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Kyungpook Mathematical Journal 2024; 64(4): 591-606

Published online December 31, 2024 https://doi.org/10.5666/KMJ.2024.64.4.591

Copyright © Kyungpook Mathematical Journal.

Investigation of Toeplitz Determinants in Ma-Minda Classes of Starlike and Convex Functions using q-Calculus

Pradeep Kumar and Amit Soni∗

Department of Mathematics, Government Engineering College, Bikaner-334001, Rajasthan, India
e-mail : prdeepbhadu007@gmail.com and aamitt1981@gmail.com

Received: July 22, 2023; Accepted: November 16, 2023

This research paper focuses on obtaining sharp bounds for Toeplitz determinants whose entries are the coefficients of starlike functions satisfying specific conditions. The conditions are related to the quantity zDqf(z)f(z), which is required to lie in a specific subset of the right half-plane. The study presents new special cases and known results in this context. The research approach involves using Toeplitz determinants and their properties, and the results are obtained through a combination of mathematical techniques, including q-calculus.

Keywords: Univalent functions, Subordination, Toeplitz determinants, Fekete-Szegȍ inequality, q-Calculus

The paragraph discusses classes of analytic functions in the open unit disk U denoted by H(U) and A. The functions in A are normalized by f(0)=f'(0)-1=0 and have a specific form as given by equation

f(z)=z+n=2anzn(zU).

Denote subclass S of A contains all univalent functions in U. The concept of subordination of two functions in H(U) is introduced where f is said to be subordinate to g in U, denoted by f(z)g(z), if there exists a Schwarz function ω satisfying certain conditions such that f(z)=g(ω(z)) in U. The paragraph cites a reference [12], for more information on the Schwarz function. Let P be the set of analytic functions p in the open unit disk U such that p(0)=1 and (p(z))>0, and let pP if and only if p(z)(1+z)/(1-z). It is a well-established fact that a function fS is said to be starlike (f) or convex (fC) if there exists a function pP such that p can be expressed as zf'(z)/f(z)=p(z) or 1+zf”(z)/f'(z)=p(z), respectively, for all zU. For more information on these classes and their properties, refer to [1] and [12].Ma and Minda [24] developed a unified framework for several subclasses of starlike and convex functions. They introduced the classes S*(φ) and C(φ) of Ma-Minda starlike and Ma-Minda convex functions, respectively. These classes are characterized by the conditions zf'(z)/f(z)φ(z) and 1+zf''(z)/f'(z)φ(z), where φP is a superordinate function satisfying φ(0)=1, φ'(0)>0, and φ(U) is symmetric with respect to the real axis. Furthermore, φ can be represented by a series expansion of the form φ(z)=1+B1z+B2z2+B3z3+, where all coefficients are real and B1>0.

The field of quantum calculus, also known as q-calculus, does not rely on limits in its computations. Its widespread use in mathematics and physics stems from its application in various areas, including but not limited to, ordinary fractional calculus, basic hypergeometric functions, orthogonal polynomials, and combinatorics. Its importance is due to its versatility and ability to address complex problems in a variety of disciplines.

Let q(0,1). The q-derivative (or q-difference) operator, introduced by Jackson [16], is defined as

(Dqf)(z)=f(z)-f(qz)(1-q)z,(z0).

We note that limq1-(Dqf)(z)=f'(z) if f is differentiable at z. For a function f of the form 1.1, we observe that

(Dqf)(z)=1+n=2[n]qanzn-1,

where [n]q=1-qn1-q,q(0,1). Clearly, when q1-, we have [n]qn.

For definitions and properties of q-calculus, one may refer to [9], [16], [18].

Definition 1.1. The class Sq* of q-starlike functions was introduced and studied by Ismail et al. in [15]. A function fA is said to be in the class Sq* if and only if

Sq*(φ)={fA:zDqf(z)f(z)φ(z),q(0,1),zU}.

In the limiting case q1-, the class Sq* reduces to the class S*.

The class Cq of q-convex functions was introduced by Ahuja et al. in [2]. A function fA is said to be in the class Cq(φ) if and only if

Cq(φ)={fA:Dq(zDqf(z))Dqf(z)φ(z),q(0,1),zU}.

In the limiting case q1-, the class Cq reduces to the class C.

