Hankel Determinant for a Subclass of Starlike Functions with Respect to Symmetric Points Subordinate to the Exponential Function

Abstract

Let $S_s^{*}\left(e^z\right)$ denote the class of starlike functions with respect to symmetric points subordinate to the exponential function, i.e., the functions which satisfy in the unit disk $\mathbb{U}$ the condition $\frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)-f(-z)}\prec e^z(z\in \mathbb{U})$. We obtained the sharp estimate of the second-order Hankel determinants $H_{2,3}\left(f\right)$ and improved the estimate of the third-order $H_{3,1}\left(f\right)$ for this functions class $S_s^{*}\left(e^z\right)$.

1. Introduction and Definitions

Let the notation $\mathcal{A}$ denote the class of all analytic functions $f\left(z\right)$ in the unit disk $\mathbb{U}=\{z\in \mathbb{C}:|z|<1\}$, which are normalized so that the Taylor expansion has the form

$$f\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n.$$

(1)

If $f\left(z\right)$ is univalent and has the form (1), it is referred to as a normalized univalent function. The class of all $f\left(z\right)$ is denoted as S.

In 1916, Bieberbach [1] studied the coefficients of class S and proposed the well-known coefficient conjecture, known as the “Bieberbach conjecture”, that if $f\left(z\right)\in S$, then $|a_n|\le n$ for all $n\ge 2.$ Although de Branges [2] proved this conjecture in 1985, research on the coefficient estimation of univalent analytic functions continues to increase. In 1959, Sakaguchi [3] introduced the class $S_s^{*}$ of starlike functions with respect to symmetric points, which consist of functions $f\left(z\right)\in S$, satisfying the condition $Re\frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)-f(-z)}>0$, and obtained coefficient estimates $|a_n|\le 1$ for all $n\ge 2.$ Subsequently, the class $S_s^{*}$ has been studied by several authors [4,5,6]. In 1991, Goodman [6] proved that if $f\left(z\right)\in S_s^{*},$ then for $n\ge 2$, $|a_n|\le \frac{2}{n}.$

Let $B_0$ denote the class of Schwarz functions in the unit disk $\mathbb{U}=\{z:|z|<1\}$, i.e., $w\left(z\right)\in B_0$ if and only if $w\left(z\right)={\displaystyle \sum _{n=1}^{\infty }}{c}_n{z}^n$ is analytic and $|w(z)|<1$ on $\mathbb{U}$. Let $f\left(z\right)$ and $g\left(z\right)$ be analytic functions in the unit disk $\mathbb{U}$. If there exists a Schwarz function $w\left(z\right)$ in the unit disk that satisfies $w\left(0\right)=0,|w(z)|<1$ and $f\left(z\right)=g\left(w\right(z\left)\right)$, then $f\left(z\right)$ is said to be subordinate to $g\left(z\right)$ and denoted by $f\left(z\right)\prec g\left(z\right).$

Ravichandran [7] introduced a class $S_s^{*}\left(\varphi \right)$ of symmetric starlike functions, which is given by

$$S_s^{*}\left(\varphi \right)=\{f\left(z\right)\in S,\frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)-f(-z)}\prec \varphi \left(z\right),z\in \mathbb{U}\},$$

where $\varphi \left(z\right)=1+B_{1}z+B_2{z}^2+\cdots$ is a univalent starlike function with respect to 1, $B_1>0$ and $Re\varphi \left(z\right)>0.$

Taking different parameters for the coefficients of the function $\varphi \left(z\right)$, we can obtain the following subclasses:

(i)

$S_s^{*}\left(\frac{1+Az}{1+Bz}\right)=S_s^{*}(A,B),(-1\le B<A<1)$ (see [8,9,10,11,12,13]; for $S_s^{*}\left(\frac{1+z}{1-z}\right)=S_s^{*}$, see [3]);

(ii)

$S_s^{*}\left(\frac{1+(1-2\alpha )z}{1-z}\right)=S_s^{*}\left(\alpha \right),(0\le \alpha <\frac{1}{2})$ (see [9,14]);

(iii)

$S_s^{*}\left(\frac{1+\beta z}{1-\alpha \beta z}\right)=S_s^{*}(\alpha ,\beta )$ (see [10,15]);

(iv)

$S_s^{*}((1-\beta ){\left[p\left(z\right)\right]}^{\alpha }+\beta )={\overline{S}}_{s,\beta }^{*}\left(\alpha \right)$, where $p\in P$, the class of all functions $p\left(z\right)$ analyzed in $\mathbb{U}$, for which $Rep\left(z\right)>0$ and $p\left(z\right)=1+c_{1}z+c_2{z}^2+\cdots ,0<\alpha \le 1,0\le \beta <1$ (see [16]);

(v)

$S_s^{*}\left(\frac{2}{1+e^{-z}}\right)=S{S}_{sG}^{*}$ (see [17]).

