Sharp Bounds on Hankel Determinants for Starlike Functions Defined by Symmetry with Respect to Symmetric Domains

Abstract

This work investigates the behavior of the coefficients of analytic functions within certain subclasses characterized by inherent symmetric structures. By leveraging deep connections with functions exhibiting positive real part properties, the approach introduces a modern analytical framework that links the studied coefficients to those of auxiliary functions with regulated behavior. This connection allows for the derivation of sharp estimates and facilitates computational treatment. The proposed method builds upon certain classical and modern coefficient inequalities. The study focuses on obtaining precise bounds for specific determinant expressions associated with initial, inverse, and inverse logarithmic coefficients, all within a subclass of starlike functions exhibiting internal symmetry aligned with a recently introduced canonical structure. This symmetric perspective reveals how geometric properties can lead to refined quantitative outcomes that enhance contemporary analytic theory.

1. Introduction

Let $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ represent the open unit disk, and consider class $\mathbb{A}$, consisting of all functions f that are analytic on $\mathbb{D}$ and satisfy the standard normalization conditions $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1$. Every function $f\in \mathbb{A}$ admits a power series expansion of the form

$$f\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n,a_n\in \mathbb{C},z\in \mathbb{D}.$$

(1)

Denote, using $\mathcal{S}$, the family of univalent functions (one-to-one) in $\mathbb{D}$ that belong to $\mathbb{A}$. From (1), we present three important and fundamental related forms. For $f\in \mathcal{S}$, the inverse of the function f is denoted by $f^{-1}$ (see [1]), which can be represented by the following expression:

$$f^{-1}\left(u\right)=u+\sum _{n=2}^{\infty }{\mathcal{A}}_n{u}^n,{\mathcal{A}}_n\in \mathbb{C},\left|u\right|<\frac{1}{4},$$

(2)

where

$$\begin{array}{cc}\hfill {\mathcal{A}}_2& =-a_2,\hfill \\ \hfill {\mathcal{A}}_3& =2a_2^2-a_3,\hfill \\ \hfill {\mathcal{A}}_4& =5a_2{a}_3-a_4-5a_2^3,\hfill \\ \hfill {\mathcal{A}}_5& =6a_2{a}_4-21a_2^2{a}_3+3a_3^2+14a_2^4-a_5.\hfill \end{array}$$

(3)

The logarithmic coefficients are defined as

$$F_f\left(z\right)=log\left({\displaystyle \frac{f\left(z\right)}{z}}\right)=2\sum _{n=1}^{\infty }{\gamma }_n{z}^n,z\in \mathbb{D},$$

(4)

and hence the coefficients are given by

$$\begin{array}{cc}\hfill {\gamma }_1& ={\displaystyle \frac{1}{2}}{a}_2,\hfill \\ \hfill {\gamma }_2& ={\displaystyle \frac{1}{2}}\left(a_3-{\displaystyle \frac{1}{2}}{a}_2^2\right),\hfill \\ \hfill {\gamma }_3& ={\displaystyle \frac{1}{2}}\left(a_4-a_2{a}_3+{\displaystyle \frac{1}{3}}{a}_2^3\right),\hfill \\ \hfill {\gamma }_4& ={\displaystyle \frac{1}{2}}\left(a_5-a_2{a}_4+a_2^2{a}_3-{\displaystyle \frac{1}{2}}{a}_3^2-{\displaystyle \frac{1}{4}}{a}_2^4\right),\hfill \\ \hfill {\gamma }_5& ={\displaystyle \frac{1}{2}}\left(a_6-a_2{a}_5-a_3{a}_4+a_2{a}_3^2+a_2^2{a}_4-a_2^3{a}_3+{\displaystyle \frac{1}{5}}{a}_2^5\right).\hfill \end{array}$$

(5)

Similarly, the logarithmic coefficients of the inverse function $f^{-1}$ are defined by

$$F_{f^{-1}}\left(u\right)=log\left({\displaystyle \frac{f^{-1}\left(u\right)}{u}}\right)=2\sum _{n=1}^{\infty }{\Gamma }_n{u}^n,\left|u\right|<{\displaystyle \frac{1}{4}},$$

(6)

where

$$\begin{array}{cc}\hfill {\Gamma }_1& ={\displaystyle \frac{1}{2}}{\mathcal{A}}_2,\hfill \\ \hfill {\Gamma }_2& ={\displaystyle \frac{1}{2}}\left({\mathcal{A}}_3-{\displaystyle \frac{1}{2}}{\mathcal{A}}_2^2\right),\hfill \\ \hfill {\Gamma }_3& ={\displaystyle \frac{1}{2}}\left({\mathcal{A}}_4-{\mathcal{A}}_2{\mathcal{A}}_3+{\displaystyle \frac{1}{3}}{\mathcal{A}}_2^3\right).\hfill \end{array}$$

(7)

Now, let us denote using $\mathbb{B}$ the set of Schwarz functions, i.e., the class of all analytic mappings $\omega :\mathbb{D}\to \mathbb{D}$ that satisfy the condition $\omega \left(0\right)=0$. Each function $\omega \in \mathbb{B}$ can be expanded into a power series of the following form:

$$\omega \left(z\right)=\sum _{n=1}^{\infty }{\sigma }_n{z}^n.$$

(8)

