# A Study of Third and Fourth Hankel Determinant Problem for a Particular Class of Bounded Turning Functions

Authors: Gaganpreet Kaur; Gurmeet Singh; Muhammad Arif; Ronnason Chinram; Javed Iqbal
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this present paper, a new generalized classℛp,q from the family of function with bounded turning was introduced by using p,q- derivative operator. Our aim for this class is to find out the upper bound of third- and fourth-order Hankel determinant. Moreover, the upper bounds for two-fold and three-fold symmetric functions for this class are also obtained.


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## Body

## 1. Introduction and Motivation

In order to better explain the terminology included in our key observations, some of the essential relevant literatures on geometric function theory need to be provided and discussed here. We start with symbolA which represents the class of holomorphic functions in the region of open unit disc U=z:z<1 and satisfy the relationship g0=g′0−1=0 for g∈A. That is; if g∈A, then it has the following Taylor series form:(1)gz=z+∑λ=2∞δλzλ,z∈U.Also, letΨ⊂A represent all univalent functions in U. Next, we are going to define the most useful class of geometric function theory known as the class P of Caratheodory functions and is defines as; a holomorphic function h belongs to P if it satisfies Rehz>0 along with the series expansion:(2)hz=1+∑λ=1∞dλzλ,z∈U.Using the Caratheodory functions family, we consider the following basics subclasses ofΨ as(3)S∗=gz∈A:zg′zgz∈P,z∈U,K=gz∈A:zg′z′g′z∈P,z∈U,ℛ=gz∈A:g′z∈P,z∈U.The investigation ofq-calculus (q stands for quantum) fascinated and inspired many scholars due its use in various areas of the quantitative sciences. Jackson [1, 2] was among the key contributors of all the scientists who introduced and developed the q-calculus theory. Just like q-calculus was used in other mathematical sciences, the formulations of this idea are commonly used to examine the existence of various structures of function theory. Though the link between certain geometric nature of the analytic function and the q-derivative operator was established by the authors in [3], but, for the usage of q-calculus in function theory, a solid and comprehensive foundation is given in [4] by Srivastava. After this development, many researchers introduced and studied some useful operators in q-analog with the applications of convolution concepts. For example; Kanas and Răducanu [5] established the q-differential operator and then examined the behavior of this operator in function theory. This operator was generalized further for multivalent analytic functions by Arif et al. [6]. Analogous to q-differential operator, Arif et al. and Khan et al. contributed the integral operators for analytic and multivalent functions in [7, 8], respectively. Similarly, in the article [9], the authors developed and analyzed the operator in q-analog for meromorphic functions. Also, see the survey type article [10] on quantum calculus and their applications. With the use of these operators, many researchers were contributed some good articles in this direction in the field of geometric function theory, see [11, 12]. The substitution of q by q/p in q-calculus has given us p,q-calculus which is the extension of q-calculus. Chakrabarti and Jagannathan [13] considered p,q-integer.For0<q<p≤1 and z≠0, the p,q-derivative operator Dp,qg of the function gz∈A is defined as(4)Dp,qgz=gpz−gqzp−qz.From the use (1) and (4), the following formulae are easily obtained:(5)Dp,qg1z+g2z=Dp,qg1z+Dp,qg2z,and for a constant c,(6)Dp,qcgz=cDp,qgz.Here, we also note thatDp,qgz=g′z when we take p=1 and q⟶1−.