# Fourth Hankel Determinant for a Subclass of Starlike Functions Based on Modified Sigmoid

Authors: Wali Khan Mashwani; Bakhtiar Ahmad; Nazar Khan; Muhammad Ghaffar Khan; Sama Arjika; Bilal Khan; Ronnason Chinram
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In our present investigation, we obtain the improved third-order Hankel determinant for a class of starlike functions connected with modified sigmoid functions. Further, we investigate the fourth-order Hankel determinant, Zalcman conjecture, and also evaluate the fourth-order Hankel determinants for 2-fold, 3-fold, and 4-fold symmetric starlike functions.


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## Body

## 1. Introduction and Motivation

Denoted byA, the class of functions f which are analytic in
(1)U=z:z∈ℂandz<1,and are of the form
(2)fz=z+∑n=2∞anzn∀z∈U.Also, letS be a subclass of class A, containing all univalent functions in U, and be normalized by the conditions
(3)f0=0,f′0=1.In 1916, working on the coefficientsan of class S, Bieberbach conjectured that
(4)an≤nn=2,3,⋯,,which was proved by De Branges in 1984 (see [1]). From 1916 to 1984, for some subclasses of S, many researchers used different techniques and established a number of results. The classes S∗, K, and R, namely, the classes of convex, starlike, and bounded turning functions, respectively, are some major subclasses of the class S. These classes are defined as follows:
(5)S∗=f∈S:Rezf′zfz>0∀z∈U,K=f∈S:Rezf′z′f′z>0∀z∈U,R=f∈S:Ref′z>0∀z∈U.Furthermore, we say that an analytic functionf1z is subordinated to f2z in U and is symbolically written as
(6)f1z≺f2z∀z∈U,if there exists a Schwartz function uz with properties that
(7)uz≤1,u0=1,such that
(8)f1z=f2uz.Moreover, iff2z is in the class S. Due to [2, 3], we get the following equivalence relation
(9)f10=f20,f1U⊆f2U.Now, by using the principle of subordination, a generalized set of the classesS∗,K, and R are given, respectively, as follows:
(10)S∗ψ=f∈S:zf′zfz≺ψz=1+z1−z∀z∈U,(11)Kψ=f∈S:zf′z′f′z≺ψz=1+z1−z∀z∈U,(12)Rψ=f∈S:f′z≺ψz=1+z1−z∀z∈U.By changing the right-hand side in (10), several familiar classes can be obtained such as if we keep ψz=1+Az/1+Bz, we get the class of starlike functions associated with the Janowski functions (see [4]). Moreover, if we take
(13)A=1,B=1−2α0<α<1,then, we have a class of starlike functions of order α (see [5]). Also, for the choice of ψz=1+2/π2log1+z/1−z2, a corresponding class of starlike functions is obtained, which was introduced by Ronning (see [6]). Furthermore, if we take ψz=1+z, the class of starlike functions related with the lemniscate of Bernoulli domain is resulted which was introduced and investigated by Jangteng et al. [7, 8]. Next, if we take ψz=1+sinz, the family of starlike functions connected with the sine function is obtained (see [9]). Mendiratta et al. [10] obtained a subclass of strongly starlike functions associated with exponential functions for the choice of ψz=ez. Sharma et al. [11] derived a class of starlike functions associated with a cardioid domain by taking ψz=1+4/3z+2/3z2.Moreover, several more subclasses of starlike functions have recently been presented in [12–15] through selecting specific functions for ψ, like functions associated with conic domains, shell-like curves associated with Fibonacci numbers, and functions related with Bell numbers.Lately, based on the techniques of Ma and Minda [16], Goel and Kumar in [17] defined the class SSG∗, based on the subordination principle, as follows:
(14)zf′zfz≺21+e−z∀z∈U,and studied its various important geometric properties.For a functionf of the form (2), Pommerenke [18, 19] defined Hankel determinant Hq,nf, parameter q, with n∈ℕ=1,2,3,⋯, as follows:
(15)Hq,nf=anan+1⋯an+q−1an+1an+2⋯an+q⋮⋮⋯⋮an+q−1an+q⋯an+2q−2.The growth ofHq,nf for fixed integer q and n was evaluated for different subfamilies of univalent functions. Jangteng et al. [7, 20] investigated the sharp bound of the determinant H2,2f=a2a4−a32 for each of the classes K,S∗, and R, while sharp estimation for the family of close-to-convex functions is still unknown (see [21]). On the other hand, Krishna et al. [22] proved the best estimate of H2,2f for the class of Bazilevic˘ functions. More detailed work on H2,2f can be seen in [23–27] and also the references cited therein.The determinant,(16)H3,1f=1a2a3a2a3a4a3a4a5,is known as third-order Hankel determinant and the estimation of this determinant H3,1f is more difficult and hence a potential attraction for a lot of researchers who focus on this field. In 1966–1967, Pommerenke defined this Hankel determinant but was an open problem till 2010. In 2010, Babalola [28] was the first researcher who worked on H3,1f and successfully obtained the upper bound of H3,1f for the functions belonging to the classes S∗,K, and R. A few mathematicians further expanded on this work to include other subclasses of holomorphic and univalent functions (see for example [29–35]). Zaprawa [36] enhanced their work in 2017 by demonstrating
(17)H3,1f≤1,f∈S∗,49540,f∈K,4160,f∈R,and asserted that these inequalities are still nonsharp. Additionally, for the sharpness, he thought about the subfamilies of S∗,K, and R comprising of functions with m-fold symmetry and acquired the sharp bounds. Recently, in 2018, Kowalczyk et al. [37] and Lecko et al. [38] got the sharp inequalities which are
(18)H3,1f≤4135,H3,1f≤19,for the classes K and S∗1/2, respectively, where the symbol S∗1/2 indicates the family of starlike functions of order 1/2.The main goal of this paper is to investigate the necessary and sufficient conditions for functions to get into the classSSG∗ in the form of coefficient inequality, convolution results, and the essential third-order Hankel determinant for this class in (6) and also for its 2-, 3-, and 4-fold symmetric functions.