Pommerenke [18] introduced the q-th order Hankel determinant of $f\left(z\right)$ of the form given by (1)

$$H_{q,n}\left(f\right):=\left|\begin{array}{cccc}{a}_n\hfill & a_{n+1}\hfill & \dots \hfill & a_{n+q-1}\hfill \\ a_{n+1}\hfill & a_{n+2}\hfill & \dots \hfill & a_{n+q}\hfill \\ ⋮\hfill & ⋮\hfill & \dots \hfill & ⋮\hfill \\ a_{n+q-1}\hfill & a_{n+q}\hfill & \dots \hfill & a_{n+2q-2}\hfill \end{array}\right|.$$

When certain special parameters are chosen for q and n, different Hankel determinants will be obtained. For example,

$$\begin{array}{c}{H}_{2,3}\left(f\right)=a_3{a}_5-a_4^2\hfill \end{array}$$

(2)

and

$$\begin{array}{c}{H}_{3,1}\left(f\right)=a_1{a}_3{a}_5+2a_2{a}_3{a}_4-a_3^3-a_2^2{a}_5-a_1{a}_4^2.\hfill \end{array}$$

(3)

The Hankel determinant plays a vital role in studies of power series with integral coefficients. Initially, the determinant was studied by researchers such as Hyman [19], Noonan [20] and Ehrenborg [21]. These studies explore the properties and behavior of power series, particularly their coefficients. It was difficult for scholars to determine the exact bounds of the Hankel determinants for a particular class of functions. After that, many papers focused on the upper-bound estimates of the second-order Hankel determinants $H_{2,1}$ and $H_{2,2}$[22,23,24]. Recently, many scholars [25,26,27,28,29,30] have investigated the third-order Hankel determinant. Some sharp upper-bound estimates of the Hankel determinant for a subclass of univalent analytic functions were obtained via complex calculations.

In 2020, Ganesh et al. introduced and investigated a class $S_s^{*}\left(e^z\right)$ of analytic functions in [31], defined as follows:

$$S_s^{*}\left(e^z\right)=\{f\in A:\frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)-f(-z)}\prec e^z,z\in \mathbb{U}\}.$$

They expressed the coefficients of functions in class $S_S^{*}\left(e^z\right)$ with the coefficients of the corresponding functions with positive real part, and these lemmas on the coefficients of the positive real part were obtained by Libera and Zlotkiewicz [32]. Using this method, the following results were obtained:

Theorem A. If $f\in S_s^{*}\left(e^z\right)$, then $|a_2|\le \frac{1}{2},|a_3|\le \frac{1}{2},|a_4|\le \frac{19}{48},|a_5|\le \frac{13}{24}.$

Theorem B. If $f\in S_s^{*}\left(e^z\right)$, then $H_{2,1}=|a_3-a_2^2|\le \frac{1}{2}.$

Theorem C. If $f\in S_s^{*}\left(e^z\right)$, then $|a_4-a_2{a}_3|\le \frac{765+59\sqrt{118}}{3468}.$

Theorem D. If $f\in S_s^{*}\left(e^z\right)$, then $H_{2,2}=2|a_2{a}_4-a_3^2|\le \frac{3}{8}.$

• By applying the above theorems together, the estimate of the third-order Hankel determinant is derived as follows:

Theorem E. If $f\in S_s^{*}\left(e^z\right)$, then $H_{3,1}=\frac{90831+1121\sqrt{118}}{166464}=0.618.$

In 2022, Zaprawa [33] researched the class $S_s^{*}\left(e^z\right)$ using a new approach, which is based on relating the coefficients of functions in a given class with the coefficients of corresponding Schwarz functions. Some coefficient estimates of the function class $S_s^{*}\left(e^z\right)$ were obtained, including logarithmic coefficient estimates, Zalcman functional estimates and upper bounds for the second- and third-order Hankel determinants, etc. The main results are as follows:

Theorem A’. If $f\in S_s^{*}\left(e^z\right)$, then $|a_4|\le \frac{1}{4},|a_5|\le \frac{1}{4}.$

