# Fourth-Order Hankel Determinants and Toeplitz Determinants for Convex Functions Connected with Sine Functions

Authors: Farah Zulfiqar; Sarfraz Nawaz Malik; Mohsan Raza; Md. Shajib Ali
Journal: Journal of Mathematics
 (2022)

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## Abstract

This article deals with the upper bound of fourth-order Hankel and Toeplitz determinants for the convex functions which are defined by using the sine function. The main tools in this study are the coefficient inequalities for the classP of functions with positive real parts. Also, the investigation of the upper bound of the fourth-order Hankel determinant for 3-fold symmetric convex functions associated with the sine function is included.


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## Body

## 1. Introduction

Let the family of all functionsf be denoted by A which are analytic in an open unit disc D=z∈ℂ:z<1 with Taylor series expansion:(1)fz=z+∑n=2∞anzn,z∈D,and S represent a family of functions f∈A which are univalent in D. Let S∗, C, and Kg denote the families of starlike, convex, and close-to-convex functions, respectively, and they are defined as(2)S∗=f∈S:ℜzf′zfz>0,z∈D,C=f∈S:ℜ1+zf″zf′z>0,z∈D,Kg=f∈S:ℜzf′zgz>0,forg∈S∗,z∈D.LetP denote the family of all analytic functions p of the form(3)pz=1+∑n=1∞cnzn,z∈D,with the positive real parts in D. As the nth coefficient for the functions belonging to the family is bounded by n, this bound helps in the study of geometric properties of functions f∈S. Specifically, the second coefficient a2 helps in finding the distortion and growth properties of a normalized univalent function. Likewise, the problems involving power series with integral coefficients and investigating the singularities are successfully handled by using Hankel determinants. Pommerenke [1, 2] introduced the idea of Hankel determinants, and he defined those for univalent functions f∈S of form (7) as follows:(4)Hq,nf=anan+1…an+q−1an+1an+2…an+q::…:an+q−1an+q…an+2q−2.In the theory of analytic functions, finding the upper bound ofHq,nf is one of the most studied problems. Several researchers found the above-mentioned bound for different subfamilies of univalent functions for fixed values of q and n. A few remarkable contributions in this regard are included here for reference. For the subfamilies S∗,C, and Kz=R (the class of functions with bounded turnings) of the set S, the sharp bounds of H2,2f were investigated by Janteng et al. [3, 4]. They proved the bounds as follows:(5)H2,2f≤1forf∈S∗,18forf∈C,49forf∈R.The accurate estimate ofH2,2f was obtained by Krishna et al. [5] for the family of Bazilevic functions. For subfamilies of S, more studies regarding H2,2f can be seen in [6–12]. According to Thomas’ conjecture [13], if f∈S, then Hq,2f≤1, but it was shown by Li and Srivastava in [14] that this conjecture is not true for n≥4. Also, Raducanu and Zaprawa [15] showed that it is false for n=2. Rather, they showed that maxH2,2f:f∈S≥1.175. As compared to H2,2f, estimation of H3,1f is much more difficult. Babalola [16] published the first paper on H3,1f in 2010 in which he obtained the upper bound of H3,1f for subfamilies of S∗,C, and R. After that, for different subfamilies of analytic and univalent functions, few other authors [17–25] also published their work regarding H3,1f. Zaprawa [26] improved the results of Babalola [16] recently in 2017, by showing(6)H3,1f≤1forf∈S∗,49540forf∈C,4160forf∈R.He claimed that these bounds are not sharp. Furthermore, he considered the subfamilies ofS∗,C, and R for sharpness, having functions with m-fold symmetry, and obtained the sharp bounds. Arif et al. [27–30] made a remarkable contribution in studying the fourth- and fifth-order Hankel determinants H4,1f and H5,1f for certain subfamilies of univalent functions. Mashwani et al. [31] have studied the fourth-order Hankel determinant for starlike functions related to sigmoid functions, whereas Kaur et al. [32] studied the same problem for a subclass of bounded turning functions. Wang et al. [33] studied the problem for bounded turning functions related to the lemniscate of Bernoulli. Recently, Zhang and Tang [34] have studied the fourth-order Hankel determinant for the class of starlike functions related to sine functions. Motivated by the above-mentioned work, we intend to add some contributions to the fourth-order Hankel determinant for the class of convex functions associated with sine functions. Recently, the following class Cs of convex functions was introduced, which is associated with the sine function:(7)Cs=f∈A:zf′z′f′z≺1+sinzz∈D,where ≺ is a subordination symbol and it also implies that the region defined by zf′D′/f′D lies in the eight-shaped region in the right-half plane. For different subfamilies of univalent functions, growth of Hq,nf has been studied for fixed values of q and n. Particularly, we have(8)H4,1f=a1a2a3a4a2a3a4a5a3a4a5a6a4a5a6a7n=1,q=4.Also, Thomas and Halim defined the symmetric Toeplitz determinantTqn as follows:(9)Tq,nf=anan+1…an+q−1an+1an…an+q::…:an+q−1an+q…ann≥1,q≥1.The Toeplitz determinants are closely related to Hankel determinants. As Hankel matrices consist of constant entries along the reverse diagonal, the Toeplitz matrices consist of constant entries along the diagonal.As a special case, whenn=1 and q=4, we have(10)T4,2f=a1a2a3a4a2a1a4a3a3a4a1a2a4a3a2a1.In this paper, we intend to find the upper bound ofH4,1f and T4,2f for the class of functions defined by (6). The following sharp results would be useful for investigating our main results.Lemma 1.
Ifp∈P and p is of form (2), then for each n,k,m,l∈ℕ=1,2,…, the following sharp inequalities hold:(11)cn≤2,cn+k−μcnck≤2,for0≤μ≤122μ−1, elsewhere,(12)cn+2k−cnck2≤6,(13)cnck−cmcl≤4,forn+k=m+l.Inequalities (10)–(12) are proved in [26, 35, 36], respectively. Inequality (13) is obvious.Libera and Złotkiewicz proved the following result [37].Lemma 2.
Letp∈P be of form (2). Then, the modulus of the expressions(14)A3=c13−2c1c2+c3,A4=c14+c22+2c1c3−3c12c2−c4,A5=c15+3c1c22+3c12c3−4c13c2−2c1c4−2c2c3+c5,A6=c16+6c12c22+4c13c3+2c1c5+2c2c4+c32−c23−5c14c2−3c12c4−6c1c2c3−c6are all bounded by 2.


