# Sharp Bounds of the Coefficient Results for the Family of Bounded Turning Functions Associated with a Petal-Shaped Domain

Authors: Omar M. Barukab; Muhammad Arif; Muhammad Abbas; Sher Afzal Khan
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The goal of this article is to determine sharp inequalities of certain coefficient-related problems for the functions of bounded turning class subordinated with a petal-shaped domain. These problems include the bounds of first three coefficients, the estimate of Fekete-Szegö inequality, and the bounds of second- and third-order Hankel determinants.


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## Body

## 1. Preliminary Concepts

Let the family of holomorphic (or analytic) functions in the region (or domain) of unit discD=z∈ℂ:z<1 be described by the symbol HD and let A be the subfamily of HD which is defined by
(1)A=f∈HD:fz=∑k=1∞akzkwitha1=1.Further, the setS⊂A contains all normalized univalent functions in D. For two functions F1,F2∈HD, we say that F1 is subordinate to F2, written symbolically by F1≺F2, if there exists a Schwarz function v with v0=0 and vz<1 that is analytic in D such that fz=gvz,z∈D. However, if F2 is univalent in D, then the following relation holds:
(2)F1z≺F2z,z∈D⇔F10=F20andF1D⊂F2D.In geometric function theory, the most basic and important subfamilies of the setS are the family S∗ of starlike functions and the family C of convex functions which are defined as follows:
(3)C=f∈A:zf′z′f′z≺Λzz∈D≔CΛ,S∗=f∈A:zf′zfz≺Λzz∈D≔S∗Λ,with
(4)Λz=1+2∑n=2∞zn≔1+z1−z,z∈D.By varying the functionΛz in (18), we get some subfamilies of the set S∗ which have significant geometric sense. For example,
(i)
If we takeΛz=1+Mz/1+Nz with −1≤N<M≤1, then the deduced family(5)S∗M,N≡S∗1+Mz1+Nzis described by the functions of the Janowski starlike family established in [1] and later studied in different directions in [2, 3]
(ii)
The familySL∗≡S∗Λz with Λz=1+z was developed in [4] by Sokól and Stankiewicz. The image of the function Λz=1+z demonstrates that the image domain is bounded by the Bernoulli’s lemniscate right-half plan specified by ∣w2−1∣<1(iii)
By selectingΛz=1+sinz, the class S∗Λz lead to the family Ssin∗ which was explored in [5] while Se∗≡S∗ez has been produced in the article [6] and later studied in [7](iv)
The familySc∗≔S∗Λz with Λz=1+4/3z+2/3z2 was contributed by Sharma and his coauthors [8] which contains function f∈A such that zf′z/fz is located in the region bounded by the cardioid given by(6)9x2+9y2−18x+52−169x2+9y2−6x+1=0(v)
The familySR∗≡S∗Λz with Λz=1+z/√2+1√2+1+z/√2+1−z is studied in [9] while Scos∗≔S∗cosz and Scosh∗≔S∗coshz were recently examined by Bano and Raza [10] and Alotaibi et al. [11], respectively(vi)
If we considerΛz=1sinh−1z, then the class Sρ∗≔S∗1+sinh−1z was provided by Kumar and Arora [12] and is defined as a function f∈A which is in the family Sρ∗ if (18) holds for the function Λz=ρz, where(7)ρz=1+sinh−1zClearly, the functionρz is a multivalued function and has the branch cuts about the line segments −i∞,−i∪i,i∞, on the imaginary axis, and hence, it is holomorphic in D. In a geometric point of view, the function ρz maps the unit disc D onto a petal-shaped region Ωρ,(8)Ωρ=w∈ℂ:sinhw−1<1.Using this idea, we now consider a subfamilyBTs of analytic functions as