(7)limq⟶1−Dp,qgz=limq⟶1−gz−gqz1−qz=g′z.Further by using (1) and (4), we have(8)Dp,qgz=1+∑λ=2∞λp,qδλzλ−1,with(9)λp,q=pλ−qλp−q=pλ−11+∑l=1λ−1qpl.Let us define the classℛp,q, which consists all the functions g∈A, satisfying(10)ReDp,qgz>0,z∈U.The Hankel determinantℋDn,λg with n,λ∈ℕ=1,2… and δ1=1 for a function g∈Ψ of the series form (1) was given by Pommerenke [14, 15], as(11)ℋDn,λg=δλδλ+1⋯δλ+n−1δλ+1δλ+2⋯δλ+n⋮⋮⋮⋮δλ+n−1δλ+n⋯δλ+2n−2.In particular, the following determinants are known as the first-, second-, and third-order Hankel determinants, respectively.(12)ℋD2,1g=1δ2δ2δ3=δ3−δ22,ℋD2,2g=δ2δ3δ3δ4=δ2δ4−δ32,ℋD3,1g=1δ2δ3δ2δ3δ4δ3δ4δ5=δ3δ2δ4−δ32−δ4δ4−δ2δ3+δ5δ3−δ22.There are comparatively few observations in literature in relation to the Hankel determinant for the functiong which belongs to the general family Ψ. For the function g∈Ψ, the best established sharp inequality is ℋD2,λg≤μλ, where μ is the absolute constant, which is due to Hayman [16]. Further for the same class Ψ, it was obtained in [17] that(13)ℋD2,2g≤μ1,for1≤μ1≤113,ℋD3,1g≤μ2,for49≤μ2≤32+28515.In a given family of functions, the problem of calculating the bounds, probably sharp, of Hankel determinants attracted the minds of several mathematicians. For example, the sharp bound ofℋD2,2g, for the subfamilies K, S∗, and ℛ (family of bounded turning functions) of the set Ψ, was calculated by Janteng et al. [18, 19]. These estimates are(14)ℋD2,2g≤18,forf∈K,1,forf∈S∗,49,forf∈ℛ.For the following two families,S∗β0≤β<1 of star-like functions of order β and for SS∗β0<β≤1 of strongly star-like functions of order β, the authors computed in [20, 21] that ℋD2,2g is bounded by 1−β2 and β2, respectively. The exact bound for the family of Ma-Minda star-like functions was measured in [22], see also [23]. For more work on ℋD2,2g, see references [24–28].It is quite clear from the formulae given in (12) that the calculation of ℋD3,1g is far more challenging compared with finding the bound of ℋD2,2g. Babalola was the first mathematician who investigated the bounds of third=order Hankel determinant for the families of K, S∗, and ℛ in an article [29] published in 2010. Using the same approach, later, several authors [30–34] published their articles regarding ℋD3,1g for certain subfamilies of analytic and univalent functions. After this study, Zaprawa [35] improved the findings of Babalola in 2017 by applying a new methodology. He obtained the following bounds:(15)ℋD3,1g≤49540,forg∈K,1,forg∈S∗,4160,forg∈ℛ.He argued that such limits are indeed not the best. Later in 2018, Kwon et al. [36] strengthened the Zaprawa’s result for g∈S∗ and showed that ℋD3,1g≤8/9 and this bound was further improved by Zaprawa et al. [37] in 2021. They got ℋD3,1g≤5/9 for g∈S∗. Recently in 2018, Kowalczyk et al. [38] and Lecko et al. [39] succeeded in finding the sharp bounds of ℋD3,1g for the families K and S∗1/2, respectively, where S∗1/2 indicate the star-like functions family of order 1/2. These results are given as(16)ℋD3,1g≤4135,forg∈K,19,forg∈S∗12.The estimation of fourth Hankel determinantℋD4,1g for the bounded turning functions has been obtained by Arif et al. [40] and they proved the following bounds for g∈ℛ:(17)ℋD4,1g≤0.78050.After that, Kaur and Singh [41] proved fourth Hankel determinant for bounded turning function of order α. For more contributions, see [42–46]. Recently, Srivastava et al. [47] consider a family of normalized analytic functions with bounded turnings in the open unit disk which are connected with the cardioid domains and they obtained the estimates of fourth Hankel determinant.