Theorem B’. If $f\in S_s^{*}\left(e^z\right)$, then $H_{2,2}=2|a_2{a}_4-a_3^2|\le \frac{1}{4}.$

Theorem C’. If $f\in S_s^{*}\left(e^z\right)$, then $|a_4-a_2{a}_3|\le \frac{1}{4},|a_5-a_3^2|\le \frac{1}{4}.$

Theorem D’. If $f\in S_s^{*}\left(e^z\right)$, then $H_{2,3}\le \frac{1}{8}.$

Theorem E’. If $f\in S_s^{*}\left(e^z\right)$, then $H_{3,1}=\frac{13}{128}.$

Comparing references [31,33], it is evident that Zaprawa’s research results improved upon some of Ganesh’s conclusions.

In this paper, we re-examined the upper bounds of Hankel determinants $H_{2,3}$ and $H_{3,1}$ of the class $S_s^{*}\left(e^z\right)$ and presented a complete proof. We proposed a proof method which differed from that of Zaprawa. This method is based on the use of a lemma derived by Carlson [34] about the coefficients of the Schwarz function. In particular, for the Hankel determinant $H_{3,1},$ we obtained a more precise upper-bound estimate than in [33].

These proofs require the following lemma.

Lemma 1.

[34] Let $w\left(z\right)\in B_0$. Then

$$|c_2|\le 1-|c_1{|}^2,|c_3|\le 1-|c_1{|}^2-\frac{|c_2{|}^2}{1+|c_1|},|c_4|\le 1-|c_1{|}^2-{|c_2|}^2.$$

2. Main Results and Proofs

Theorem 1.

If $f\in S_s^{*}\left(e^z\right)$, then $|H_{2,3}\left(f\right)|\le \frac{1}{8}.$ The equality is sharp.

Proof. 

Since $f\in S_s^{*}\left(e^z\right)$, we have

$$\frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)-f(-z)}=e^{w\left(z\right)}$$

where $w\left(z\right)\in B_0.$

Comparing the coefficients on both sides yields

$$\left\{\begin{array}{c}{a}_2=\frac{c_1}{2},\hfill \\ a_3=\frac{c_2}{2}+\frac{c_1^2}{4},\hfill \\ a_4=\frac{c_3}{4}+\frac{3}{8}{c}_1{c}_2+\frac{5}{48}{c}_1^3,\hfill \\ a_5=\frac{c_4}{4}+\frac{1}{4}{c}_1{c}_3+\frac{1}{4}{c}_2^2+\frac{1}{4}{c}_1^2{c}_2+\frac{1}{24}{c}_1^4.\hfill \end{array}\right.$$

(4)

Based on (2) and (4), we have

$$|H_{2,3}\left(f\right)|=\frac{1}{2304}|-c_1^6+12c_1^4{c}_2+108c_1^2{c}_2^2+288c_2{c}_4+144c_1^2{c}_4+288c_2^3+24c_1^3{c}_3-144c_1{c}_2{c}_3-144c_3^2|.$$

(5)

Applying the triangle inequality and Lemma 1 to (5) yields

$$\begin{array}{c}\hfill |H_{2,3}\left(f\right)|\le \frac{1}{2304}[|c_1{|}^6+12|c_1{|}^4|c_2|+108|c_1{|}^2|c_2{|}^2+288|c_2|(1-|c_1{|}^2-|c_2{|}^2)+144|c_1{|}^2(1-|c_1{|}^2-|c_2{|}^2)\\ \hfill +288|c_2{|}^3+24|c_1{|}^3\left(\right(1-|c_1{|}^2-\frac{|c_2{|}^2}{1+|c_1|})+144|c_1||c_2|(1-|c_1{|}^2-\frac{|c_2{|}^2}{1+|c_1|})+144(1-|c_1{|}^2-\frac{|c_2{|}^2}{1+|c_1|}{)}^2]\\ \hfill =\frac{1}{2304}[|c_1{|}^6-24|c_1{|}^5+12|c_2||c_1{|}^4+144|c_1{|}^4-144|c_2||c_1{|}^3-120|c_1{|}^3-36|c_2{|}^2|c_1{|}^2-288|c_2||c_1{|}^2\\ \hfill -144|c_1{|}^2+288|c_2{|}^2|c_1|+144|c_2||c_1|-288|c_2{|}^2+288|c_2|+144+\frac{144|c_2{|}^4}{(1+|c_1{|)}^2}-\frac{144|c_1||c_2{|}^3}{1+|c_1|}-\frac{24|c_1{|}^3{|c_2|}^2}{1+|c_1|}].\end{array}$$