(9)BTs=f∈A:f′z≺Λ~z,andΛ~zisgivenby8.If we take the functionΛz, given by (4), instead of Λ~z in (9), we get the familiar class R of bounded turning functions. From the statement of the Nashiro-Warschowski theorem, it follows that the functions in R are univalent in D. The properties of this class was studied extensively by the researchers, see [13–16].The Hankel determinantHDq,nfwithq,n∈ℕ=1,2,⋯anda1=1 for a function f∈S of the series form (1) was given by Pommerenke [17, 18] as
(10)HDq,nf≔anan+1⋯an+q−1an+1an+2⋯an+q⋮⋮⋯⋮an+q−1an+q⋯an+2q−2.Specifically, the first-, second-, and third-order Hankel determinants, respectively, are(11)HD2,1f=1a2a2a3=a3−a22,HD2,2f=a2a3a3a4=a2a4−a32,HD3,1f=1a2a3a2a3a4a3a4a5=a3a2a4−a32−a4a4−a2a3+a5a3−a22.In literature, there are relatively few findings in relation to the Hankel determinant for the functionf belongs to the general family S. For the function f∈S, the best established sharp inequality is HD2,nf≤λn, where λ is absolute constant, which is due to Hayman [19]. Further, for the same class S, it was obtained in [20] that
(12)HD2,2f≤λ,for1≤λ≤113,HD3,1f≤μ,for49≤μ≤32+28515.The growth ofHDq,nf has often been evaluated for different subfamilies of the set S of univalent functions. For example, the sharp bound of HD2,2f, for the subfamilies C, S∗, and R of the set S, was measured by Janteng et al. [21, 22]. These bounds are
(13)HD2,2f≤18,forf∈C,1,forf∈S∗,49,forf∈R.The exact bound for the collection of close-to-convex functions of such a specific determinant is still unavailable (see [23]). On the other hand, for the set of Bazilevic˘ functions, the best estimate of HD2,2f was proved by Krishna and RamReddy [24]. For more work on HD2,2f, see References [25–29].It is very obvious from the formulae provided in (11) that the estimate of HD3,1f is far more complicated compared with finding the bound of HD2,2f. In the first paper on HD3,1f, published in 2010, Babalola [30] obtained the upper bound of HD3,1f for the families of C, S∗, and R. He obtained the following bounds:
(14)HD3,1f≤0.714⋯,forf∈C,16,forf∈S∗,0.742⋯,forf∈R.Later on, using the same methodology, some other authors [31–35] published their work concerning HD3,1f for different subfamilies of analytic and univalent functions. In 2017, Zaprawa [36] improved Babalola’s [30] results by applying a new technique which is given as
(15)HD3,1f≤49540,forf∈C,1,forf∈S∗,4160,forf∈R.He argues that such limits are indeed not the best ones. After that, in 2018, Kwon et al. [37] enhanced Zaprawa’s bound for f∈S∗ and showed that HD3,1f≤8/9, but it is still not the best possible. The firstly examined papers in which the authors obtained the sharp bounds of HD3,1f came to the reader’s hands in 2018. Such papers have been written by Kowalczyk et al. [38] and Lecko et al. [39]. These results are given as
(16)HD3,1f≤4135,forf∈C,19,forf∈S∗12,where S∗1/2 indicates the starlike function family of order 1/2. We would also like to acknowledge the research provided by Mahmood et al. [40] in which they examined the third Hankel determinant in the q-analog for a subfamily of starlike functions and for more contribution of such type families, see [41, 42]. In the present article, our aim is to calculate the sharp bounds of some of the problems related to Hankel determinant for the class BTs of bounded turning functions connected with a petal-shaped domain.