## 2. A Set of Results

In order to investigateℋD4,1g, we need the following results.Lemma 1.
(see [48]). If hz∈P, having the form (2), then(18)dλ|2;forλ∈ℕ,(19)dλ+k−μdλdk≤2;for0≤μ≤1,(20)dpdq−drds≤4;forp+q=r+s.Theorem 1.
(see [40]). Let gz=z+∑λ=2∞δλzλ∈S∗ and for real μ,(21)δ22δ3−μδ22=43−4μ,forμ≤58;122μ−1,forμ∈58,34;141−μ,forμ∈34,78;44μ−3,forμ≥78.


## 3. Bounds of Third Hankel Determinant

The third Hankel determinantℋD3,1g is a polynomial of four variables, as:(22)ℋD3,1g=δ3δ5−δ42+δ2δ3δ4−δ2δ5+δ3δ2δ4−δ32.In order to solveℋD3,1g, we need to know the correspondence between g and h∈P.(23)gz∈ℛp,q⇔Dp,qgz∈P.Thus,(24)1+∑λ=2∞λp,qδλzλ−1=1+∑λ=1∞dλzλ.By simplifying, we yield(25)δλ=dλ−1λp,q.Now, using the above coefficients in (22), we obtain(26)ℋD3,1g=13p,q5p,qd2d4−14p,q2d32+22p,q3p,q4p,qd1d2d3−12p,q25p,qd12d4−13p,q3d23.Rearranging the above terms,(27)ℋD3,1g=12p,q25p,qd4d2−d12−14p,q2d3d3−d1d2+13p,q3d2d4−d22−22p,q3p,q4p,q−14p,q2d2d4−d1d3+13p,q5p,q−12p,q25p,q−13p,q3+22p,q3p,q4p,q−14p,q2d2d4.Using triangular inequality and the results (18) and (19) of lemma of Section 2, we get(28)|ℋD3,1g|≤442p,q3p,q4p,q+13p,q5p,q−14p,q2.Remark 1.
As we know by (7), if p=1 and q⟶1−, then the above result (28) coincides with Zaprawa [35].


## 4. Bounds of Fourth Hankel Determinant

Firstly,ℋD4,1g is the fourth Hankel determinant of the form (11) with six coefficients which can be written in the form,(29)ℋD4,1g=δ7ℋD3,1g−δ6Λ1+δ5Λ2−δ4Λ3,where Λ1,Λ2, and Λ3 are third-order determinants given as(30)Λ1=δ3δ6−δ4δ5−δ2δ2δ6−δ3δ5+δ4δ2δ4−δ32;(31)Λ2=δ4δ6−δ52−δ2δ3δ6−δ4δ5+δ3δ3δ5−δ42;(32)Λ3=δ2δ4δ6−δ52−δ3δ3δ6−δ4δ5+δ4δ3δ5−δ42.Theorem 2.
Ifg∈ℛp,q, then(33)ℋD4,1g≤813p,q7p,qσ1+14p,q2σ2+15p,q2σ3+16p,qσ4,where(34)σ1=42p,q4p,q+15p,q,σ2=22p,q6p,q−17p,q+43p,q5p,q,σ3=22p,q4p,q+23p,q2−15p,q,σ4=23p,q24p,q+24p,q5p,q+13p,q6p,q.Proof.
By using definition (10), we have gz∈ℛp,q. Thus, Dp,qgz=hz∈P..(35)1+∑λ=2∞λp,qδλzλ−1=1+∑λ=1∞dλzλ.
Therefore,(36)λp,qδλ=dλ−1.
Substituting (36) in (30) and (31) and in (32) yields that(37)Λ1=13p,q6p,qd2d5−14p,q5p,qd3d4−12p,q26p,qd12d5+12p,q3p,q5p,qd1d2d4+12p,q4p,q2d1d32−13p,q24p,qd22d3;Λ2=14p,q6p,qd3d5−15p,q2d42−12p,q3p,q6p,qd1d2d5+12p,q4p,q5p,qd1d3d4+13p,q25p,qd22d4−13p,q4p,q2d2d32;Λ3=12p,q4p,q6p,qd1d3d5−12p,q5p,q2d1d42−13p,q26p,qd22d5+23p,q4p,q5p,qd2d3d4−14p,q3d33.
Now, rewrite the above equations as follows:(38)Λ1=12p,q26p,qd5d2−d12+13p,q24p,qd3d4−d22−12p,q4p,q2d3d4−d1d3−14p,q5p,q+13p,q24p,q−12p,q4p,q2d4d3−d1d2+14p,q5p,q+13p,q24p,q−12p,q4p,q2−12p,q3p,q5p,qd2d5−d1d4+16p,q3p,q−16p,q2p,q2−14p,q5p,q−13p,q24p,q+12p,q4p,q2+12p,q3p,q5p,qd2d5;Λ2=12p,q3p,q6p,qd5d3−d1d2−13p,q25p,qd4d4−d22+13p,q4p,q2d3d5−d2d3−12p,q4p,q5p,q−15p,q2+13p,q25p,qd3d5−d1d4−15p,q2−13p,q25p,qd4d4−d1d3+14p,q6p,q−12p,q3p,q6p,q−13p,q4p,q2+12p,q4p,q5p,q−15p,q2+13p,q25p,qd3d5;Λ3=13p,q26p,qd5d4−d22−12p,q4p,q6p,qd5d4−d1d3+14p,q3d3d6−d32−14p,q3d3d6−d2d4+12p,q5p,q2d4d5−d1d4−23p,q4p,q5p,q−14p,q3d4d5−d2d3+12p,q4p,q6p,q+23p,q4p,q5p,q−13p,q26p,q−12p,q5p,q2−14p,q3d4d5.
Using the triangular inequality with the inequalities (18) and (19) of lemma on the above equations, we obtain(39)Λ1≤423p,q24p,q+14p,q5p,q+13p,q6p,q;(40)Λ2≤422p,q4p,q5p,q+23p,q25p,q+14p,q6p,q−15p,q2;(41)Λ3≤422p,q4p,q6p,q+43p,q4p,q5p,q.
Now, using the values (39)–(41) and (28) along with the inequality δλ≤2/λp,q in (29), we get our desired result.