## 2. Main Results

### 2.1. Bounds ofH4,1f and T4,2f for the Set Cs Connected with the Sine Function

Following (7), we can write H4,1f, where f∈S and a1=1, as(15)H4,1f=a7H3,1f−a6R1+a5R2−a4R3,where(16)H3,1f=a3a5−a42−a2a2a5−a3a4+a3a2a4−a32and R1,R2, and R3 are determinants of order 3, given by(17)R1=a3a6−a4a5−a2a2a6−a3a5+a4a2a4−a32,(18)R2=a4a6−a52−a2a3a6−a4a5+a3a3a5−a42,(19)R3=a2a4a6−a52−a3a3a6−a4a5+a4a3a5−a42.Also,(20)T4,1f=a1C1−a2C2+a3C3−a4C4,where(21)C1=a1a12−a22−a4a1a4−a2a3+a3a4a2−a1a3,(22)C2=a2a12−a22−a4a1a3−a2a4+a3a2a3−a1a4,C3=a2a1a4−a2a3−a3a12−a32+a4a1a2−a3a4,C4=a2a2a4−a1a3−a3a1a2−a3a4+a4a12−a42.As, from (7), H4,1f is a polynomial of six coefficients of function f of the given class, these coefficients are taken as a2,a3,a4,a5,a6, and a7. However, there is a connection between these coefficients and the coefficients of function p in the class P in many problems. Consider that f∈Cs has form (1); then, there is a Schwartz function wz with w0=0 and wz<1, such that(23)zf′z′f′z=1+sinwz.Now,(24)zf′z′f′z=1+∑n=2∞n2anzn−11+∑n=2∞nanzn−1=1+2a2z+6a3−4a22z2+−18a2a3+12a4+8a23z3+−32a2a4+20a5−18a32+48a3a22−16a24z4+−50a2a5+30a6−60a4a3+80a4a22+90a2a32−120a3a23+32a25z5+…Consider(25)pz=1+wz1−wz=1+c1z+c2z2+c3z3+⋯.Since we havep∈P,(26)wz=pz−11+pz=c1z+c2z2+c3z3+⋯2+c1z+c2z2+c3z3+⋯.Also,(27)1+sinwz=1+12c1z+12c2−14c12z2+12c3−12c1c2+548c13z3+516c2c12−132c14+12c4−12c1c3−14c22z4+13840c15+516c1c22−18c2c13−12c2c3+516c3c12−12c1c4+12c5z5+⋯.On comparing coefficients between (25) and (28), we get(28)a2=14c1,a3=112c2,a4=−196c1c2−1576c13+124c3,a5=−1960c2c12+11152c14−1120c1c3−1160c22+140c4,a6=−1128800c15+7134560c2c13+11152c22−1160c4c1−7720c2c3+160c5,and(29)a7=239914515200c16−347241920c2c14+1756c3c13+13360c4+2916128c22c12+−1210c5+2310080c2c3c1+184c6−1252c32+58064c23−5672c4c2.By using these coefficients, we can write (16)–(19) in the following way:(30)H3,1f=−19331776c16+134560c2c14+2334560c3c13+−1146080c22−1640c4c12+115760c3c1c2+112c2140c4−1160c22−11728c23−1576c32,(31)R1=28911059200c17−29230400c2c15−111520c3c14+1871658880c22+12304c4c13+15760c2c3−1960c5c12+112c211152c22−1160c4+11280c32+132c2140c4−1160c22+113824c23c1+112c2−7720c2c3+160c5−13456c3c22−124c3140c4−1160c22,+112c2−7720c2c3+160c5−13456c3c22−124c3140c4−1160c22,(32)R2=−3166355200c18+10112441600c2c16+312764800c3c15+−1974147200c22−123040c4c14+764800c2c3−134560c5c13+−16400c32+738400c2c4−37921600c23c12+15400c22c3−11920c2c5+12400c3c4c1+11440c3c5+714400c22c4−11600c42−19230400c24−1934560c2c23,and(33)R3=−834777574400c19+317796262400c2c17−4966355200c3c16+−1122880c4−169132710400c22c15−1138240c5+2098294400c2c3c14+612764800c2c4−232764800c32−2993199065600c23c13+−123040c2c5+115529600c22c3+125600c3c4c12+112800c22c4−418294400c24+15760c3c5−16400c42−29276480c2c32c1−18640c22c5+141472c23c3−113824c33+15760c2c3c4.Similarly, in case of Toeplitz determinants,(34)C1=−1331776c16−19216c14c2+16912c13c3+−116−59216c22c12+1384c1c2c3+1−1576c32−1144c22,(35)C2=11327104c17+1110592c15c2−127648c14c3+136864c22−164+13456c2c13−14608c12c2c3+14+1288c22+12304c32c1−1144c3c2,(36)C3=−13981312c16c2+−11152−1331776c22c14+182944c13c3c2+−1110592c23−196c2c12+148c3+113824c22c3c1+11728c23−16912c32c2−112c2,(37)C4=1191102976c19+110616832c17c2−12654208c16c3+11769472c22−19216c15−1221184c14c2c3+1884736c23−11536c2−182944c22+1110592c32−1576c13+1384c3−173728c22c3c12+−1596c2+118432c32c2−113824c23c1+124c3−113824c33+13456c22c3.By using the previous computations, we prove the following.Theorem 1.
If the functionf∈Cs and is of form (1), then(38)H4,1f≤11226715959715049359360000≈0.0074599⋯.Proof.
