# Fourth Toeplitz Determinants for Starlike Functions Defined by Using the Sine Function

Authors: Hai-Yan Zhang; Huo Tang
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, we aim to study the upper bounds of the fourth Toeplitz determinantT42 for the function class Ss∗, which are connected with the sine function.


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## Body

## 1. Introduction

Suppose thatA represents the class of analytic functions f which in the open unit disk D=z:∣z∣<1 of the form
(1)fz=z+a2z2+a3z3+⋯z∈D,and suppose that S is the subclass of A consisting of univalent functions.LetP denotes the class of analytic functions p normalized by
(2)pz=1+c1z+c2z2+c3z3+⋯,and meeting the condition Rpz>0z∈D. Let f and g be analytic functions in D. Then, we say that the function g is subordinate to the function f, and we write
(3)gz≺fzz∈D,if there exists a Schwarz function ωz with ω0=0 and ∣ωz∣<1, such that (see [1])
(4)gz=fωzz∈D.In 2018, Cho et al. [2] introduced the following function class Ss∗:
(5)Ss∗≔f∈A:zf′zfz≺1+sinzz∈D,which means that the quantity zf′z/fz lies in an eight-shaped region in the right-half plane.Thomas and Halim [3] defined the symmetric Toeplitz determinant Tqn as follows:
(6)Tqn=anan+1⋯an+q−1an+1an⋯an+q−1⋮⋮⋮an+q−1an+q+2⋯ann≥1,q≥1.As a special case, we have(7)T42=a2a3a4a5a3a2a3a4a4a3a2a3a5a4a3a2n=2,q=4.That is,(8)T42=a22−a322+2a32−a2a4a2a4−a3a5−a2a3−a3a42+a42−a3a52−a3a4−a2a52.Many and many researchers have studied several Hankel and Toeplitz determinants for various classes of functions. For example, Janteng et al. [4, 5] investigated second Hankel determinant for a function with a positive real part derivative and starlike and convex functions, respectively; Bansal [6] and Lee et al. [7] discussed the second Hankel determinant for certain analytic functions; Bansal et al. [8], Zaprawa [9], Zhang et al. [10] and Babalola [11] derived third-order Hankel determinant for certain different univalent functions; Raza et al. [12] and Shi et al. [13, 14] studied upper bounds of the third Hankel determinant for some classes of analytic functions related to lemniscate of Bernoulli, cardioid domain and exponential function; Mahmood et al. [15] found third Hankel determinant for a subclass of q-starlike functions. Following the above work, Zhang et al. [16] recently considered fourth-order Hankel determinants of starlike functions related to the sine function. On the other hand, Thomas et al. [3] and Ali et al. [17] studied Toeplitz matrices whose elements are the coefficients of starlike, close-to-convex, and univalent functions. Besides, Tang et al. [18] studied third-order Hankel and Toeplitz determinant for a subclass of multivalent q-starlike functions of order α; Zhang et al. [19] considered third-order Hankel and Toeplitz determinants of starlike functions, which are defined by using the sine function; Ramachandran et al. [20] derived an estimation for the Hankel and Topelitz determinant with domains bounded by conical sections involving Ruscheweygh derivative; Srivastava et al. [21] found the Hankel determinant and the Toeplitz matrices for this newly-defined class of analytic q-starlike functions. Based on the work of Shi et al. [14], Zhang and Tang [16], Thomas and Halim [3], and Ali et al. [17], in the present paper, we aim to investigate the fourth-order Toeplitz determinant T42 for this function class Ss∗ associated with sine function and obtain the upper bounds for the determinants T42.