## 2. A Set of Lemmas

Definition 1.
LetPrepresent the class of all functionspthat are holomorphic inDwithRepz>0and has the series representation(17)pz=1+∑n=1∞cnznz∈D.For the proofs of our key findings, we need the following lemma. It contains the well-known formula forc2, see [43], the formula for c3 due to Libera and Zlotkiewicz [44], and the formula for c4 proved in [45].Lemma 2.
Letp∈Phas the series form ((17)). Then, forx,σ,ρ∈D¯=D∪1,
(18)2c2=c12+x4−c12,(19)4c3=c13+24−c12c1x−c14−c12x2+24−c121−x2σ,(20)8c4=c14+4−c12xc12x2−3x+3+4x−44−c121−x2cx−1z+x¯σ2−1−σ2ρ.Lemma 3.
Ifp∈Pand has the series form ((17)), then(21)cn+k−μcnck≤2max1,2μ−1,(22)cn≤2forn≥1,(23)Jc13−Kc1c2+Lc3≤2J+K−2J+2J−K+L,with J,K,L,μ∈ℂ and for B∈0,1 with B2B−1≤D≤B, we have
(24)c3−2Bc1c2+Dc13≤2.The inequalities (21), (22), (23), and (24) in the above lemma are taken from [43, 46], [47–49], and [50], respectively.


