈C,4160,forf∈ℛ.He also claimed that these bounds were still not sharp. Furthermore, he obtained the sharp bounds form-fold symmetric functions. Later on, Kwon et al. [26] proved that(7)H3,1f≤89,for f∈S∗. For some recent work on the bound of H3,1f, refer [27–30]. Arif et al. and Arif et al. [31, 32] computed the bound on the fourth- and fifth-order Hankel determinants for the functions with bounded turning. Cho and Kumar [33] investigated the bound on H4,1f for the class of functions associated with lune. Recently, Arif et al. [34] have determined the bound on the fourth Hankel determinant for the class of star-like functions related to lemniscate of Bernoulli.In this paper, we make a contribution to the subject by finding the fourth Hankel determinant for the class of star-like functions. We also study this determinant for the subclasses consisting of 2-fold and 3-fold symmetric functions.


## 2. Lemmas for ClassP

In the proofs of our results, we need the following sharp estimates for functions in the classP.Lemma 1.
Ifp∈P and p is of form (3), then the sharp estimates(8)cn≤2,(9)cn+k−μcnck≤2,for0≤μ≤1,(10)cn+2k−cnck2≤6,(11)cnck−cmcl≤4,forn+k=m+lhold for each n,k,m,l∈ℕ=1,2,….
Inequalities (8), (9), and (10) are proved in [23, 35, 36], respectively. Inequality (11) is obvious.
Libera and Złotkiewicz proved the following result, see [37].Lemma 2.
Letp∈P be of form (3). Then, the expressions(12)c3∗=c13−2c1c2+c3,c4∗=c14+c22+2c1c3−3c12c2−c4,c5∗=c15+3c1c22+3c12c3−4c13c2−2c1c4−2c2c3+c5,c6∗=c16+6c12c22+4c13c3+2c1c5+2c2c4+c32−c23−5c14c2−3c12c4−6c1c2c3−c6,are all bounded by 2.Lemma 3.
(See [37]). Let p∈P be given by (3) and c1∈0,2. Then,(13)2c2=c12+4−c12x,4c3=c13+24−c12c1x−4−c12c1x2+24−c121−x2z,for each complex numbers x, z such that x≤1, z≤1.