## 2. A Set of Lemmas

LetP be the family of functions p that are holomorphic in U with Repz>0 and its series form is as follow:
(19)pz=1+∑n=1∞cnzn∀z∈U.Lemma 1.
Ifp∈Pand it is of the form ((19)), then,(20)cn≤2n≥1,(21)cn+k−δcnck≤20≤δ≤1,(22)c2−ξc12≤2max1;2ξ−1ξ∈ℂ.Further results related to Lemma1 can be found in [39, 40].Lemma 2 (see [41]).
Letp∈Phave the series expansion of the form ((19)). Then, forx, σ, ρ∈U¯=U∪1,(23)2c2=c12+x4−c12,4c3=c13+24−c12c1x−c14−c12x2+24−c121−x2σ,8c4=c14+4−c12xc12x2−3x+3+4x−44−c12·1−x2cx−1σ+x¯σ2−1−σ2ρ.Lemma 3 (see [42]).
Letm,n,lsatisfy the inequalities0<m<1,0<r<1,and(24)8r1−rmn−2l2+mr+m−n2+m1−mn−2rm2≤4m21−m2r1−r.
Ifp∈Pand has power series (19), then,(25)lp14+rp22+2mp1p3−32np12p2−p4≤2.


## 3. Improve Upper BoundH3,1f for the Class SSG∗

To prove Theorem6, we need the following two lemmas (Lemma 4 and Lemma 5).Lemma 4 (see [43]).
Iff∈SSG∗and is of the form (2), then,(26)a3−a22≤14.Lemma 5 (see [43]).
Letf∈SSG∗be of the form (2), and then,(27)a2a3−a4≤16.We now state and prove Theorem6.Theorem 6.
Letf∈SSG∗ be of the form (2), and then,
(28)a6≤2179757600≃0.37842,a7≤14244292073600≃0.68694,a2a4−a32≤116.
The result is sharp for function(29)fz=zexp∫0zet2−1tet2+1dt=z+14z3+⋯.Proof.
Sincef∈SSG∗, then, there exists a Schwarz function wz, given in (7) such that
(30)zf′zfz=21+e−wz.
Let(31)Ψwz=21+e−wz,(32)kz=1+c1z+c2z2+⋯=1+wz1−wz.
Obviously, the functionkz∈P and
(33)wz=kz−1kz+1.
This gives(34)wz=kz−1kz+1=c1z+c2z2+c3z3+⋯2+c1z+c2z2+c3z3+⋯.
From (31) and (34), we have
(35)21+e−wz=1+14c1z+14c2−18c12z2+11192c13−14c2c1+14c3z3+−3128c14+1164c12c2−14c3c1−18c22+14c4z4+⋯,while
(36)zf′zfz=1+a2z+2a3−a22z2+a23−3a2a3+3a4z3+−a24+4a22a3−4a2a4−2a32+4a5z4+5a6−5a2a5+a25−5a3a4−5a23a3+5a22a4+5a2a32z5+6a7−6a2a6+6a22a5−6a3a5+12a2a3a4−a26−6a23a4−3a42+2a33−9a22a32+6a24a3z6+⋯.On equating coefficients of (35) and (36), we get