Asf∈Cs, then by using (30)–(33) in (15), we get(39)H4,1f=38723481579499520000c112+232322432c34−11596967296000c26−41696967296000c17c5−1927869184c6c16−191451520c6c23−148384c6c32−176967296c32c23+2539580608000c24c4+11792000c22c42+157600c12c52−143200c2c52−164000c43+1619580608000c13c3c2c4−167200c12c5c2c3−153760c1c5c2c4−71348384000c22c1c3c4+11483840c6c2c1c3+24758060800c2c1c33+589145152000c15c5c2+5336864000c2c14c32+232903040c6c13c3+56332256000c12c42c2+164000c1c42c3+291036800c22c3c5+128800c5c3c4−913348364800c13c5c22−432419200c1c5c32−30191032192000c22c15c3−113870720c6c12c22+679676800c1c5c23+12903040c6c2c14+15196768000c12c4c32+251139345920c16c2c4+1073580608000c12c4c23−1107755738368000c17c2c3+140320c6c2c4−13628800c14c5c3−19191741824000c24c1c3−2920736000c15c3c4−153760c6c12c4−307034644864000c22c14c4+19531341803776000c13c3c23+46992322432000c12c22c32−172419200c13c5c4−137800c32c2c4−13629120065812480000c110c2+142011003290624000c19c3+671391337720832000c18c22−777962624000c18c4−400731003290624000c16c23+84199289728000c16c32+852083668860416000c24c14−59471741824000c13c33+12867200c14c42−391217728000c12c25.(40)H4,1f=148157949952000038723c112−3270984c110c2+6816480c19c3+24170040c22−4656960c4c18+−288161280c5−95705280c2c3c17+−328320000c6+436440960c22−19235040c23+867456000c4c2c16+−673505280c3c4−1408544640c3c22+1954160640c2c5c15+613499760c24+165888000c6c2−3183287040c22c4+167961600c42−132710400c3c5+692375040c2c32c14+−1644226560c33+1342863360c2c3c4+2250005760c23c3+3815424000c6c3−3384115200c5c4−1262131200c22c5c13+8360755200c52−8957952000c6c4−7166361600c5c2c3−1368576000c6c22+8405544960c42c2+751472640c4c32−864829440c25+974384640c22c32+889989120c4c23c12+3334348800c5c23+7524679680c42c3+10948608000c6c2c3+2048716800c2c33−7096688640c22c3c4−8957952000c5c2c4−530565120c24c3−8559820800c5c32c1−1175040000c32c23+2105948160c24c4−12740198400c32c2c4+11943936000c6c2c4+16721510400c5c3c4−7524679680c43+4769280000c34−80110080c26−9953280000c6c32−11147673600c2c52+13470105600c22c3c5−6303744000c6c23+268738560c22c42.
After rearranging the terms, we get(41)H4,1f=1481579499520000−38723A6c2−c123−2961200c2A52+6661588c3A3A6−448912A6c3−c1c22−45407915c4A3A5−288238726c1A6c5−1288238726c2c3−328281275A6c6−403543209328281275c32−A6c6−827940501c2c4−A6c6−45022105c23+519121748c2A5c5−1592920565519121748c2c3+635357352c3A4c5−c1c4−1475634554c2A4c6−28575923661475634554c32−371563976c5A3c4−c1c3+158880018c42A3c1+769228159c24A3c1+1052977475c22c4A3c1+5135210696c2−c12c4−c1c3c6−32899116885135210696c32−3872726611c2−c12c4−25207640343872726611c22c6−c1c5−4015593821A3c2c4c3−4121711992A3c3c23−8680311746c4c12c6−112371344638680311746c2c4−24834316c32c12c4−1175763436024834316c22−8937232652c5c2−c12c5−77613412908937232652c2c3−1898862450c23c12c4−4300446431898862450c22−1305360709c22c12c6+17052974791c3c2c1c6−1368336869017052974791c2c4−13635892999c5c2c1c4−393730930013635892999c22+−4485418959c2c33−4542515027c24c3+6721360728c42c3c1+368323828c5c1c6−10733372524368323828c32−2726601496c2c5c5−76927262192726601496c2c3+10690320416c2c4c6−423363437810690320416c32+2843921897c24c4−351266982843921897c22−9215242838c32c6−43595241159215242838c32−7632171375c6c23−2720144201c32c23+17732972519c5c3c4−328281277c62−7524679680c43−31507775c2c17c3−1387142442c22c42.
After using triangular inequalities and lemmas, we get the following expression:(42)H4,1f=148157949952000038723×2×23+42961200×2+4×6661588×2+448912×2×22+2×45407915×2×2+2288238726×2×2+957610286328281275×328281275×2+2×3311762002+2×3311762002+2×519121748×2×1333359691129780437+2×635357352×2×2+2×1475634554×2×4239550178737817277+371563976×2×2×2+158880018×4×2×2+769228159×16×2×2+1052977475×4×2×2×2+5135210696×2×2×5159331091700191+872726611×2×2×2+4015593821×16+4121711992×32+8680311746×8×137939571804340155873+24834316×16×117452172026208579+8937232652×2×2×2+1898862450×32×2+1305360709×32+17052974791×8×2++13635892999×8×2+4485418959×16+4542515027×32+6721360728×8×2+368323828×4×1054921061092080957+2726601496×4×6329425471681650374+10690320416×4×2+2843921897×16×2+9215242838×4×2+7632171375×16+2720144201×32+17732972519×8+328281277×4+7524679680×8+31507775×512+1387142442×16.