Toeplitz matrices and their determinants are of significant importance in various areas of mathematics, and they find many applications in diverse fields. (See [32]). A survey article by Ye and Lim [34] provides information on the application of Toeplitz matrices in several areas of pure and applied mathematics. It is worth noting that Toeplitz symmetric matrices are those matrices that have constant entries along the diagonal. For a given function f(z)=z+n=2anzn, we can associate a determinant TQ(n) defined by:

TQ(n)=anan+1...an+q-1an+1an...an+q an+q-1an+q...an

The bounds of |T2(n)|, |T3(1)|, and |T3(2)| for the class S* and C were investigated by Ali et al. [3] in 2017, where m and n are natural numbers. Following this, several researchers examined the bounds of |Tm(n)| for different subclasses of S for small values of m and n in recent years ([5], [8], [10], [23]), taking inspiration from the work of Ali et al. [3]. Recent research has been conducted on coefficient problems by various authors including Cho et al. [6], Cudna et al. [8], Kowalczyk et al. [21], and Lecko et al. [23].

This article presents sharp estimates for Toeplitz determinants T2(2) and T3(1) for functions that belong to the Sq*(φ) and Cq(φ) classes. The article also introduces the functions Kφ and Hφ, which are defined as follows:

zK'φ(z)Kφ(z)=φ(iz),Kφ(0)=K'φ(0)-1=0

and

1+zH''φ(z)H'φ(z)=φ(iz),Hφ(0)=H'φ(0)-1=0

respectively, belong to the classes S*(φ) and K(φ). We will use these functions to establish the sharpness of our results in certain cases. It is a well-known fact from Grenander and Szegő [14] that for a function p in P with p(z)=1+c1z+c2z2+c3z3+...,we have|cn|2.

The primary outcomes of this research are established based on the aforementioned estimate by associating coefficients of the functions in the classes S*(u) and K(u) to the functions in the class P. Moreover, we will utilize the Fekete-Szego¨ functional estimates for the two classes S*(u) and K(u) from Ali et al. [4] and Ma and Minda [24]. Ma and Minda [24] employed the symmetry of the image of φ to ensure that the coefficients of φ are real, and we have adopted this notion for the first two coefficients. In addition, they utilized the univalence in defining the function p1.

p1(z)=1-φ-1(p(z))1+φ-1(p(z)).

The condition can be waived by using the definition of p1 provided in equation (2.5).

The sharp bound for T2(2)=a32-a22 is given by Theorem 2.1 and Theorem 2.2 for functions fSq*(φ) and fCq(φ), respectively.

Theorem 2.1. If f Sq*(φ) and φ(z)=1+B1z+B2z2+ with 0<B1B2+B12, then the Toeplitz determinant T2(2) satisfies the sharp bound:

T2(2)1q2[2]q2B12q+B22+B12q.

Proof. Since fS*(φ), there is a function w in the class Ω of Schwarz functions satisfying that

zDqf(z)f(z)=φ(w(z)).

Corresponding to the function w, define the function p1 : DC by

p1(z)=1+w(z)1-w(z)=1+c1z+c2z2+,

so that

w(z)=p1(z)-1p1(z)+1=12c1z+12c2-12c12z2+.

Clearly, the function p1 is analytic in D with p1(0)=1. Since wΩ, it follows that p1P. Using (2.6) and the Taylor series of φ given by φ(z)=1+B1z+B2z2+B3z3+, we get

φ(w(z))=1+12B1c1z+12B1c2-12c12+14B2c12z2+.

Since f(z)=z+a2z2+a3z3+, the Taylor series expansion of the function zDqf/f is given by

zDqf(z)f(z)=1+a2([2]q-1)z+-a22([2]q-1)+a3([3]q-1)z2+.

Using (2.4), (2.7), (2.8) the coefficients a2 and a3 can be expressed as a function of the coefficients ci of pP and Bi of φ as follows:

a2=12([2]q-1)B1c1

and

a3=14([3]q-1)B12([2]q-1)-B1+B2c12+2B1c2

.

The equations (2.9) and (2.10) (see Ali et al. [4] for a general result for p-valent functions) readily shows that

|a3-νa22|{1q2[2]q(qB2+B12-μ[2]qB12)    [2]qB12qB2+B12-B1,B1q2[2]q    qB2+B12-B1[2]qB12qB2+B12+B11q2[2]q(-qB2-B12+μ[2]qB12)    qB2+B12+B1[2]qB12, .

Since cn2, the equation (2.9) shows that

a2B1([2]q-1)

and, when B1B2+B12, the equation (2.10) readily yields

a31([3]q-1)B12[2]q-1+B2.