## 2. Main Results

Due to prove our desired results, we require the following lemmas.Lemma 1 (see [22]).
Ifpz∈P, then exists some x,z with ∣x∣≤1,∣z∣≤1, such that
(9)2c2=c12+x4−c12,(10)4c3=c13+2c1x4−c12−4−c12c1x2+24−c121−x2z.Lemma 2 (see [23]).
Letpz∈P, then
(11)∣c14+c22+2c1c3−3c12c2−c4∣≤2;(12)∣c15+3c1c22+3c12c3−4c13c2−2c1c4−2c2c3+c5∣≤2;(13)∣c16+6c12c22+4c13c3+2c1c5+2c2c4+c32−c23−5c14c2−3c12c4−6c1c2c3−c6∣≤2;(14)∣cn∣≤2,n=1,2,⋯.Lemma 3 (see [24]).
Letpz∈P, then, we have
(15)∣c2−c122∣≤2−c122;(16)∣cn+k−μcnck∣<2,0≤μ≤1;(17)∣cn+2k−μcnck2∣≤21+2μ.The following are the main conclusions of this paper and related proof.Theorem 1.
Suppose thatfz∈Ss∗ and of the form (1), then
(18)∣a2∣≤1,∣a3∣≤12,∣a4∣≤0.344,∣a5∣≤38,∣a6∣≤67120,∣a7∣≤558710800.Proof.
Becausefz∈Ss∗, by the definition of subordination, so there exists a Schwarz function ωz with ω0=0 and ∣ωz∣<1, such that
(19)zf′zfz=1+sinωz.☐Now(20)zf′zfz=z+∑n=2∞nanznz+∑n=2∞anzn=1+∑n=2∞nanzn−11−a2z+a22−a3z2−a23−2a2a3+a4z3+a24−3a22a3+2a2a4−a5z4+⋯=1+a2z+2a3−a22z2+a23−3a2a3+3a4z3+4a5−a24+4a22a3−4a2a4−2a32z4+5a6−5a2a5+a25−5a3a4−5a23a3+5a22a4+5a2a32z5+6a7−6a2a6+6a22a5−6a3a5+12a2a3a4−a26−6a23a4−3a42+2a33−9a22a32+6a24a3z6+⋯.Define a function(21)pz=1+ωz1−ωz=1+c1z+c2z2+⋯.Apparently so,pz∈P and
(22)ωz=pz−11+pz=c1z+c2z2+c3z3+⋯2+c1z+c2z2+c3z3+⋯.On the other hand,(23)1+sinωz=1+12c1z+c22−c124z2+5c1348+c3−c1c22z3+c4−c1c32+5c12c216−c224−c1432z4+c5−c1c4−c2c32+5c12c3+c1c2216−c13c28+c153840z5+c6−c1c5−c2c42+5c1c2c38+5c2348−c324+5c16512+c14c2768−3c12c2216+5c12c416−c13c38z6+⋯.Comparing the coefficients ofz,z2,z3,z4,z5,z6 between the equations (20) and (23), we obtain
(24)a2=c12,a3=c24,a4=c36−c1c224−c13144,a5=c48−c1c324+5c141152−c12c2192−c2232,(25)a6=−3c1c480−7c2c3120−11c154800−43c1c22960+71c13c25760+c510,(26)a7=c12c4480+c1c2c3480+833c16691200−41c12c223840−109c14c211520−c1c530−5c2c496+5c231152+c612+c13c3144.By virtue of Lemma2, we can obtain
(27)∣a2∣≤1,∣a3∣≤12,(28)∣a4∣=c36−c1c224−c13144=16c3−c1c23+c172c2−c122.Letc1=c,c∈0,2 and using Lemma 3, we get
(29)∣a4∣=16c3−c1c23+c172c2−c122≤13+c2−c2/272,setting
(30)Fc=13+c2−c2/272,It can be easily verified thatFc takes its maximum value at c=23/3, that is
(31)∣a4∣≤F233=13+3162≈0.344,(32)∣a5∣=c48−c1c324+5c141152−c12c2192−c2232=18c4−c1c33−c12576c2−c122−c232c2−c122−7c12c2576.Letc1=c,c∈0,2 from Lemma 3, we obtain
(33)∣a5∣≤14+5c22−c2/2576+1162−c22+7c2288,taking
(34)Fc=14+5c22−c2/2576+1162−c22+7c2288,It can be easily verified that maximum ofFc occurs at c=0, that is,