Hence,(43)H4,1f≤3592549107104481579499520000≈0.0074599,which completes the proof.Theorem 2.
If the functionf∈Cs and is of form (1), then(44)T4,1f≤501434459286654464≈1.7493.Proof.
Asf∈Cs, using (34)–(37) in (20), we get(45)T4,1f=1+1256c14−1288c32−1165888c16−18c12−172c22+5576c1c2c3+19216c13c3c2+13456c13c3−1221184c16c2+1497664c13c3c22−15308416c13c23c3+182944c1c23c3−1331776c1c2c33+1884736c12c22c32−110616832c15c22c3−163700992c17c2c3+12654208c14c2c32−134608c12c22+155296c15c3−173728c14c22−14608c12c32−123887872c16c22−11990656c14c23−1663552c12c24−141472c22c32+184934656c14c24+1127401984c16c23+1509607936c18c22+14586471424c110c2−11146617856c19c3+131850496c16c32−11990656c13c33−12654208c18+120736c24+1110075314176c112.
Rearranging the terms, we may write(46)T4,1f=1110075314176c112+24c110c2−96c19c3+−41472+216c22c18−1728c17c2c3+−663552+3456c32−497664c2+864c23−4608c22c16+−10368c22c3+1990656c3c15+−1492992c22+41472c32c2−55296c23−39813120c2+1296c24+429981696c14+11943936c2c3+31850496c3−55296c33−20736c23c3+221184c3c22c13+−310542336c22−13759414272+124416c22c32−165888c24−23887872c32c12+1327104c23c3+955514880c2c3−331776c2c33c1−1528823808c22−382205952c32+331776c34+5308416c24+110075314176−2654208c22c32.
After rearranging the terms, we get(47)T4,1f=1110075314176−A6c2−c123−100A3A6c3−32100c1c2−2c1A6c5−32c1c4−2570c1c2A6c3−4162570c1c2−41472A3A5+2156544A5c3−7464962156544c1c2−22704c22A5c3−246722704c1c2−312c4A5c3−100312c1c2+29c2A4c6−6829c1c5+−663552+c6−4608c22+3955c32A6+22311936c2A3c3−402278422311936c1c2−902c22A3c5−1389902c1c4−12A3c4c5−912c1c4−104c3A3c6−208104c1c5−70616c32A3c3−6947070616c1c2−102891c23A3c3−9058102891c1c2+239616c22A3c3−78336239616c1c2+41472A3c5−8294441472c1c4+12498c3c4c12c3−834412498c1c2+5c6c2−4313088c2+429981696A4+34504704c3A3−633c42c1c3−212633c1c2+4313088c3c1c4−302330884313088c1c3−188c5c2c1c4−5216188c1c3+1857024c23c1c3−2949121857024c1c2−144138c24c1c3−11076144138c1c2−859963392c1.(48)c3−1289945088859963392c1c2−4c6c1c5−64c1c4−517010c2c32c1c3−293955517010c1c2+746496c2c1c5−1492992746496c1c4+9216c22c1c5−138249216c1c4−2608c6c2c1c3−4512608c1c2+40974336c22c1c3−605491240974336c1c2−40985c22c4c1c3−670240985c1c2+1106804736c2c1c3−4359536641106804736c1c2+4c52−1990656c4−13759414272c12+23606c22c5c3−420323606c1c2+3958c32c6−79163958c1c5−41803776c2c4−4246732841803776c22+429981696c4−1958805504429981696c22+9216c23c4−46089216c22+32c2c6c4−3432c22+2c24c4−12c22−8334c32c2c4−613398334c22−663552c6−1327104663552c1c5−4608c6c22+398537c34+5308416c24+110075314176−416047104c32+324c4c5c3+c62−18081792c2c32−2889216c22c32−2156544c5c3.
After using the triangular inequality and above-stated lemmas, we get(49)T4,1f=11100753141762×23+100×2×2×2+2×2×2×4+2570×2×2×2×2+41472×2×2+2156544×2×2+22704×4×2×2+312×2×2×2+29×2×2×21429+663552+2+4608×4+3955×4×2+22311936×2×2×2+902×4×2×1876451+12×2×22+104×2×2×6+70616×4×2×2+102891×8×2×2+2×239616×4×2+6×41472×2+2×2×12498×2×4+5×4+43130880×2+429981696×2+34504704×2×2+2×633×4×2+2×4313088×2×67726+512247×2×188×2×2+2×1857024×8×2+2×144138×16×2+4×859963392×2+2×4×2×4+2×4×517010×2×2+2×746496×2×6+2×9216×4×4+2×2×2608×2×2+2×40974336×4×2+2×2×40985×4×2+2×1106804736×2×2+4×4+1990656×2+13759414272×4+23606×4×2×2+3958×4×6+41803776×2×13063+429981696×1469+9216×8×2+32×2×2×94+2×16×2+8334×4×2×1143444167+663552×6+4608×2×4+398537×16+5308416×16+110075314176+416047104×4+324×2×2×2+4+18081792×2×4+2889216×4×4+2156544×2×2.
This reduces to(50)T4,1f≤192550832256110075314176=501434459286654464≈1.7493⋯,which completes the proof.