Using these estimates for the second and third coefficients given in (2.12) and (2.13) , we have

a32-a22a32+a221([3]q-1)2B12[2]q-1+B22+B12[2]q-1.

The result is sharp for the function Kφ given by (1.2). The Taylor series can be obtained by noting that Kφ corresponds to the function f given by (2.4) when w(z)=iz. In this case, p1(z)=1+2iz-2z2+.... With c1=2i and c2=-2, we get a2=B1q and a3=-(B2+B12/q)q[2]q Clearly, for the function Kφ, we have

a32-a22=1([3]q-1)2B12[2]q-1+B22+B12[2]q-1.

proving the sharpness.

Theorem 2.2. If fCq(φ) and φ(z)=1+B1z+B2z2+ with 0<B1B2+B12, then the Toeplitz determinant T2(2) satisfies the sharp bound given by

T2(2)1q2[2]q2[3]q2(B12q+B22+1q2[2]q2B12.

Proof. Let f(z)=z+a2z2+a3z3+ and φ(z)=1+B1z+B2z2+. Since fK(φ), there is a function w in the class Ω of Schwarz functions such that

1+qzDq(Dqf(z))Dqf(z)=φ(w(z)).

The Taylor series expansion of the function f given by f(z)=z+a2z2+a3z3+ shows that

1+qzDq(Dqf(z))Dqf(z)=1+a2[2]q([2]q-1)z+-a22[2]q2([2]q-1)+a3[3]q([3]q-1)z2+.

Then using (2.14), (2.15) and (2.7), the coefficients a2 and a3 can be expressed as a function of the coefficients ci of pP given by

a2=12[2]q([2]q-1)B1c1.

Using the well-known estimate cn2 for the function p1 with positive real part, it follows that

a2B1q[2]q.

For a function fCq(φ), we can easily see that

|a3-νa22|{1q2[2]q[3]q(B12+qB2-[3]qμB12[2]q)  [3]q[2]qμB12B12+qB2-B1B1q2[2]q[3]q    B12+qB2-B1[3]q[2]qμB12B12+qB2+B11q2[2]q[3]q(-B12-qB2+[3]qμB12[2]q)    B12+qB2+B1[3]q[2]qμB12 .
a32-a22a32+a221q2[2]q2[3]q2B12q+B22+1q2[2]q2B12.

Since B1|B2+B12|, the inequality (2.18) readily gives

|a3|B2+B12qq[2]q[3]q.

The result is sharp for the function Hφ defined in (1.3). Indeed, for this function Hφ, we have a2=B1i/q[2]q and a3= -B12/q+B2/q[2]q[3]q and hence

a32-a22=1q2[2]q2[3]q2B12q+B22+1q2[2]q2B12.

proving the sharpness of the result.

Theorems 2.3 and 2.4 give the sharp bound for the Toeplitz determinant T3(1) for functions, respectively, in the classes Sq*(φ) and Cq(φ).

Theorem 2.3. If f Sq*(φ) and φ(z)=1+B1z+B2z2+, with B1>0 and B1-B12B23B12-B1, then the Toeplitz determinant T3(1) satisfies the sharp bound:

T3(1)1+2B12q2+1q2[2]q2B2+B12q(2[2]q-1)B12-B2.

Proof. Since

T3(1)=1a2a3a21a2a3a21=1-2a22-a3a3-2a22,

it follows that

T3(1)1+2a22+a3a3-2a22.

Since B1B12+B2, the inequality (2.13) gives

a31[3]q-1B12q+B2.

Since B1+B23B12, the equation (2.11) readily yields

a3-2a221q[2]q(2[2]q-1)B12-B2.

Using these estimates for the second and third coefficients given in (2.12) and (2.21), and the bound for a3-2a22 given by (2.22) in (2.20), we obtain

T3(1)1+2B12/q2+1q2[2]q2B2+B12/q(2[2]q-1)B12-B2.

Theorem 2.4. If f Cq(φ) and φ(z)=1+B1z+B2z2+, with B1>0 and B1-B12B22B12-B1, then the Toeplitz determinant T3(1) satisfies the sharp bound:

T3(1)1+2B12/q2[2]q2+1q2[2]q2[3]q2B2+B12/q(2[3]q-[2]q)B12q[2]q-B2.

Proof. The given conditions on B1 and B2 is the same as B1B12+B2 and, B1+B22B12. Since B1B12+B2, the inequality (2.18) gives

|a2|B1q[2]q

and

a31q[2]q[3]qB12/q+B2.