## 3. Coefficient Inequalities for the ClassBTs

We begin this section by finding the absolute values of the first three initial coefficients for the function of classBTs.Theorem 4.
Iff∈BTsand has the series representation ((1)), then(25)a2≤12,a3≤13,a4≤14.
These bounds are sharp.Proof.
Letf∈BTs. Then, (9) can be written in the form of the Schwarz function as
(26)f′z=1+sinh−1wz,z∈D.
Now, ifp∈P, then it may be written in terms of the Schwarz function w by
(27)pz=1+wz1−wz=1+c1z+c2z2+c3z3+⋯,equivalently,
(28)wz=pz−1pz+1=c1z+c2z2+c3z3+c4z4+⋯2+c1z+c2z2+c3z3+c4z4+⋯.
From (1), we easily get
(29)f′z=1+2a2z+3a3z2+4a4z3+5a5z4+⋯.
By simplification and using the series expansion (28), we obtain
(30)1+sinh−1wz=1+12c1z+12c2−14c12z2+12c3+548c13−12c1c2z312c4−14c22−132c14+516c12c2−12c1c3z4+⋯.
Comparing (29) and (30),we get
(31)a2=14c1,(32)a3=1312c2−14c12,(33)a4=1412c3+548c13−12c1c2,(34)a5=1512c4−14c22−132c14+516c12c2−12c1c3.
Fora2, implementing (22) in (31), we obtain
(35)a2≤12.
Fora3, reordering (32), we get
(36)a3=16c2−12c1c1,and using (21), we have
(37)a3≤13.
Fora4, we can rewrite (33) as
(38)a4=14548c13−12c1c2+12c3.
Application of triangle inequality plus (23), we get
(39)a4≤142548+212−2548+2548−12+12.
By simple calculations, we obtain(40)a4≤14.
These outcomes are sharp. For this, we consider a function(41)fn′z=1+sinh−1zn,forn=1,2,3.
Thus, we have(42)f1z=∫0z1+sinh−1tdt=z+12z2−124z4+⋯,f2z=∫0z1+sinh−1t2dt=z+13z3−142z7+⋯,f3z=∫0z1+sinh−1t3dt=z+14z4−160z10+⋯.Now, we discussed about the Hankel determinant problem, which is explicitly related to the Fekete-Szegö functional which is an extraordinary instance of the Hankel determinant.Theorem 5.
Iffof the form ((1)) belongs toBTs,then(43)a3−γa22≤max1,3γ4,forγ∈ℂ.
This inequality is sharp.Proof.
Employing (31) and (32), we may write
(44)a3−γa22=c26−c1212−γc1216.
By rearranging, it yields(45)a3−γa22=16c2−3γ+48c2.
Application of (21) leads us to
(46)a3−γa22≤16max2,223γ+48−1.
After the simplification, we obtain(47)a3−γa22≤13max1,3γ4.
The required result is sharp and is determined by(48)f2z=∫0z1+sinh−1t2dt=z+13z3−142z7+⋯.Theorem 6.
Iffhas the form ((1)) belongs toBTs, then(49)a2a3−a4≤14.
This inequality is sharp.Proof.
Using (31), (32), and (33), we have
(50)a2a3−a4=18c3−223c1c2+724c13.
From (24), we have
(51)0≤B=23≤1,B=23≥D=724,and also satisfy
(52)B2B−1=2343−1≤D=724.
Thus, by using (24), we have
(53)a2a3−a4≤14.
Equality is achieved by using(54)f3z=∫0z1+sinh−1t3dt=z+14z4−160z10+⋯.Next, we will determine the second-order Hankel determinantHD2,2f for f∈BTs.Theorem 7.
Iffbelongs toBTs, then the second Hankel determinant(55)HD2,2f=a2a4−a32≤19.
This result is the best possible.Proof.
From (31), (32), and (33), we have
(56)HD2,2f=−12304c14−1288c12c2+132c1c3−136c22.
Using (18) and (19) to express c2 and c3 in terms of c1 and noting that without loss in generality we can write c1=c, with 0≤c≤2, we obtain
(57)HD2,2f=−1768c4−1128c24−c2x2−11444−c22x2+164c4−c21−x2z,with the aid of the triangle inequality and replacing z≤1, x=b, with b≤1. So,
(58)HD2,2f≤1768c4+1128c24−c2b2+11444−c22b2+164b4−c21−b2≔ϕc,b.
It is a simple exercise to show thatϕ′c,b≥0 on 0,1, so that ϕc,b≤ϕc,1. Putting b=1 gives
(59)HD2,2f≤1768c4+1128c24−c2+11444−c22≔ϕc,1.
Also,ϕ′c,1<0, and so ϕc,1 is a decreasing function. Thus, the maximum value at c=0 is
(60)HD2,2f≤16144=19.
The required second Hankel determinant is sharp and is obtained by(61)f2z=∫0z1+sinh−1t2dt=z+13z3−142z7+⋯.