(35)∣a5∣≤F0=38,(36)∣a6∣=−3c1c480−7c2c3120−11c154800−43c1c22960+71c13c25760+c510=124c5−9c1c410+7120c5−c2c3+11c132400c2−c122−43c1c2960c2−c122−211c13c214400.Assumec1=c,c∈0,2, by Lemma 3, we get
(37)∣a6∣≤760+112+11c32−c2/22400+432402−c22+211c37200,putting
(38)Fc=760+112+11c32−c2/22400+432402−c22+211c37200,it is demonstrable that maximum of Fc occurs at c=0, that is,
(39)∣a6∣≤F0=67120.(40)∣a7∣=c12c4480+c1c2c3480+833c16691200−41c12c223840−109c14c211520−c1c530−5c2c496+5c231152+c612+c13c3144=−37c16691200−25c12c225760−c1c530+c12c4−c22480+c1c2c3−c1c2480+c13c3−c1c2144−29c14c2−c12/211520+5c22c2−c12/21152+c6−5/8c2c412.Letc1=c,c∈0,2 and applying Lemma 3, we get
(41)∣a7∣≤16+c2240+9c120+29c42−c2/211520+37c6691200+c372+25c21440+52−c2/2288,showing
(42)Fc=16+c2240+9c120+29c42−c2/211520+37c6691200+c372+25c21440+52−c2/2288,further, we get
(43)F′c≥0.So, the functionFc takes its maximum value at c=2, that is,
(44)∣a7∣≤F2=558710800.Theorem 2.
Suppose thatfz∈Ss∗ and of the form (1), then, we get
(45)∣a32−a22∣≤54.Proof.
According to equation (26), we have
(46)∣a32−a22∣=c2216−c124.
By applying Lemma1, we get
(47)∣a32−a22∣=c1464+x24−c12264+c12x4−c1232−c124.
Letx=t,t∈0,1,c1=c,c∈0,2. Then, by the triangle inequality, we obtain
(48)∣a32−a22∣≤c2t4−c232+t24−c2264+c464+c24.
Suppose that(49)Fc,t=c2t4−c232+t24−c2264+c464+c24,then ∀t∈0,1,∀c∈0,2, the upper bound of Fc,t corresponds to t=1,c=2. Hence,
(50)∣a32−a22∣≤F1,2=54.☐Theorem 3.
Suppose thatfz∈Ss∗ and of the form (1), then, we have
(51)∣a2a3−a3a4∣≤2536.Proof.
From (26), we have
(52)∣a2a3−a3a4∣=∣c1c28+c13c2576−c2c324+c1c2296∣=c24c12−c3−c1c2/46+c13144.
If we insertc1=c,c∈0,2 and according to Lemma 3, we get
(53)∣a2a3−a3a4∣≤12c2+13+c3144.
Assume that(54)Fc=12c2+13+c3144.
Therefore, we have∀c∈0,2(55)F′c=14+c296>0,namely, the maximum value of Fc can be obtained at c=2, that is,
(56)∣a2a3−a3a4∣≤F2=2536.☐Theorem 4.
Suppose thatfz∈Ss∗ and of the form (1), then, we get
(57)∣a2a4−a32∣≤14.Proof.
Suppose thatfz∈Ss∗, then, through equation (26), we get
(58)∣a2a4−a32∣=c1c312−c12c248−c14288−c2216.☐Now, according to Lemma1, we obtain
(59)∣a2a4−a32∣=c1c312−c12c248−c14288−c2216=−5c14576−x2c124−c1248−x24−c12264+c14−c121−x2z24.If we insertc1=c,c∈0,2,∣x∣=t,t∈0,1. Then, by the triangle inequality, we get
(60)∣a2a4−a32∣≤t2c24−c248+1−t2c4−c224+t24−c2264+5c4576.Putting(61)Fc,t=t2c24−c248+1−t2c4−c224+t24−c2264+5c4576,then, ∀t∈0,1,∀c∈0,2, the upper bound of Fc,t corresponds to t=1,c=0. Hence,
(62)∣a2a4−a32∣≤F0,1=14.Theorem 5.
Suppose thatfz∈Ss∗ and of the form (1), then, we get
(63)∣a2a5−a3a4∣≤1136.Proof.
Assume thatfz∈Ss∗, then, on the basis of equation (26), we obtain