Since B12B12-B2, the inequality (2.18) gives

a3-2a221q[2]q[3]q(2[3]q-[2]q)q[2]qB12-B2.

Using the bound for a2 and a3 given by (2.16) and (2.23) and the bound for a3-2a2 given by (2.25) in (2.20), we get the desired result.

T3(1)1+2B12/q2[2]q2+1q2[2]q2[3]q2B2+B12/q(2[3]q-[2]q)B12q[2]q-B2.

The result is sharp for the function Hφ defined in (1.3). Indeed, for this function Hφ, we have a2=B1i/2 and a3= -B12+B2/6 and hence 1-2a22-a3a3-2a22=1+12B12+136B12+B22B12-B2 proving the sharpness of the result.

3.1. The Ma and Minda classes of starlike and convex functions encompass various notable subclasses that have been extensively examined by different researchers (see, for instance, Kargar et al. [20] and Mahzoon [25]). Theorems 2.1 to 2.4 provide precise estimates for T2(2) and T3(1) in specific subclasses within these categories.

For -1B<A1, the classes S*[A,B] and C[A,B] are defined as S*[A,B]=S*((1+Az)/(1+Bz)) and C[A,B]=C((1+Az)/(1+Bz)), respectively. These classes, known as Janowski starlike functions and Janowski convex functions, were initially introduced and investigated by Janowski [17]. The series expansion of φ(z)=(1+Az)/(1+Bz) yields the following expression:

φ(z):=1+Az1+Bz=1+(A-B)z+B(B-A)z2+B2(A-B)z3+,

which implies B1=(A-B) and B2=-B(A-B). If |A-2B|1, then, for fS*[A,B],

T2(2)A2+(q-2)AB+(1-q)B22q4[2]q2+(A-B)2q

and for fC[A,B], we have

|T2(2)q2[3]q2(A-B)2+A2+(q-2)AB+(1-q)B22q4[2]q2[3]q2.

If Bmin{(A-1)/2,(3A-1)/2}, then for fS*[A,B], we have

T3(1)1+2(A-B)2q2+(2[2]q-1)A2+(-4[2]q+3)AB+2qB2A2-(2+q)AB+(1+q)B2q2[2]q2.

If (A+B)0 and B(A-1)/2, then, for fC[A,B]

T3(1)1+2(A-B)2q2[2]q2+2[3]q-[2]qq[2]q(A-B)2+AB-B2A2-(2+q)AB+(1+q)B2q2[2]q2[3]q2.

The classes S*(α):=S*[1-2α,-1] and C(α):=C[1-2α,-1] were introduced and analyzed by Robertson [29]. These classes consist of starlike functions of order α and convex functions of order α, respectively. For fS*(α), we have the following expression:

T2(2)4(1-α)21q+(2+q-2α)2q6[2]q2

and

T3(1)1+8(1-α)2q2+4(1-α)2(2+q-2α)4[2]q-3-(4[2]q-2)αq3[2]q2,α2/3.

For fC(α), we have

T2(2)4(1-α)2q2[2]q22+q-2αq[3]q2+1

and

T3(1)1+4(1-α)2)q2[2]q22+(2+q-2α)q[3]q2(2[3]q-[2]q)(2-2α)q[2]q-1,α1/2.

In particular, as q1-, for fS:=S(0), the bound T2(2)13 and T3(1)24 holds. Similarly, as q1-, for fK:=K(0), the bound T2(2)2 and T3(1)4 holds. These bounds for starlike and convex functions were recently derived in the work of Ali et al. [3].

3.2. Mendiratta et al. [26] introduced and studied the class Sα,e*=S*ez. In a broader context, Khatter et al. [19] introduced and investigated the classes Sα,e*:=S*(α+(1-α)ez) and Cα,e:=Cα+(1-α)ez, where 0α<1. When α=0, these classes correspond to Se* and Ce, respectively. The Taylor series expansion of φ is given by

φ(z)=α+(1-α)ez=1+(1-α)z+12(1-α)z2+16(1-α)z3+

shows that B1=(1-α) and B2=(1-α)/2. For 0α1/2, we get

T2(2)(1-α)24q[2]q2(2+q-2α)2q2+4[2]q2

and

T3(1)(4q2[2]q2+(1-α)28[2]q2+(2+q-2α)(4[2]q-3-(4[2]q-2)α)4q2[2]q2

for fSα,e** For 0α1/2, we get

T2(2)(1-α)2(2+q-2α)2+4[3]q24q4[2]q2[3]q2

and

|T3(1)|(1-α)24q2[2]q[3]q2+(2+q-2α){4[3]q-[2]q(2+q)-(4[3]q-2[2]q)α}/22q4[2]q3[3]q2+2q4[2]q3[3]q22q4[2]q3[3]q2

For fCα,e, particularly as q1-, when fSe*, we obtain T2(2)25/161.5625 and T3(1)63/16 3.9375. Similarly, when q1- and fCe, we have T2(2)5/16 0.3125 and T3(1)25/161.5625.