### 2.2. Bounds ofH4,1 for the Set Cs3

Letn∈ℕ=1,2,3,…. Rotation of a domain D about the origin through an angle of 2π/n containing D onto itself is said to be n-fold symmetric. An analytic function f is n-fold symmetric in D if(51)fe2πi/nz=e2πi/nfzholds for any z∈D. Denote Sn as the set of n-fold univalent functions which have the following Taylor series form:(52)fz=z+∑k=1∞ank+1znk+1z∈D.DenoteCn as the subfamily of Sn of n-fold symmetric convex functions. We can see that an analytic function f of form (52) belongs to the family Cn, if and only if(53)zf′z′f′z=pz,where p∈Pn. The family Pn is defined as(54)Pn=p∈P:pz=1+∑k=1∞cnkznk,z∈D.Now, consider the following.Theorem 3.
Letf∈Cs3 be of form (52). Then,(55)H4,1f≤16048≈0.00016534.Proof.
Letf∈Cs3 of form (52). Consider the function p∈P3 as(56)pz=1+wz1−wz=1+c3z3+c6z6+c9z9+⋯.
Now,(57)wz=pz−1pz+1.
The classCs3 which is associated with the sine functions can be written in the following form:(58)zf′zf′z=1+sinwz,z∈D.
By expanding and equating them, we get the following expression:(59)1+12a4z3+42a7−48a42z6+⋯=1+12c3z3+12c6−14c32z6+⋯.
This implies(60)a4=124c3,a7=−1252c32+184c6.
By using these coefficients, we can getH3,1f,R1,R2, and R3 as(61)H3,1=−1576c32,R1=0,R2=0,R3=−113824c33.
By using these values inH4,1f, we get(62)H4,1f=a7H3,1f−a6R1+a5R2−a4R3=−1252c32+184c6−1576c32−124c3−113824c33=232322432c34−148384c32c6=−148384c32c6−2348c32.
The triangle inequality and the application of Lemma1 lead us to(63)H4,1f=16048≈0.00016534,which completes the proof.