Let $x=|c_1|\in [0,1],y=|c_2|\in [0,1-x^2].$ Then,

$$\begin{array}{c}\hfill 2304|H_{2,3}\left(f\right)|\le x^6-24x^5+(12y+144)x^4+(-144y-120)x^3+(-36y^2-288y-144)x^2\\ \hfill +(288y^2+144y)x-288y^2+288y+144+\frac{144y^4}{{(1+x)}^2}-\frac{144x{y}^3}{1+x}-\frac{24x^3{y}^2}{1+x}=T(x,y).\end{array}$$

We need to find the maximum value 288 of $T(x,y)$ on $\mathsf{\Omega }=\{(x,y):0\le x\le 1,0\le y\le 1-x^2\}$. First, we find the possible extreme points of $T(x,y)$ inside $\mathsf{\Omega }$. Let

$$\left\{\begin{array}{c}\frac{\partial T}{\partial x}=6x^5-120x^4+(48y+576)x^3+(-432y-360)x^2+(-72y^2-576y-288)x+288y^2+144y\hfill \\ -\frac{288y^4}{{(1+x)}^3}-\frac{144y^3}{{(1+x)}^2}-\frac{24y^2(3x^2+2x^3)}{{(1+x)}^2}=0,\hfill \\ \frac{\partial T}{\partial y}=12x^4-144x^3+(-72y-288)x^2+(576y+144)x-576y+288\hfill \\ +\frac{576y^3}{{(1+x)}^2}-\frac{432x{y}^2}{1+x}-\frac{48x^{3}y}{1+x}=0.\hfill \end{array}\right.$$

In this case, it is equivalent to

$$\left\{\begin{array}{c}{x}^8-17x^7+(8y+39)x^6+(-48y+169)x^5+(-20y^2-288y+40)x^4+(-8y^2-472y-180)x^3\hfill \\ +(96y^2-288y-156)x^2+(-24y^3+132y^2-24y-48)x-48y^4-24y^3+48y^2+24y=0,\hfill \\ x^6-10x^5+(-10y-47)x^4+(32y-48)x^3+(-36y^2+42y+24)x^2\hfill \\ +(36y^2-48y+60)x+48y^3-48y+24=0.\hfill \end{array}\right.$$

By solving the system of binary nonlinear equations, we find that

$$\left\{\begin{array}{c}{x}_1=-1\hfill \\ y_1=0,\hfill \end{array}\right.\left\{\begin{array}{c}{x}_2=0.1126\cdots \hfill \\ y_2=-1.3016\cdots ,\hfill \end{array}\right.\left\{\begin{array}{c}{x}_3=1.0265\cdots \hfill \\ y_3=-0.1197\cdots ,\hfill \end{array}\right.\left\{\begin{array}{c}{x}_4=13.2848\cdots \hfill \\ y_4=-0.9468\cdots ,\hfill \end{array}\right.\left\{\begin{array}{c}{x}_5=-0.3053\cdots \hfill \\ y_5=0.8011\cdots ,\hfill \end{array}\right.$$

$$\left\{\begin{array}{c}{x}_6=-0.4422\cdots \hfill \\ y_6=-0.5837\cdots ,\hfill \end{array}\right.\left\{\begin{array}{c}{x}_7=-1.4723\cdots \hfill \\ y_8=-0.0182\cdots ,\hfill \end{array}\right.\left\{\begin{array}{c}{x}_8=-1.4765\cdots \hfill \\ y_8=-0.2346\cdots ,\hfill \end{array}\right.\left\{\begin{array}{c}{x}_9=-4.4675\cdots \hfill \\ y_9=-6.4232\cdots .\hfill \end{array}\right.$$

Obviously, there are no extreme points of T inside $\mathsf{\Omega }$.

It is assumed that the extreme points lie on the edges of $\mathsf{\Omega }$.

(a) When $x=0,T(0,y)=144y^4-288y^2+288y+144.$

Since $T^{\prime }(0,y)=576y^3-576y+288>0,$$T(0,y)\le T(0,1)=288.$

(b) When $y=0,$$T(x,0)=x^6-24x^5+144x^4-120x^3-144x^2+144.$

Let $T^{\prime }(x,0)=6x^5-120x^4+576x^3-360x^2-288x=0.$ A simple calculation provides the stationary point $x=0.$

From $T^{″}(x,0)=30x^4-488x^3+1728x^2-720x-288,$ we obtain $T^{″}(0,0)=-288<0.$ This implies $T(x,0)\le T(0,0)=1448.$

Therefore, we have $|H_{2,3}\left(f\right)|\le \frac{288}{2304}=\frac{1}{8}$. Considering $f\left(z\right)=\frac{z^3}{2-z^2}=\frac{1}{2}{z}^3+\frac{1}{4}{z}^5+\cdots$, the equality condition in the theorem’s statement is true. □

Theorem 2.