(64)∣a2a5−a3a4∣=5c152304+c1c416−c1c22192−c12c348−c13c21152−c2c324=−c13c2−c12/21152−c3c2−c12/224+c1c4−c1c324+c15576+c1c4−1/4c2248.☐If we insertc1=c,c∈0,2, from Lemma 3, we obtain
(65)∣a2a5−a3a4∣≤c32−c2/21152+2−c2/212+c8+c5576.Taking(66)Fc=c32−c2/21152+2−c2/212+c8+c5576.Then, easy to show that maximum ofFc occurs at c=2,∀c∈0,2, also which is
(67)∣a2a5−a3a4∣≤F2=1136.Theorem 6.
Suppose thatfz∈Ss∗ and in the form (1), then, we get
(68)∣a3a5−a2a4∣≤916.Proof.
Assume thatfz∈Ss∗, then, according to equation (26), we get
(69)∣a3a5−a2a4∣=c23128+c1c312−c12c248−c14288−5c14c24608−c2c432+c1c2c396+c12c22768=c1c3−c1c2/412+5c12c2c2−c12/22304−c2c4−1/3c1c332+c22c2−c12/2128+7c12c222304−c14288.☐If we insertc1=c,c∈0,2 and in view of Lemma 3, we have
(70)∣a3a5−a2a4∣≤c6+18+5c22−c2/21152+2−c2/232+7c2576+c4288.Taking(71)Fc=c6+18+5c22−c2/21152+2−c2/232+7c2576+c4288.Then,∀c∈0,2, the demonstrable function Fc obtains the maximum value at c=2, that is,
(72)∣a3a5−a2a4∣≤F2=916.Theorem 7.
Suppose thatfz∈Ss∗ and of the form (1), then, we get
(73)∣a5a3−a42∣≤97324.Proof.
Suppose thatfz∈Ss∗, then, by the equation (26), we obtain
(74)a5a3−a42=7c14c213824+c2c432+c1c2c3288−c23128+c13c3432−7c12c222304−c3236−c1620736=c2c4−c1c3/932−c3c3−c1c2/436−c22c2−c12/2128−c12c2c2−c12/2144+c13c3−31/32c1c2432−5c14c26912−c1620736.☐If we insertc1=c,c∈0,2 and by Lemma 3, we obtain
(75)∣a5a3−a42∣≤18+19+2−c2/232+c22−c2/272+c3216+5c43456+c620736.Putting(76)Fc=18+19+2−c2/232+c22−c2/272+c3216+5c43456+c620736.∀c∈0,2,F′c>0, Then, maximum of Fc occurs at c=2, that is
(77)∣a5a3−a42∣≤F2=97324.Theorem 8.
Suppose thatfz∈Ss∗ and of the form (1), then, we get
(78)∣T42∣≤263384.5104976≈2.51.Proof.
Since(79)T42=a22−a322+2a32−a2a4a2a4−a3a5−a2a3−a3a42+a42−a3a52−a3a4−a2a52,then, by applying the triangle inequality, we get
(80)∣T42∣≤a22−a322+2a32−a2a4a2a4−a3a5+a2a3−a3a42+a42−a3a52+a3a4−a2a52.
Now, substituting (18), (45)–(73) into (80), we easily obtain the desired assertion (78).☐


## 3. Conclusion

In this paper, based on the paper [15], we continuously discuss the problem of the fourth-order Toeplitz determinant of starlike functions, which are connected with the sine function and get the upper bounds of the determinant. In the next step, we can consider the fourth-order Toeplitz determinant of other function classes defined by various linear or nonlinear operators and also make the related discussion on the fifth-order Toeplitz determinant for certain function classes.



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