## 3. Bound ofH4,1f for the Set S∗

From (4), we can write H4,1f as(14)H4,1f≔a7H3,1f−a6Δ1+a5Δ2−a4Δ3,where(15)H3,1f=a3a5−a42−a2a2a5−a3a4+a3a2a4−a32,(16)Δ1=a3a6−a4a5−a2a2a6−a3a5+a4a2a4−a32,(17)Δ2=a4a6−a52−a2a3a6−a4a5+a3a3a5−a42,(18)Δ3=a2a4a6−a52−a3a3a6−a4a5+a4a3a5−a42.We observe thatH4,1f is a polynomial of six successive coefficients a2, a3, a4, a5, a6, and a7 of a function f in a given class. Let f∈S∗ be of form (1). Then, for p∈P given in (3), we have(19)zf′zfz=pz.Using (19), we get(20)n−1an=∑i=1n−1aicn−i.From (20), we obtain that(21)a2=c1,a3=12c12+c2,a4=16c13+3c1c2+2c3,a5=124c14+6c12c2+8c1c3+3c22+6c4,a6=1120c15+10c13c2+15c1c22+20c12c3+20c2c3+30c1c4+24c5,a7=1720c16+15c14c2+40c13c3+45c12c22+90c12c4+120c1c2c3+90c2c4+144c1c5+15c23+40c32+120c6.By using the abovementioned formulae, we write (15), (16), (17), and (18) in the following way:(22)H3,1f=1144−c16+3c14c2+8c13c3−9c12c22−18c12c4+24c1c2c3−9c23+18c2c4−16c32,(23)Δ1=1240−c17+10c14c3−5c13c22+c15c2−15c1c23−10c22c3+20c12c2c3−10c13c4−20c3c4+30c1c2c4+24c2c5−24c12c5,(24)Δ2=1960−c18−10c14c22+16c15c3−80c1c22c3+15c24+120c12c2c4−60c42−64c13c5+64c3c5,(25)Δ3=18640−c19−360c12c22c3+24c16c3−18c15c22−144c14c5+576c1c3c5−540c1c42+135c1c24+360c13c2c4−432c22c5+720c2c3c4−320c33.Now, we are ready to formulate our main result.Theorem 1.
Iff∈S∗ and is of form (1), then(26)H4,1f≤457571129600=3.5306….Proof.
Sincef∈S∗, using (22), (23), (24), and (25) in (14), we get(27)H4,1f=11036800c112−6c110c2−40c19c3+45c18c22+90c18c4+288c17c5−60c16c23−1080c16c2c4+240c16c32−1200c16c6+1440c15c3c4+2592c15c2c5+720c15c22c3+3600c14c2c6−5760c14c3c5−3600c14c2c32−2700c14c22c4−225c14c24+2700c14c42+4800c13c23c3−4320c13c22c5+9600c13c3c6−8640c13c4c5+3200c13c33+16200c12c2c42−3600c12c22c32+5400c12c23c4−21600c12c4c6−7200c12c32c4−10800c12c22c6−1350c12c25+20736c12c52−1800c1c24c3+9600c1c2c33+12960c1c23c5−23040c1c32c5+21600c1c3c42−21600c1c22c3c4+28800c1c2c3c6−25920c1c2c4c5−21600c2c32c4+34560c3c4c5+17280c22c3c5+21600c2c4c6−1350c24c4+1200c23c32−10800c23c6+8100c22c42−19200c32c6−20736c2c52+675c26+6400c34−16200c43.
After rearranging the terms, we get (27) in the following form:(28)H4,1f=11036800−c6∗c2−c123+2c2c5∗2−4c3c3∗c6∗−40c3c4∗c5∗+37c6∗c3−c1c22+93c4c3∗c5∗+228c1c6∗c5−c2c3+525c6∗c6−c2c4+422c4∗c4−c1c32−116c6∗c6−c23−1608c6−581608c1c5c6∗+1780c3c5∗c4−52445c22−6856c5c4∗c3−40256856c1c2+1792c2c4∗c6−c32+2464c4c4∗c4−c22−8873c3∗c3−c1c2c6−c2c4+155c22c4∗c4−c22+9662c6c3∗c3−c1c2+4920c3∗c4−c22c5−c2c3−10895c2−c12c4−c22c6−c1c5−2468c2−c12c4−c1c3c6−c32+13850c3∗c4c5−c6c3+3401c3∗c2c4c3−c6c1+10152c2−c12c4−c22c6−c32+29921c6c2−c12c4−c1c3+22903c2−c12c4−1775922903c22c6−c2c4−18574c4c2−c12c6−c32−10823c4c2−c12c6−c1c5−23586c3−c1c22c6−492723586c2c4−15237c5c2−c12c5−c2c3+1958c3c4c5−c2c3−6282c5c3−c1c2c4−c1c3+15882c3c5c4−c1c3+2483c2c3−c1c22c4−c22+15984c2c4−c1c3c6−c32−10179c3−168377c1c2c4−c22c5−c2c3−18810c4c4−c22c4−c1c3+28485c4c3−c1c2c5−c2c3−21909c4c5c3−c1c2+14570c5c3−c1c2c4−c22+574c3c4−c2c5c5−c2c3+6367c6−c322−11535c4c4−c222−945c4−c22+12965c42−c3c5c4−c22+3939c2c4c6−c2c4+16515c6c6−c2c4+2078c62+4293c42c4−c22+7939c3c3−c1c2c6−c2c4−2684c6c6−c1c5+3340c2c2c4−c6c4−c22−23475c6c6−c32−5c23c3c3−c1c2+c32c23−c6+c14c2c2c4−c1c5+310c14c24+718c43−4927c2−c12c5−c1c4c5−c2c3−3113c3c4c5−4981c4c6c12−c2+7c1c24c3−4927c4c2−c12c3−c1c22+4927c3c4c2−c12c3−c1c2.
Let(29)δ1=−c6∗c2−c123+2c2c5∗2−4c3c3∗c6∗−40c3c4∗c5∗+37c6∗c3−c1c22+93c4c3∗c5∗+228c1c6∗c5−c2c3+525c6∗c6−c2c4+422c4∗c4−c1c32,δ2=−116c6∗c6−c23−1608c6−581608c1c5c6∗+1780c3c5∗c4−52445c22−6856c5c4∗c3−40256856c1c2+1792c2c4∗c6−c32+2464c4c4∗c4−c22,δ3=−8873c3∗c3−c1c2c6−c2c4+155c22c4∗c4−c22+9662c6c3∗c3−c1c2+4920c3∗c4−c22c5−c2c3−10895c2−c12c4−c22c6−c1c5,δ4=−2468c2−c12c4−c1c3c6−c32+13850c3∗c4c5−c6c3+3401c3∗c2c4c3−c6c1+10152c2−c12c4−c22c6−c32+29921c6c2−c12c4−c1c3,δ5=+22903c2−c12c4−1775922903c22c6−c2c4−18574c4c2−c12c6−c32−10823c4c2−c12c6−c1c5−23586c3−c1c22c6−492723586c2c4,δ6=−15237c5c2−c12c5−c2c3+1958c3c4c5−c2c3−6282c5c3−c1c2c4−c1c3+15882c3c5c4−c1c3,δ7=+2483c2c3−c1c22c4−c22+15984c2c4−c1c3c6−c32−10179c3−168377c1c2c4−c22c5−c2c3−18810c4c4−c22c4−c1c3,δ8=+28485c4c3−c1c2c5−c2c3−21909c4c5c3−c1c2+14570c5c3−c1c2c4−c22+574c3c4−c2c5c5−c2c3+6367c6−c322,δ9=−11535c4c4−c222−945c4−c223+12965c42−c3c5c4−c22+3939c2c4c6−c2c4+16515c6c6−c2c4+2078c62+4293c42c4−c22+7939c3c3−c1c2c6−c2c4,δ10=−2684c6c6−c1c5+3340c2c2c4−c6c4−c22−23475c6c6−c32+7c1c24c3−5c23c3c3−c1c2+c32c23−c6+c14c2c2c4−c1c5+310c14c24+718c43,δ11=−4927c2−c12c5−c1c4c5−c2c3−3113c3c4c5−4981c4c6c12−c2−4927c4c2−c12c3−c1c22+4927c3c4c2−c12c3−c1c2.
Applying the triangle inequality and Lemma1 and Lemma 2 in (28), we have(30)δ1≤9204,δ2≤110960,δ3≤277280,δ4≤505544,δ5≤607088,δ6≤314872,δ7≤399512,δ8≤549772,δ9≤407300,δ10≤217204,δ11≤61832.
Hence our stated bound forH4,1f is obtained.