(37)a2=14c1,(38)a3=18c2−132c12,(39)a4=71152c13−596c2c1+112c3,(40)a5=−1718432c14+7384c12c2−124c3c1−3128c22+116c4,(41)a6=−2571843200c15−10723040c13c2+311920c3c12+1397680c1c22−11320c4c1−19480c3c2+120c5,(42)a7=33599132710400c16−732211840c14c2−7917280c13c3−1451184320c12c22+1097680c4c12+471440c1c2c3−7240c5c1+559216c23−13384c4c2−5288c32+124c6.Now from (41), we have
(43)a6=196c5−910c1c4+19480c5−c1c4+311920c12c3−107372c1c2−2571843200c15+1397680c1c2≤196c5−910c1c4+19480c5−c1c4+311920c12c3−107372c1c2+2571843200c15+1397680c1c2.By applying (20) and (21) to above we get
(44)a6≤2179757600.Now from (42), we have
(45)a7=124c6−512c32−13384c2c4−55312c22−7240c1c5−109224c1c4+471440c1c2c3−14516016c1c2+33599132710400c16−732211840c14c2−7917280c13c3≤124c6−512c32+13384c2c4−55312c22+7240c1c5−109224c1c4+471440c1c2c3−14516016c1c2+33599132710400c16+732211840c14c2+7917280c13c3.By applying (20) and (21) to the above, we get
(46)a7≤14244292073600.Now, from (37)–(39), we have
(47)a2a4−a32=59216c14−1192c12c2+148c3c1−164c22.Using Lemma2, we get
(48)a2a4−a32=−79216c14−1192c12x24−c12−1256x24−c122+196c14−c121−x2z.Letx=y,y∈0,1,c1=c,c∈0,2, and z=1, along with triangle inequality, and we have
(49)a2a4−a32≤79216c4+1192c2y24−c2+1256y24−c22+196c4−c21−y2=Gc,y.Differentiating (49) partially with respect to y, we have
(50)∂Gc,y∂y=c2−8c+124−c2y384>0,showing thatGc,yis an increasing function in intervaly∈0,1,c∈0,2,so the maximum is attained aty=1,that is,
(51)maxGc,y=Gc,1=79216c4+1192c24−c2+12564−c22=Fc.Now(52)F′c=72304c3+196c4−c2−196c3−164c4−c2=72304c3−1192c4−c2−196c3,since F′c=0 has root at c=0 and also
(53)F′′c=−148<0,so the maximum is attained atc=0; therefore, we have
(54)Fc=116.Hence,(55)a2a4−a32≤Fc=116.For the third Hankel determinant, we need the following result.Lemma 7 (see [17]).
Letf∈SSG∗be of the form (2). Then,(56)a2≤12,a3≤14,a4≤16,a5≤18.Theorem 8.
Letf∈SSG∗ be of the form (2). Then,
(57)H3,1f≤43576≃0.07465.Proof.
Since from (16), we have
(58)H3,1f=a3a2a4−a32−a4a4−a2a3+a5a3−a22,by applying triangle inequality, we obtain
(59)H3,1f≤a3a2a4−a32+a4a4−a2a3+a5a3−a22.
Now, using Lemmas4–7 and Theorem 6 in conjunction with (59), we can get the required result.