Let $f\left(z\right)\in S_s^{*}\left(e^z\right)$. Then, $|H_{3,1}\left(f\right)|\le \frac{203.5083\cdots }{2304}=0.08832\cdots .$

Proof. 

From (3) and (4), we have

$$|H_{3,1}\left(f\right)|=\frac{1}{2304}|-c_1^6-12c_1^4{c}_2-36c_1^2{c}_2^2+288c_2{c}_4+24c_1^3{c}_3+144c_1{c}_2{c}_3-144c_3^2|.$$

(6)

Applying the triangle inequality and Lemma 1 to Equation (6), we obtain

$$\begin{array}{cc}\hfill |H_{3,1}\left(f\right)|& \le \frac{1}{2304}[|c_1{|}^6+12|c_1{|}^4|c_2||+36|c_1{|}^2|c_2{|}^2+288|c_2|(1-|c_1{|}^2-|c_2{|}^2)+24|c_1{|}^3(1-|c_1{|}^2-\frac{|c_2{|}^2}{1+|c_1|})\hfill \\ & +144|c_1||c_2|(1-|c_1{|}^2-\frac{|c_2{|}^2}{1+|c_1|})+144(1-|c_1{|}^2-\frac{|c_2{|}^2}{1+|c_1|}{)}^2]\hfill \\ & =\frac{1}{2304}[|c_1{|}^6-24|c_1{|}^5+12|c_2||c_1{|}^5+144|c_1{|}^4-144|c_2||c_1{|}^3+24|c_1{|}^3+36|c_2{|}^2{|c_1|}^2\hfill \\ & -288|c_2||c_1{|}^2-288|c_1{|}^2+288|c_2{|}^2|c_1|+144|c_2||c_1|-288|c_2{|}^3-288|c_2{|}^2+288|c_2|\hfill \\ & +144+\frac{144|c_2{|}^4}{(1+|c_1{|)}^2}-\frac{144|c_1||c_2{|}^3}{1+|c_1|}-\frac{24|c_1{|}^3{|c_2|}^2}{1+|c_1|}.\hfill \end{array}$$

Setting $x=|c_1|\in [0,1]$ and $y=|c_2|\in [0,1-x^2],$ we obtain

$$\begin{array}{cc}\hfill 2304|H_{3,1}\left(f\right)|& \le x^6-24x^5+(12y+144)x^4+(-144y+24)x^3+(36y^2-288y-288)x^2+(288y^2+144y)x\hfill \\ & -288y^3-288y^2+288y+144+\frac{144y^4}{{(1+x)}^2}-\frac{144x{y}^3}{1+x}-\frac{24x^3{y}^2}{1+x}=\mathsf{\Psi }(x,y).\hfill \end{array}$$

Next, we will seek the maximum value of $\mathsf{\Psi }$ on $\mathsf{\Omega }=\{(x,y):0\le x\le 1,0\le y\le 1-x^2\}$.

First, we need to find the possible extreme points inside $\mathsf{\Omega }$ for $\mathsf{\Psi }$. Let

$$\left\{\begin{array}{c}\frac{\partial \mathsf{\Psi }}{\partial x}=6x^5-120x^4+(48y+576)x^3+(-432y+72)x^2+(72y^2-576y-576)x+288y^2+144y\hfill \\ -\frac{288y^4}{{(1+x)}^3}-\frac{144y^3}{{(1+x)}^2}-\frac{24y^2(3x^2+2x^3)}{{(1+x)}^2}=0,\hfill \\ \frac{\partial \mathsf{\Psi }}{\partial y}=12x^4-144x^3+(72y-288)x^2+(576y+144)x-864y^2-576y+288\hfill \\ +\frac{576y^3}{{(1+x)}^2}-\frac{432x{y}^2}{1+x}-\frac{48x^{3}y}{1+x}=0.\hfill \end{array}\right.$$