## 2.1. Bounds ofH4,1f and T4,2f for the Set Cs Connected with the Sine Function

Following (7), we can write H4,1f, where f∈S and a1=1, as(15)H4,1f=a7H3,1f−a6R1+a5R2−a4R3,where(16)H3,1f=a3a5−a42−a2a2a5−a3a4+a3a2a4−a32and R1,R2, and R3 are determinants of order 3, given by(17)R1=a3a6−a4a5−a2a2a6−a3a5+a4a2a4−a32,(18)R2=a4a6−a52−a2a3a6−a4a5+a3a3a5−a42,(19)R3=a2a4a6−a52−a3a3a6−a4a5+a4a3a5−a42.Also,(20)T4,1f=a1C1−a2C2+a3C3−a4C4,where(21)C1=a1a12−a22−a4a1a4−a2a3+a3a4a2−a1a3,(22)C2=a2a12−a22−a4a1a3−a2a4+a3a2a3−a1a4,C3=a2a1a4−a2a3−a3a12−a32+a4a1a2−a3a4,C4=a2a2a4−a1a3−a3a1a2−a3a4+a4a12−a42.As, from (7), H4,1f is a polynomial of six coefficients of function f of the given class, these coefficients are taken as a2,a3,a4,a5,a6, and a7. However, there is a connection between these coefficients and the coefficients of function p in the class P in many problems. Consider that f∈Cs has form (1); then, there is a Schwartz function wz with w0=0 and wz<1, such that(23)zf′z′f′z=1+sinwz.Now,(24)zf′z′f′z=1+∑n=2∞n2anzn−11+∑n=2∞nanzn−1=1+2a2z+6a3−4a22z2+−18a2a3+12a4+8a23z3+−32a2a4+20a5−18a32+48a3a22−16a24z4+−50a2a5+30a6−60a4a3+80a4a22+90a2a32−120a3a23+32a25z5+…Consider(25)pz=1+wz1−wz=1+c1z+c2z2+c3z3+⋯.Since we havep∈P,(26)wz=pz−11+pz=c1z+c2z2+c3z3+⋯2+c1z+c2z2+c3z3+⋯.Also,(27)1+sinwz=1+12c1z+12c2−14c12z2+12c3−12c1c2+548c13z3+516c2c12−132c14+12c4−12c1c3−14c22z4+13840c15+516c1c22−18c2c13−12c2c3+516c3c12−12c1c4+12c5z5+⋯.On comparing coefficients between (25) and (28), we get(28)a2=14c1,a3=112c2,a4=−196c1c2−1576c13+124c3,a5=−1960c2c12+11152c14−1120c1c3−1160c22+140c4,a6=−1128800c15+7134560c2c13+11152c22−1160c4c1−7720c2c3+160c5,and(29)a7=239914515200c16−347241920c2c14+1756c3c13+13360c4+2916128c22c12+−1210c5+2310080c2c3c1+184c6−1252c32+58064c23−5672c4c2.By using these coefficients, we can write (16)–(19) in the following way:(30)H3,1f=−19331776c16+134560c2c14+2334560c3c13+−1146080c22−1640c4c12+115760c3c1c2+112c2140c4−1160c22−11728c23−1576c32,(31)R1=28911059200c17−29230400c2c15−111520c3c14+1871658880c22+12304c4c13+15760c2c3−1960c5c12+112c211152c22−1160c4+11280c32+132c2140c4−1160c22+113824c23c1+112c2−7720c2c3+160c5−13456c3c22−124c3140c4−1160c22,+112c2−7720c2c3+160c5−13456c3c22−124c3140c4−1160c22,(32)R2=−3166355200c18+10112441600c2c16+312764800c3c15+−1974147200c22−123040c4c14+764800c2c3−134560c5c13+−16400c32+738400c2c4−37921600c23c12+15400c22c3−11920c2c5+12400c3c4c1+11440c3c5+714400c22c4−11600c42−19230400c24−1934560c2c23,and(33)R3=−834777574400c19+317796262400c2c17−4966355200c3c16+−1122880c4−169132710400c22c15−1138240c5+2098294400c2c3c14+612764800c2c4−232764800c32−2993199065600c23c13+−123040c2c5+115529600c22c3+125600c3c4c12+112800c22c4−418294400c24+15760c3c5−16400c42−29276480c2c32c1−18640c22c5+141472c23c3−113824c33+15760c2c3c4.Similarly, in case of Toeplitz determinants,(34)C1=−1331776c16−19216c14c2+16912c13c3+−116−59216c22c12+1384c1c2c3+1−1576c32−1144c22,(35)C2=11327104c17+1110592c15c2−127648c14c3+136864c22−164+13456c2c13−14608c12c2c3+14+1288c22+12304c32c1−1144c3c2,(36)C3=−13981312c16c2+−11152−1331776c22c14+182944c13c3c2+−1110592c23−196c2c12+148c3+113824c22c3c1+11728c23−16912c32c2−112c2,(37)C4=1191102976c19+110616832c17c2−12654208c16c3+11769472c22−19216c15−1221184c14c2c3+1884736c23−11536c2−182944c22+1110592c32−1576c13+1384c3−173728c22c3c12+−1596c2+118432c32c2−113824c23c1+124c3−113824c33+13456c22c3.By using the previous computations, we prove the following.Theorem 1.
If the functionf∈Cs and is of form (1), then(38)H4,1f≤11226715959715049359360000≈0.0074599⋯.Proof.
Asf∈Cs, then by using (30)–(33) in (15), we get(39)H4,1f=38723481579499520000c112+232322432c34−11596967296000c26−41696967296000c17c5−1927869184c6c16−191451520c6c23−148384c6c32−176967296c32c23+2539580608000c24c4+11792000c22c42+157600c12c52−143200c2c52−164000c43+1619580608000c13c3c2c4−167200c12c5c2c3−153760c1c5c2c4−71348384000c22c1c3c4+11483840c6c2c1c3+24758060800c2c1c33+589145152000c15c5c2+5336864000c2c14c32+232903040c6c13c3+56332256000c12c42c2+164000c1c42c3+291036800c22c3c5+128800c5c3c4−913348364800c13c5c22−432419200c1c5c32−30191032192000c22c15c3−113870720c6c12c22+679676800c1c5c23+12903040c6c2c14+15196768000c12c4c32+251139345920c16c2c4+1073580608000c12c4c23−1107755738368000c17c2c3+140320c6c2c4−13628800c14c5c3−19191741824000c24c1c3−2920736000c15c3c4−153760c6c12c4−307034644864000c22c14c4+19531341803776000c13c3c23+46992322432000c12c22c32−172419200c13c5c4−137800c32c2c4−13629120065812480000c110c2+142011003290624000c19c3+671391337720832000c18c22−777962624000c18c4−400731003290624000c16c23+84199289728000c16c32+852083668860416000c24c14−59471741824000c13c33+12867200c14c42−391217728000c12c25.(40)H4,1f=148157949952000038723c112−3270984c110c2+6816480c19c3+24170040c22−4656960c4c18+−288161280c5−95705280c2c3c17+−328320000c6+436440960c22−19235040c23+867456000c4c2c16+−673505280c3c4−1408544640c3c22+1954160640c2c5c15+613499760c24+165888000c6c2−3183287040c22c4+167961600c42−132710400c3c5+692375040c2c32c14+−1644226560c33+1342863360c2c3c4+2250005760c23c3+3815424000c6c3−3384115200c5c4−1262131200c22c5c13+8360755200c52−8957952000c6c4−7166361600c5c2c3−1368576000c6c22+8405544960c42c2+751472640c4c32−864829440c25+974384640c22c32+889989120c4c23c12+3334348800c5c23+7524679680c42c3+10948608000c6c2c3+2048716800c2c33−7096688640c22c3c4−8957952000c5c2c4−530565120c24c3−8559820800c5c32c1−1175040000c32c23+2105948160c24c4−12740198400c32c2c4+11943936000c6c2c4+16721510400c5c3c4−7524679680c43+4769280000c34−80110080c26−9953280000c6c32−11147673600c2c52+13470105600c22c3c5−6303744000c6c23+268738560c22c42.