3.3. Sharma et al. [30] defined and studied the class of functions defined by SC*=S*φc(z), where φc(z)=1+(4/3)z+(2/3)z2. The geometrical interpretation is that a function f belongs to the class SC* if zDqf(z)f(z) lies in the region Ωc bounded by the cardioid i.e., φc(U):=x+iy:9x2+9y2- 18x+5)2-169x2+9y2-6x+1=0. The convex analogous class of the above mentioned class is KC:=Kφc(z). Its geometrical interpretation is that a function f belongs to the class KC if 1+qzDqDqf(z)Dqf(z) lies in the region Ωc bounded by the cardioid i.e., φc(D):=x+iy:9x2+9y2- 18x+5)2-169x2+9y2-6x+1=0. When fSC*, it follows that zDqf(z)f(z)1+(4/3)z+ (2/3)z2 which yields B1=4/3 and B2=2/3. And therefore

T2(2)256+192q+q2(36+144q2[2]q2)81q3[2]q2

and

T3(1)81q3[2]q2+288q[2]q2+(16+6q)(32[2]q-22)81q3[2]q2forfSC*.

Whereas

T2(2)256+192q+36q2+144q2[3]q281q4[2]q2[3]q2

and

T3(1)81[3]q2+288[3]q2[2]q2+(16+6q)(32[3]q-[2]q(16+6q))q4[2]q381[3]q2forfC.

3.4. The class Ssin* was introduced and studied by Cho et al. [7]. This class consists of functions of the form Ssin=S(1+sinz). The convex analogous subclass is defined as Csin:=C(1+sinz). Consider a function fSsin*. By writing the Taylor series expansion for sinz, we have φ(z):=1+sinz=1+z-16z3+1120z5+. Thus, B1=1 and B2=0 which implies T2(2)1+q2[2]q2q3[2]q2, proved in Zhang et al. [36], and T3(1)1+2q[2]q2+2[2]q-1q3[2]q2 for fSsin *. Similarly we can obatin T2(2)1+q[3]q2q3[2]q2[3]q2 and T3(1)1+2q2[2]q[3]q2+2[3]q-[2]qq4[2]q3[3]q2 for fCsin .

3.5. The class SC*, as defined by Raina and Sokol [27], is given by SC=S(φC), where φC=z+1+z2. Its convex subclass is denoted as KC=K(φC). Both SC* and KC consist of functions for which zDqf(z)f(z) and 1+qzDqDqf(z)Dqf(z) lie in the leftmoon region ΩC, defined as φC(D)=wC:w2-1<2|w|. Hence, we can conclude that

φC(z):=z+1+z2=1+z+12z2-18z4+

Therefore, B1=1 and B2=1/2 which immediately yields T2(2)4+4q+q2(1+4[2]q2)4q3[2]q2 and T3(1) 4q(q2+2)[2]q2+(2+q)(4[2]q-3)4q3[2]q2 for fSC*. Similarly, fKC implies 1+qzDqDqf(z)Dqf(z)z+1+z2, and therefore we have T2(2)(2+q2q)2+[3]q2q2[2]q2[3]q2 and T3(1)4q4[2]q2+8q2+(2+q)(4[3]q-[2]q(2+q))[3]q2[2]q4q4[2]q2.

3.6. Ronning [28], inspired by Goodman [13], introduced and analyzed the parabolic starlike class SP and the uniformly convex class UCV, derived from the Ma-Minda class of starlike and convex functions by replacing

φ(z):=1+2π2log1+z1-z2=1+8π2z+163π2z2+18445π2z3+

This yields B1=8/π2 and B2=16/3π2 and thus we get

T2(2)51272+12π2q+4q2+9q2[2]q28π49q3[2]q2π8

and

T3(1)1+40961+2qq2[2]q21π8+10242q-1q3[2]q/3π6+1152[2]q2-256q2[2]q2/9π4

for fSP. For fUCV, we get

T2(2)(64576+96qπ2+(4q2+9[3]q2π4q4[2]q2/9[3]q2π8.