## 5. Bounds ofℋD4,1g for Two-Fold and Three-Fold Symmetric Functions

n-Fold symmetric function consists all those functions g which satisfy the following condition:(42)gεz=εgz,∀z∈U,where ε=exp2Πι/n. The set of univalent functions with n-fold symmetry (that is Ψn) has the expansion of the form,(43)Ψn=gz∈Ψ;gz=z+∑j=1∞δnj+1znj+1,z∈U.Furthermore, univalent functiongz∈Ψn belongs to ℛp,qn if and only if(44)Dp,qgz=hzwithh∈Pn,whereas(45)Pn=hz∈P;hz=1+∑j=1∞dnjznj,and function gz∈Ψn∈S∗n if Rezg′z/gz>0,z∈U.Now, ifg∈Ψ3, then gz=z+δ4z4+δ7z7+⋯; hence, ℋD4,1g=δ42δ42−δ7. In the same way, if g∈Ψ2, then gz=z+δ3z3+δ5z5+⋯; clearly, we can see that two-fold symmetric functions are odd. So, ℋD4,1g=δ3δ5δ7−δ33δ7+δ32δ52−δ53.Theorem 3.
Ifgz is three-fold symmetric bounded turning function, that is, gz∈ℛp,q3, then(46)ℋD4,1g≤177p,q.Proof.
Firstly, consider thatgz∈ℛp,q3, then ∃ is a function q˜∈S∗3 of the form z+α4z4+α7z7+⋯ such that zq˜′z/q˜z=Dp,qgz. Since gz∈ℛp,q3 and with the use of (43) for n=3, it follows that(47)1+3α4z3+6α7−3α42z6+⋯=1+4p,qδ4z3+7p,qδ7z6+⋯.
Identifying the coefficients, we get(48)3α4=4p,qδ4,6α7−3α42=7p,qδ7.
We already knowq˜∈S∗3, then ∃ function qz of the form qz=z+∑j=2∞qjzj∈S∗ such that q˜z=3qz3. Thus,(49)z+α4z4+α7z7+⋯=z+13q2z4+13q3−19q22z7+⋯.
Therefore,(50)α4=13q2,α7=13q3−19q22.
Now, by rearranging (48) and (50), we get(51)δ4=q24p,q,δ7=2q3−q227p,q.
SinceℋD4,1g=δ42δ42−δ7, this implies(52)ℋD4,1g≤24p,q27p,qq22q3−7p,q+4p,q224p,q2q22.
With the use of Theorem1 in Section 2, where λ=7p,q+4p,q2/24p,q2∈5/8,3/4 for p=1,q⟶1−, we get our theorem proved.Theorem 4.
Ifg is two-fold symmetric bounded turning function, that is, g∈ℛp,q2, then(53)ℋD4,1g≤83p,q5p,q7p,q.Proof.
By the definition of two-fold symmetric function, the Hankel determinant can be written as(54)ℋD4,1g=δ3δ5δ7−δ33δ7+δ32δ52−δ53.
Sinceg∈ℛp,q2, then ∃ is a function h∈P2 such that Dp,qgz=hz; then, the expansion of (43) and (45) for n=2 yields that(55)1+3p,qδ3z2+5p,qδ5z4+7p,qδ7z6+⋯=1+d2z2+r4z4+d6z6+⋯,∴δn=1np,qdn−1,ℋD4,1g=13p,q5p,q7p,qd2d4d6−13p,q37p,qd23d6+13p,q25p,q2d22d42−15p,q3d43≤13p,q5p,q7p,qd2d6−3p,q7p,q5p,q2d42d4−5p,q3p,q2d22.
Now, with the help of lemma in Section2, we get our desired result as asserted by the statement.Remark 2.
Forp=1,q⟶1− results of (29), (45) and (43) will coincide with results derived in [40].



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