After rearranging the terms, we get(41)H4,1f=1481579499520000−38723A6c2−c123−2961200c2A52+6661588c3A3A6−448912A6c3−c1c22−45407915c4A3A5−288238726c1A6c5−1288238726c2c3−328281275A6c6−403543209328281275c32−A6c6−827940501c2c4−A6c6−45022105c23+519121748c2A5c5−1592920565519121748c2c3+635357352c3A4c5−c1c4−1475634554c2A4c6−28575923661475634554c32−371563976c5A3c4−c1c3+158880018c42A3c1+769228159c24A3c1+1052977475c22c4A3c1+5135210696c2−c12c4−c1c3c6−32899116885135210696c32−3872726611c2−c12c4−25207640343872726611c22c6−c1c5−4015593821A3c2c4c3−4121711992A3c3c23−8680311746c4c12c6−112371344638680311746c2c4−24834316c32c12c4−1175763436024834316c22−8937232652c5c2−c12c5−77613412908937232652c2c3−1898862450c23c12c4−4300446431898862450c22−1305360709c22c12c6+17052974791c3c2c1c6−1368336869017052974791c2c4−13635892999c5c2c1c4−393730930013635892999c22+−4485418959c2c33−4542515027c24c3+6721360728c42c3c1+368323828c5c1c6−10733372524368323828c32−2726601496c2c5c5−76927262192726601496c2c3+10690320416c2c4c6−423363437810690320416c32+2843921897c24c4−351266982843921897c22−9215242838c32c6−43595241159215242838c32−7632171375c6c23−2720144201c32c23+17732972519c5c3c4−328281277c62−7524679680c43−31507775c2c17c3−1387142442c22c42.
After using triangular inequalities and lemmas, we get the following expression:(42)H4,1f=148157949952000038723×2×23+42961200×2+4×6661588×2+448912×2×22+2×45407915×2×2+2288238726×2×2+957610286328281275×328281275×2+2×3311762002+2×3311762002+2×519121748×2×1333359691129780437+2×635357352×2×2+2×1475634554×2×4239550178737817277+371563976×2×2×2+158880018×4×2×2+769228159×16×2×2+1052977475×4×2×2×2+5135210696×2×2×5159331091700191+872726611×2×2×2+4015593821×16+4121711992×32+8680311746×8×137939571804340155873+24834316×16×117452172026208579+8937232652×2×2×2+1898862450×32×2+1305360709×32+17052974791×8×2++13635892999×8×2+4485418959×16+4542515027×32+6721360728×8×2+368323828×4×1054921061092080957+2726601496×4×6329425471681650374+10690320416×4×2+2843921897×16×2+9215242838×4×2+7632171375×16+2720144201×32+17732972519×8+328281277×4+7524679680×8+31507775×512+1387142442×16.
Hence,(43)H4,1f≤3592549107104481579499520000≈0.0074599,which completes the proof.Theorem 2.
If the functionf∈Cs and is of form (1), then(44)T4,1f≤501434459286654464≈1.7493.Proof.
Asf∈Cs, using (34)–(37) in (20), we get(45)T4,1f=1+1256c14−1288c32−1165888c16−18c12−172c22+5576c1c2c3+19216c13c3c2+13456c13c3−1221184c16c2+1497664c13c3c22−15308416c13c23c3+182944c1c23c3−1331776c1c2c33+1884736c12c22c32−110616832c15c22c3−163700992c17c2c3+12654208c14c2c32−134608c12c22+155296c15c3−173728c14c22−14608c12c32−123887872c16c22−11990656c14c23−1663552c12c24−141472c22c32+184934656c14c24+1127401984c16c23+1509607936c18c22+14586471424c110c2−11146617856c19c3+131850496c16c32−11990656c13c33−12654208c18+120736c24+1110075314176c112.
Rearranging the terms, we may write(46)T4,1f=1110075314176c112+24c110c2−96c19c3+−41472+216c22c18−1728c17c2c3+−663552+3456c32−497664c2+864c23−4608c22c16+−10368c22c3+1990656c3c15+−1492992c22+41472c32c2−55296c23−39813120c2+1296c24+429981696c14+11943936c2c3+31850496c3−55296c33−20736c23c3+221184c3c22c13+−310542336c22−13759414272+124416c22c32−165888c24−23887872c32c12+1327104c23c3+955514880c2c3−331776c2c33c1−1528823808c22−382205952c32+331776c34+5308416c24+110075314176−2654208c22c32.
After rearranging the terms, we get(47)T4,1f=1110075314176−A6c2−c123−100A3A6c3−32100c1c2−2c1A6c5−32c1c4−2570c1c2A6c3−4162570c1c2−41472A3A5+2156544A5c3−7464962156544c1c2−22704c22A5c3−246722704c1c2−312c4A5c3−100312c1c2+29c2A4c6−6829c1c5+−663552+c6−4608c22+3955c32A6+22311936c2A3c3−402278422311936c1c2−902c22A3c5−1389902c1c4−12A3c4c5−912c1c4−104c3A3c6−208104c1c5−70616c32A3c3−6947070616c1c2−102891c23A3c3−9058102891c1c2+239616c22A3c3−78336239616c1c2+41472A3c5−8294441472c1c4+12498c3c4c12c3−834412498c1c2+5c6c2−4313088c2+429981696A4+34504704c3A3−633c42c1c3−212633c1c2+4313088c3c1c4−302330884313088c1c3−188c5c2c1c4−5216188c1c3+1857024c23c1c3−2949121857024c1c2−144138c24c1c3−11076144138c1c2−859963392c1.(48)c3−1289945088859963392c1c2−4c6c1c5−64c1c4−517010c2c32c1c3−293955517010c1c2+746496c2c1c5−1492992746496c1c4+9216c22c1c5−138249216c1c4−2608c6c2c1c3−4512608c1c2+40974336c22c1c3−605491240974336c1c2−40985c22c4c1c3−670240985c1c2+1106804736c2c1c3−4359536641106804736c1c2+4c52−1990656c4−13759414272c12+23606c22c5c3−420323606c1c2+3958c32c6−79163958c1c5−41803776c2c4−4246732841803776c22+429981696c4−1958805504429981696c22+9216c23c4−46089216c22+32c2c6c4−3432c22+2c24c4−12c22−8334c32c2c4−613398334c22−663552c6−1327104663552c1c5−4608c6c22+398537c34+5308416c24+110075314176−416047104c32+324c4c5c3+c62−18081792c2c32−2889216c22c32−2156544c5c3.
After using the triangular inequality and above-stated lemmas, we get(49)T4,1f=11100753141762×23+100×2×2×2+2×2×2×4+2570×2×2×2×2+41472×2×2+2156544×2×2+22704×4×2×2+312×2×2×2+29×2×2×21429+663552+2+4608×4+3955×4×2+22311936×2×2×2+902×4×2×1876451+12×2×22+104×2×2×6+70616×4×2×2+102891×8×2×2+2×239616×4×2+6×41472×2+2×2×12498×2×4+5×4+43130880×2+429981696×2+34504704×2×2+2×633×4×2+2×4313088×2×67726+512247×2×188×2×2+2×1857024×8×2+2×144138×16×2+4×859963392×2+2×4×2×4+2×4×517010×2×2+2×746496×2×6+2×9216×4×4+2×2×2608×2×2+2×40974336×4×2+2×2×40985×4×2+2×1106804736×2×2+4×4+1990656×2+13759414272×4+23606×4×2×2+3958×4×6+41803776×2×13063+429981696×1469+9216×8×2+32×2×2×94+2×16×2+8334×4×2×1143444167+663552×6+4608×2×4+398537×16+5308416×16+110075314176+416047104×4+324×2×2×2+4+18081792×2×4+2889216×4×4+2156544×2×2.
This reduces to(50)T4,1f≤192550832256110075314176=501434459286654464≈1.7493⋯,which completes the proof.