## 4. Third-Order Hankel Determinant

We will now determine the third-order Hankel determinantHD3,1f for f∈BTs.Theorem 8.
Iffbelongs toBTs, then the third Hankel determinant(62)HD3,1f≤116.
This result is sharp.Proof.
The third Hankel determinant can be written as(63)HD3,1f=a2a4−a32a3−a1a4−a2a3a4+a1a3−a22a5.
After simplification of the above equation, we have(64)HD3,1f=2a2a3a4−a33−a42+a3a5−a22a5.
Letc1=c and putting the estimations of ai’s from (31), (32), (33), and (34), we get
(65)HD3,1f=1552960−151c6+144c4c2+1584c3c3−768c2c22−8064c2c4+13824cc2c3−7168c23+9216c2c4−8640c32.
To simplify computation, lett=4−c2 in (18), (19), and (20). Now, using the simplified form of these formulae, we obtain
(66)144c4c2=72c6+c4tx,1584c3c3=396c6+792c4tx−396c4tx2+792c3t1−x2σ,768c2c22=192c6+384c4tx+192c2t2x2,8064c2c4=1008c4tx3−4032c3tx1−x2σ−4032c2tx¯1−x2σ2−3024c4tx2+4032c2t1−x21−σ2ρ+4032c3t1−x2σ+3024c4tx+1008c6+4032c2tx2,13824cc2c3=−1728c2t2x3−1728c4tx2+3456ct2x1−x2σ+3456c2t2x2+3456c3t1−x2σ+5184c4tx+1728c6,7168c23=896t3x3+2688c2t2x2+2688c4tx+896c6,9216c2c4=2304c2tx2+2304t2x3+576c6+2304c4tx+2304c3t1−x2σ+2304c2t1−x21−σ2ρ+1728c2t2x2+2304ct2x1−x2σ+2304t2x1−x21−σ2ρ−1728c4tx2−2304c2tx¯1−x2σ2−2304c3tx1−x2σ−1728c2t2x3−2304t2xx¯1−x2σ2+576c4tx3+576c2t2x4−2304ct2x21−x2σ,8640c32=540c2t2x4−2160ct2x21−x2σ−2160c2t2x3−1080c4tx2+2160t21−x22σ2+4320ct2x1−x2σ+2160c2t2x2+2160c3t1−x2σ+2160c4tx+540c6.
Substituting these expressions in (65), by simple but too long computation,
(67)HD3,1f=1552960−15c6+2304t2x3−896t3x3−1728c2tx2−432c4tx3+252c4tx2+96c4tx+36c2t2x4−1296c2t2x3+144c2t2x2−2160t21−x22σ2+360c3t1−x2σ+1728c3tx1−x2σ+1728c2tx¯1−x2σ2−1728c2t1−x21−σ2ρ−144ct2x21−x2σ−2304t2xx¯1−x2σ2+1440ct2x1−x2σ+2304t2x1−x21−σ2ρ.
Sincet=4−c2,(68)HD3,1f=1552960v1c,x+v2c,xσ+v3c,xσ2+Ψc,x,σρ,where ρ,x,σ∈D¯, and
(69)v1c,x=−15c6+4−c24−c2−1280x3−400c2x3+36c2x4+144c2x2−1728c2x2−432c4x3+252c4x2+96c4x,v2c,x=4−c21−x24−c21440cx−144cx2+1728c3x+360c3,v3c,x=4−c21−x24−c2−144x2−2160+1728c2x¯,Ψc,x,z=4−c21−x21−σ2−1728c2+2304x4−c2.
Now, by usingx=x,σ=y and utilizing the fact ρ≤1, we get
(70)HD3,1f≤1552960v1c,x+v2c,xy+v3c,xy2+Ψc,x,σ,(71)≤1552960Gc,x,y,where
(72)Gc,x,y=g1c,x+g2c,xy+g3c,xy2+g4c,x1−y2,with
(73)g1c,x=15c6+4−c24−c21280x3+400c2x3+36c2x4+144c2x2+1728c2x2+432c4x3+252c4x2+96c4x,g2c,x=4−c21−x24−c21440cx+144cx2+1728c3x+360c3,g3c,x=4−c21−x24−c2144x2+2160+1728c2x¯,g4c,x=4−c21−x21728c2+2304x4−c2.
Clearly, in the last four functions,g2c,x and g3c,x are nonnegative in the interval 0,2×0,1. So from (70) along with y=σ in the interval 0,1, we get
(74)Gc,x,y=Gc,x,1.
Therefore,(75)Gc,x,1=g1c,x+g2c,x+g3c,x+g4c,x=Fc,x.
Here, we shall maximizeFc,x over the interval 0,2×0,1. For this purpose, we consider possible cases:
(i)
By takingx=0, we have(76)Fc,0=15c6−360c5+2160c4+144c3−17280c2+34560=f1c.
Sincef1′c<0 in 0,2, so, f1c is decreasing over 0,2. Thus, f1c has its maxima at c=0 which is equal to 34560
(ii)
By takingx=1, we have(77)Fc,1=−185c6−1968c4+5952c2+20480=f2c.
Asf2′c=0 has its optimum point at c0=0.674788. Therefore, f2c is an increasing function for c≤c0 and decreasing for c0≤c. Hence, f2c has its maxima at c=c0 that is approximately equal to 22764.68167
(iii)
By takingc=0, we have(78)F0,x=−2304x4+20480x3−32256x2+34560=f3x.
Asf3′x<0 over 0,1, so, f3x is decreasing over 0,1. Thus, f3x has its maxima at x=0 which is equal to 34560. Now, by taking c=2, we obtain
(79)F2,x≤960(iv)
Whenc,x lies in 0,2×0,1. Then, some simple computation shows that there exists real solution for these equations(80)∂Fc,x∂x=0,∂Fc,x∂c=0,lies inside this region 0,2×0,1 at c,x≈0,0. Consequently, we obtain
(81)Fc,x=34560.
Thus, from all the above cases, we conclude that(82)Gc,x,y≤34560on0,2×0,1×0,1.
From (71), we can write
(83)HD3,1f≤1552960Gc,x,y≤116≈0.0625.
Iff∈BTs, then the equality is obtained from the function
(84)f3z=∫0z1+sinh−1t3dt=z+14z4−160z10+⋯.


## 5. Conclusion

For the family of bounded turning functions connected with a petal-shaped domain, we studied the problems such as the bounds of the first three coefficients, the estimate of the Fekete-Szegö inequality, and the bounds of Hankel determinants of order three. All the bounds which we investigated are sharp.



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