## 4. Bounds ofH4,1f for S∗2 and S∗3

Letm∈ℕ. A domain Λ is said to be m-fold symmetric if a rotation of Λ about the origin through an angle 2π/m carries Λ onto itself. It is said that an analytic function f is m-fold symmetric in D if(31)fe2πi/mz=e2πi/mfz,holds for any z∈D.BySm, we denote the set of m-fold univalent functions having the following Taylor series form:(32)fz=z+∑k=1∞amk+1zmk+1,z∈D.The subfamilyS∗m of Sm is the set of m-fold symmetric star-like functions. It is easy to see that an analytic function f of form (32) belongs to the family S∗m if and only if(33)zf′zfz=pz,withp∈Pm,where the set Pm is defined as follows:(34)Pm=p∈P:pz=1+∑k=1∞cmkzmk,z∈D.Now, we are ready to prove the following theorem.Theorem 2.
Letf∈S∗2 be of form (32). Then, a3a7−a52≤1/4. The estimate is sharp.Proof.
Letf∈S∗2 be of the form fz=gz2, where g∈S∗, and let f and g have the following Taylor series expansions:(35)fz=z+a3z3+a5z5+⋯,gz=z+b2z2+b3z3+⋯.
For brevity, we writeΘf instead of a3a7−a52.
From the equivalencef∈S∗2⇔g∈S∗, comparing coefficients in(36)z2+b2z4+b3z6+⋯=z+a3z3+a5z5+⋯2,we get(37)a3=12b2,a5=12b3−18b22,a7=12b4−14b2b3+116b23.
Consequently, forf∈S∗2, we have(38)Θf=14b2b4−b32+164b24.
From (20) and Lemma 3, we get(39)Θf=119216c1c3−12c22−c14=11924−c122c12x−12+c12x2+8c11−x2z.
Therefore, writingc=c1 and also assuming c in [0, 2] and r=x, we have(40)Θf≤11924−c28c+2c2r+2−c6−cr2,which takes the greatest value for r=1, so(41)Θf≤1644−c24+c2≤14.
Since the equality mentioned above holds forc=0, one can deduce that Θf=1/4 if pz=1+z2/1−z2 and, consequently, if gz=z/1−z2, so the extremal star-like function is of the form fz=z/1−z4.Our next results are given in the following two theorems.Theorem 3.
Iff∈S∗2 be of form (32); then,(42)H4,1f≤18.
The estimate is sharp.Proof.
Letf∈S∗2 be given by (32). Since a2=a4=a6=0, there is(43)H4,1f=a5−a32a3a7−a52.
Using the same argument as given in the proof of Theorem2, we obtain(44)a5−a32=12b3−34b22=14c2−12c12,which is, by Lemma 1, less than or equal to (1/2).
The functionfz=z/1−z4=z+1/2z5+3/8z9+⋯ is extremal in the estimate proven in Theorem 2 as well as in the estimate of (44). Hence, for this function, we have H4,1f=1/8.Theorem 4.
Iff∈S∗3 is of the form (32), then(45)H4,1f≤118.
The estimate is sharp.Proof.
Letf∈S∗3 be of the form fz=gz33, where g∈S∗, and let f and g be of the form(46)fz=z+a4z4+a7z7+⋯,gz=z+b2z2+b3z3+⋯.
Sincef∈S∗3⇔g∈S∗, from the equality(47)z3+b2z6+b3z9+⋯=z+a4z4+a7z7+⋯3,we get(48)a4=13b2,a7=13b3−19b22.
Consequently, forf∈S∗3, there is a2=a3=a5=a6=0 and(49)H4,1f=a42a42−a7.
From (48), (20), and Lemma 3, we get(50)H4,1f=1162c12c12−3c2=−1324c12c12+34−c12x.
Therefore, writingc=c1, we have(51)H4,1f≤1162c26−c2≤118.
The equality mentioned above holds forc=3 and x=1, so pz=α1+z/1−z+1−α1−z/1+z with α=2+3/4. For this reason, the extremal function from the class S∗ is of the form gz=z/1−z2α1+z21−α, and the corresponding function f∈S∗3 is given by(52)fz=gz33=z+13z4+12z7+⋯.



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