## 2.2. Bounds ofH4,1 for the Set Cs3

Letn∈ℕ=1,2,3,…. Rotation of a domain D about the origin through an angle of 2π/n containing D onto itself is said to be n-fold symmetric. An analytic function f is n-fold symmetric in D if(51)fe2πi/nz=e2πi/nfzholds for any z∈D. Denote Sn as the set of n-fold univalent functions which have the following Taylor series form:(52)fz=z+∑k=1∞ank+1znk+1z∈D.DenoteCn as the subfamily of Sn of n-fold symmetric convex functions. We can see that an analytic function f of form (52) belongs to the family Cn, if and only if(53)zf′z′f′z=pz,where p∈Pn. The family Pn is defined as(54)Pn=p∈P:pz=1+∑k=1∞cnkznk,z∈D.Now, consider the following.Theorem 3.
Letf∈Cs3 be of form (52). Then,(55)H4,1f≤16048≈0.00016534.Proof.
Letf∈Cs3 of form (52). Consider the function p∈P3 as(56)pz=1+wz1−wz=1+c3z3+c6z6+c9z9+⋯.
Now,(57)wz=pz−1pz+1.
The classCs3 which is associated with the sine functions can be written in the following form:(58)zf′zf′z=1+sinwz,z∈D.
By expanding and equating them, we get the following expression:(59)1+12a4z3+42a7−48a42z6+⋯=1+12c3z3+12c6−14c32z6+⋯.
This implies(60)a4=124c3,a7=−1252c32+184c6.
By using these coefficients, we can getH3,1f,R1,R2, and R3 as(61)H3,1=−1576c32,R1=0,R2=0,R3=−113824c33.
By using these values inH4,1f, we get(62)H4,1f=a7H3,1f−a6R1+a5R2−a4R3=−1252c32+184c6−1576c32−124c3−113824c33=232322432c34−148384c32c6=−148384c32c6−2348c32.
The triangle inequality and the application of Lemma1 lead us to(63)H4,1f=16048≈0.00016534,which completes the proof.


## 3. Conclusion

In this paper, we have found the upper bounds of fourth-order Hankel and Toeplitz determinants, followed by a review of such findings obtained so far for certain analytic functions. We have studied them for the convex functions associated with the function1+sinz. A similar bound of the fourth-order Hankel determinant for 3-fold symmetric convex functions associated with 1+sinz has also been investigated.



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*Source: 2871511-2022-05-25.xml*