3.7. Yunus et al. [35] investigated the class Slim*=S*1+2z+z22 associated with the limacon (4u2+4v2-8u-5)2+8(4u2+4v2-12u-3)=0. The class Klim=K1+2z+z22 was also studied. In this case, we have B1=2 and B2=12. Hence, for fSlim*, we obtain T2(2)16+8q+q2+8q2[2]q24q3[2]q2 and T3(1)4q[2]q2(q2+4)+(4+q)(8[2]q-5)4q3[2]q2. For fKlim, we have T2(2)16+8q+q2(1+8[3]q2)4q4[2]q2[3]q2 and T3(1)4q4[2]q2[3]q2+16q2[3]q2+(4+q)8[3]q-[2]q(4+q)/[2]q4q4[2]q2[3]q2.

3.8. Wani and Swaminathan [33] investigated the class of functions defined by SNe*=S*φNe(z) and KNe=KφNe, where the function φNe(z)=1+z-z33 maps the unit disk into the interior of the 2-cusped kidney-shaped nephroid. In this case, we have B1=1 and B2=0, which leads to T2(2)1+q2[2]q2q3[2]q2 and T3(1)1+2q[2]q2+2[2]q-1q3[2]q2 for fSNe*. For fKNe, we obtain T2(2)1+q[3]q2q3[2]q2[3]q2 and T3(1)1+2q2[2]q[3]q2+2[3]q-[2]qq4[2]q3[3]q2 for fSP.

3.9. S.S. Kumar and K. Gangania [22] examined the class of functions defined by SC*=S*φC, where φC=1+zez. Its convex subclass is denoted as KC:=KφC. Both SC* and KC consist of functions for which zDqf(z)f(z) and 1+qzDqDqf(z)Dqf(z) map the unit disk onto a cardioid domain. We can derive the following estimates: For fSC*: T2(2)[2]q2q2([3]q-1)2+1q, T3(1)1+21+1[2]qq2. For fKC: T2(2)1q4[3]q2+1q2[3]q2, T3(1)1+2q2[2]q2+1+q2q4[2]q2[3]q2.

3.10. J. Sokol and J. Stankiewicz [31] examined the class of functions defined by SC*=S*φC, where φC=1+z. Its convex subclass is denoted as KC:=KφC. Both SC* and KC consist of functions for which zDqf(z)f(z) and 1+qzDqDqf(z)Dqf(z) map the unit disk onto the right half of the lemniscate |ω2-1|=1. We can derive the following estimates: For fSC*: T2(2)(2-q)264q2([3]q-1)2+14q, T3(1)1+12q2+(2-q)(4q+3)64q3[2]q2. For fKC: T2(2)(2-q)264q4[2]q2[3]q2+14q2[2]q2, T3(1)1+12q2[2]q2+(2-q)(5q2+3q+2)64q4[2]q2[3]q2.

3.11.

P. Goel and S. Sivaprasad Kumar [11] investigated the class of functions defined by SC*=S*φC, where φC=21+e-z. Its convex subclass is denoted as KC:=KφC. Both SC* and KC consist of functions for which zDqf(z)f(z) and 1+qzDqDqf(z)Dqf(z) satisfy certain conditions. The modified sigmoid function φC=21+e-z maps the unit disk U onto a domain ΔSG:=ωC:|log(w2-w)|<1, which is symmetric about the real axis. Furthermore, the function G(z) defined by φC is convex and starlike with respect to G(0)=1. It also has a positive real part in D. Thus, G(z) belongs to the class of Ma-Minda functions. Therefore, the classes S(G) and C(G) naturally become subclasses of S* and C. We denote them as S*(SG) and C(SG). Analytically, a function fS*(SG) if and only if zUqf(z)f(z) lies in the region ΔSG. Using the expression 21+e-z=1+z2-z312, we can deduce the following: B1=12, B2=0, T2(2)1+14q([3]q-1)4q, T3(1)1+12q2+2q+116q3[2]q2, and T2(2)14q2[2]q2(1+14q2[3]q2), T3(1)1+12q2[2]q2+2[3]q-[2]q16q4[2]q3[3]q2.

This research paper established precise bounds T2(2) and T3(1) for Toeplitz determinants with entries derived from the coefficients of starlike functions and convex functions under certain conditions. The paper introduced new special cases and revisits known results, employing a methodology that integrates Toeplitz determinant properties and q-calculus techniques.

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