## 4. Bounds ofH4,1f for the Class SSG∗

In recent years, researchers has started to evaluate the fourth-order Hankel determinant for different subclasses of analytic functions. The trend of finding the fourth-order Hankel determinant in geometric function theory started in 2018, when Arif et al. [44] studied and obtained the upper bound for the class of bounded turning functions. Recently Zhang and Tang [31] studied the fourth-order Hankel determinant for a subclass of starlike functions associated with the sine function. Inspired from the recent research going on and from the above works, we discuss here the fourth-order Hankel determinant for the class SSG∗.From (15), we can write H4,1f as
(60)H4,1f=a7H3,1f−a6δ1+a5δ2−a4δ3,where
(61)δ1=a3a2a5−a3a4−a4a5−a2a4+a6a3−a22,(62)δ2=a3a3a5−a42−a5a5−a2a4+a6a4−a2a3,(63)δ3=a4a3a5−a42−a5a2a5−a3a4+a6a4−a2a3.Theorem 9.
Letf∈SSG∗ be of the form (2), and then,
(64)a2a5−a3a4≤536.Proof.
From equations (37), (38), (39), and (40), we get
(65)a2a5−a3a4=−124576c15+52304c13c2−1128c3c12+11536c1c22+164c4c1−196c3c2=52304c13c2−3360c12+164c1c4−12c1c3−196c2c3−116c1c2≤52304c13c2−3360c12+164c1c4−12c1c3+196c2c3−116c1c2.
Now, making use of (20), (21), and (22) in conjunction with (65), we can get the required result.Theorem 10.
Letf∈SSG∗ be of the form (2), and then,
(66)a5−a2a4≤18.Proof.
From equations (37), (39), and (40), we get
(67)a5−a2a4=1165128c14−12c12c2+c3c1+38c22−c4.
Applying Lemma3 to the last term, we get the required result.Theorem 11.
Letf∈SSG∗ be of the form (2), and then,
(68)a3a5−a42≤740582944.Proof.
From equations (37), (39), and (40), we get
(69)a3a5−a42=−435308416c16−23442368c14c2+13456c13c3+1136864c12c22−1512c4c12+1288c1c2c3−31024c23+1128c4c2−1144c32=−435308416c16−23442368c14c2−1144c3c3−12c1c2−31024c22c2−11108c12+1128c4c2−14c12+13456c13c3−23128c1c2≤435308416c16+23442368c14c2+1144c3c3−12c1c2+31024c22c2−11108c12+1128c4c2−14c12+13456c13c3−23128c1c2.
Now, making use of (20), (21), and (22) in conjunction with (69), we can get the required result.Theorem 12.
Letf∈SSG∗ be of the form (2), and then,
(70)H4,1f≤16431024581119439360000≃0.13757.Proof.
From (15), we have
(71)H4,1f=a7H3,1f−a6δ1+a5δ2−a4δ3,where δ1,δ2, and δ3 are defined in (61), (62), and (63), respectively. Now, using triangle inequalities, we have
(72)H4,1f≤a7H3,1f+a6δ1+a5δ2+a4δ3,since
(73)δ1=a3a2a5−a3a4−a4a5−a2a4+a6a3−a22≤a3a2a5−a3a4+a4a5−a2a4+a6a3−a22.
By applying Lemmas4 and 7 and Theorems 6, 9, and 10, we get
(74)δ1≤34597230400.
And also,(75)δ2=a3a3a5−a42−a5a5−a2a4+a6a4−a2a3≤a3a3a5−a42+a5a5−a2a4+a6a4−a2a3.
Using Lemmas5 and 7 and Theorems 6, 10, and 11 we get
(76)δ2≤8378538294400.
Also, again,(77)δ3=a4a3a5−a42−a5a2a5−a3a4+a6a4−a2a3≤a4a3a5−a42+a5a2a5−a3a4+a6a4−a2a3,
Using Lemmas5 and 7 and Theorems 6, 9, and 11, we get
(78)δ3≤118581712441600.
Now, using the values of (74), (76), and (78) along with Theorem 6 and Lemma 7 to (72), we get the desired the estimate.


## 5. Zalcman Conjecture for ClassSSG∗

One of the main conjectures in the geometric function theory, suggested by Lawrence Zalcman in 1960, is that the coefficients of classS satisfy the inequality
(79)∣an2−a2n−1∣≤n−12.Only the well-known Koebe functionkz=z/1−z2 and its rotations have equality in the above form. For the popular Fekete-Szego inequality, when n=2, the equality holds. Recently, Khan et al. [43] evaluated the Zalcman conjecture for the class of starlike functions with respect to symmetric points associated with the sine function. Many researchers have studied the Zalcman function in the literature [45–47].Theorem 13.
Letf∈SSG∗ be of the form (2), and then,
(80)a5−a32≤18.
The result is sharp for function(81)fz=zexp∫0zet4−1tet4+1dt=z+18z5+⋯.Proof.
From equations (37) and (40), we get
(82)a5−a32=116−351152c14+512c12c2−23c3c1−58c22+c4.
Using Lemma3 and equation (82), we can get the required result.