Then, it is equivalent to

$$\left\{\begin{array}{c}{x}^8-17x^7+(8y+39)x^6+(-48y+241)x^5+(4y^2-288y+208)x^4+(64y^2-472y-156)x^3\hfill \\ +(168y^2-288y-276)x^2+(-24y^3+156y^2-24y-96)x-48y^4-24y^3+48y^2+24y=0,\hfill \\ x^6-10x^5+(-2y-47)x^4+(56y-48)x^3+(-108y^2+54y+24)x^2\hfill \\ +(-180y^2-48y+60)x+48y^3-72y^2-48y+24=0.\hfill \end{array}\right.$$

Solving this nonlinear system of equations gives

$\left\{\begin{array}{c}{x}_0=-1\hfill \\ y_0=0,\hfill \end{array}\right.$$\left\{\begin{array}{c}{x}_1=0.1084\cdots \hfill \\ y_1=0.3801\cdots ,\hfill \end{array}\right.$$\left\{\begin{array}{c}{x}_2=1.0099\cdots \hfill \\ y_2=-0.0399\cdots ,\hfill \end{array}\right.$$\left\{\begin{array}{c}{x}_3=-0.2586\cdots \hfill \\ y_3=-0.7152\cdots ,\hfill \end{array}\right.$$\left\{\begin{array}{c}{x}_4=-0.4558\cdots \hfill \\ y_4=0.6786\cdots .\hfill \end{array}\right.$

Thus, the extreme point inside $\mathsf{\Omega }$ is $\left\{\begin{array}{c}{x}_1=0.1084\cdots \hfill \\ y_1=0.3801\cdots \hfill \end{array}\right.$

• and accordingly,

$$\mathsf{\Psi }(0.1084\cdots ,0.3801\cdots )=203.5083\cdots .$$

Second, we discuss the extreme values of $\mathsf{\Psi }$ on the boundary of $\mathsf{\Omega }$ separately.

(a) When $x=0$, $\mathsf{\Psi }(0,y)=144y^4-288y^3-288y^2+288y+144={\varphi }_1\left(y\right).$

Let ${\varphi }_1^{\prime }\left(y\right)=576y^3-864y^2-576y+288=0.$ Then, $y=0.3554.$

Since ${\varphi }_1^{″}\left(y\right)=1728y^2-1728y-576,$${\varphi }_1^{″}\left(0.3554\right)=-971.8690<0.$ Thus, ${\varphi }_1\left(y\right)\le {\varphi }_1\left(0.3554\right)=199.3471.$

(b) When $y=0,$ $\mathsf{\Psi }(x,0)=x^6-24x^5+144x^4+24x^3-288x^2+144={\varphi }_2\left(x\right).$

Let ${\varphi }_2^{\prime }\left(x\right)=6x^5-120x^4+576x^3+72x^2-576x=0.$ Then, $x=0.$

Since ${\varphi }_2^{″}\left(x\right)=30x^4-480x^3+1728x^2+144x-576,$${\varphi }_2^{″}\left(0\right)=-576<0.$ Thus, ${\varphi }_2\left(x\right)\le {\varphi }_2\left(0\right)=144.$

(c) When $y=1-x^2,$$\mathsf{\Psi }(x,1-x^2)=577x^6-1188x^4+612x^2={\varphi }_3\left(x\right).$

Let ${\varphi }_3^{\prime }\left(x\right)=3462x^5-4752x^3+1224x=0.$ Then, $x=\frac{\sqrt{228492-1154\sqrt{9777}}}{577}=0.5862\cdots .$

Since ${\varphi }_3^{″}\left(x\right)=17310x^4-14256x^2+1224,$${\varphi }_3^{″}(0.5862\cdots )=-1630.8\cdots <0.$ Thus, ${\varphi }_3\left(x\right)\le {\varphi }_3\left(0.5862\right)=93.4332\cdots .$

Therefore,

$$|H_{3,1}\left(f\right)|\le \frac{203.5083\cdots }{2304}=0.08832\cdots .$$

This completes the proof of Theorem 2. □

3. Conclusions

In this paper, upper-bound estimates for the Hankel determinant of the class $S_S^{*}\left(e^z\right)$ are studied using a new Lemma and method. Theorem 5 in reference [33] is proved once again, and the extremal function is obtained. The upper-bound estimate of $|H_{3,1}\left(f\right)|$ for the functions class $S_s^{*}\left(e^z\right)$ is more accurate than Theorem 6 in reference [33]. The proof of Theorem 2 indicates that the exact upper-bound estimate of $|H_{3,1}\left(f\right)|$ cannot be obtained at the boundary of $\mathsf{\Omega }$.