## 6. Bounds ofH4,1f for 2-Fold, 3-Fold, and 4-Fold Symmetric Functions

Letm∈ℕ=1,2,3,⋯,. It is called m-fold symmetric if a rotation of domain U about the origin through an angle 2π/m carries itself on the domain U. It is obvious that in U, an analytic function f is m-fold symmetric if
(83)fe2πmz=e2πmfz∀z∈U.The set ofm-fold symmetric univalent functions with the following series:
(84)fz=z+∑k=2∞amk+1zmk+1∀z∈U,is referred to asSm.The subclassSSG∗m is a collection of m-fold symmetric starlike functions associated with the modified sigmoid function. More precisely, an analytic function f of the form (84) belongs to class SSG∗m if and only if
(85)zf′zfz=21+e−pz−1/pz+1p∈Pm,where the set Pm is defined by
(86)Pm=p∈P:pz=1+∑k=1∞cmkzmk∀z∈U.Theorem 14.
Iff∈SSG∗2 and be of the form (84), then,
(87)a3a7−a52≤75121504.Proof.
Sincef∈SSG∗2, therefore, there exists a function p∈P2 such that
(88)zf′zfz=21+e−pz−1/pz+1.
Using the series forms (84) and (86), when m=2 in the above relation, we have
(89)a3=18c2,(90)a5=116c4−3128c22,(91)a7=5510752c23−13448c4c2+128c6.
Now, using (89), (90), and (91), we get
(92)a3a7−a52=31344064c24−57168c22c4+1224c6c2−1256c42=31344064c24+1224c2c6−532c2c4−1256c42≤31344064c24+1224c2c6−532c2c4+1256c42.
Now, using (20) and (21) to the above, we get the required result.Theorem 15.
Iff∈SSG∗2 and be of the form (84), then,
(93)a5−a32≤18.Proof.
Using (89) and (90), we have
(94)a5−a32=116c4−58c22.
Using (21) to the above, we get the required result.Theorem 16.
Iff∈SSG∗2 and be of the form (84), then,
(95)H4,1f≤751172032.Proof.
Sincef∈SSG∗2, therefore, a2=a4=a4=0 and we have
(96)H4,1f=a5−a32a3a7−a52.
Then,(97)H3,1f=a5−a32a3a7−a52,
Using (87) and (93), we get the required result.Theorem 17.
Iff∈SSG∗3 and be of the form (84), then,
(98)H4,1f≤1432.Proof.
Sincef∈SSG∗3, therefore, there exists a function p∈P3 such that
(99)zf′zfz=21+e−pz−1/pz+1.
Using the series forms (84) and (86), when m=3 in the above relation, we have
(100)a4=112c3,a7=124c6−5288c32.
Now,(101)H4,1f=a42a42−a7.
Therefore,(102)H4,1f=−13456c32c6−712c32=13456c32c6−712c32.
Using (20) and (21), we get the desired result.Theorem 18.
Iff∈SSG∗4 and be of the form (84), then,
(103)H4,1f≤164.Proof.
Sincef∈SSG∗4, therefore, there exists a function p∈P4 such that
(104)zf′zfz=21+e−pz−1/pz+1.
Using the series forms (84) and (86), when m=4 in the above relation, we have
(105)a5=116c4.
Sincef∈SSG∗4, therefore, a2=a3=a4=a6=a7=0 and we have
(106)H4,1f=a52,H4,1f=1256c42.
Now,(107)H4,1f=1256c42.
Using (20) to the above, we get the required result.


## 7. Conclusion

In our present investigation, we have obtained the improved third-order Hankel determinant for a class of starlike functions connected with modified sigmoid functions. Furthermore, we have investigated the fourth-order Hankel determinant and Zalcman conjecture and also evaluated fourth-order Hankel determinants for 2-fold, 3-fold, and 4-fold symmetric starlike functions.



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