Let f and g be analytic functions defined on the unit disk $\mathbb{D}$. We say that f is subordinate to g in $\mathbb{D}$, denoted by $f\prec g$, if there exists a Schwarz function $\omega \in \mathbb{B}$ such that $f\left(z\right)=g\left(\omega \right(z\left)\right),z\in \mathbb{D}.$ In addition, when g is univalent in $\mathbb{D}$, the condition $f\prec g$ is equivalent to the pair of conditions $f\left(0\right)=g\left(0\right)$ and $f\left(\mathbb{D}\right)\subseteq g\left(\mathbb{D}\right)$.

In this work, we focus on the class written as

$${\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)=\left\{f\in \mathcal{S}:{\displaystyle \frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)-f(-z)}}={\phi }_H\left(\omega \left(z\right)\right),\exists \omega \in \mathbb{B},\forall z\in \mathbb{D}\right\},$$

where

$${\phi }_H\left(z\right)=1+z+{\displaystyle \frac{1}{3}}{z}^2-{\displaystyle \frac{1}{9}}{z}^3,(z\in \mathbb{D}).$$

We have $\Re {\phi }_H^{\prime }\left(z\right)>0$, so ${\phi }_H$ is univalent in $\mathbb{D}$; see Figure 1 and paper [2]. The example below demonstrates that there exists an infinite family of functions belonging to the class ${\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$. Consider the function

$$f_{\rho }\left(z\right)={\displaystyle \frac{z}{1+\rho z}},\left(\right|\rho | \le 1),$$

which is well-known to be a member of the class $\mathcal{S}$ for $\left|\rho \right| \le 1$. Through straightforward calculation, we obtain

$${\Theta }_{\rho }\left(z\right):={\displaystyle \frac{2z{f}_{\rho }^{\prime }\left(z\right)}{f_{\rho }\left(z\right)-f_{\rho }(-z)}}={\displaystyle \frac{1-\rho z}{1+\rho z}},z\in \mathbb{D}.$$

If we take $\left|\rho \right| \le 0.378$, then ${\Theta }_{\rho }\left(z\right)\prec {\phi }_H\left(z\right)$. Therefore,

$$f_{\rho }\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)\mathrm{whenever}\left|\rho \right| \le 0.378.$$

This geometric inclusion can be observed in Figure 2.

Using the same computational approach as outlined above, we also observe that the functions

$$g_1\left(z\right)=e^{{\displaystyle \frac{1}{2}}z}+{\displaystyle \frac{1}{2}}z-1\mathrm{and}{g}_2\left(z\right)=z-{\displaystyle \frac{1}{6}}{z}^2$$

are additional examples of functions that belong to the class ${\mathcal{S}}_{\mathrm{Sym}}^{*}$. In particular, for function $g_1$, we consider the associated expression

$$A\left(z\right)={\displaystyle \frac{z+z{e}^{\frac{1}{2}z}}{z+e^{\frac{1}{2}z}-e^{-\frac{1}{2}z}}},z\in \mathbb{D},$$

which satisfies the subordination relation $A\left(z\right)\prec {\phi }_H\left(z\right)$. Similarly, the function $g_2$ corresponds to

$$B\left(z\right)=1-{\displaystyle \frac{z}{3}},z\in \mathbb{D},$$

and it also holds that $B\left(z\right)\prec {\phi }_H\left(z\right)$. These observations further support the richness of the class ${\mathcal{S}}_{\mathrm{Sym}}^{*}$ and demonstrate the variety of functions it contains.

The intricate relationships between analytic functions and their corresponding coefficient structures have long captured the attention of researchers due to their theoretical and practical significance. Motivated by this, we delve into a modern framework that not only refines classical results but also uncovers sharper bounds by connecting analytic functions with their associated Schwarz functions, ultimately contributing to a deeper understanding of coefficient-related problems in complex analysis. In this study, we introduce a methodology that centers on establishing relationships between the coefficients of functions belonging to a specified analytic class and those of the associated Schwarz functions. In many situations, this framework facilitates the prediction of precise bounds for functionals, thereby simplifying the computational process.

By utilizing several key lemmas, most notably those formulated by Libera and Złotkiewicz and by Prokhorov and Szynal, along with additional analytical tools and a careful refinement of computations, we succeed in obtaining improved, and in most cases sharp, coefficient estimates.

We begin by presenting some essential lemmas concerning Schwarz functions. The first of these is a foundational result due to Prokhorov and Szynal.

Lemma 1

([3]). Let $\omega \in \mathbb{B}$ be of the form (8). Then, for every pair of real numbers α and β satisfying

$$(\alpha ,\beta )\in \left\{\left|\alpha \right| \le {\displaystyle \frac{1}{2}},-1 \le \beta \le 1\right\}\cup \left\{{\displaystyle \frac{1}{2}} \le \left|\alpha \right| \le 2,{\displaystyle \frac{4}{27}}{\left(\right|\alpha |+1)}^3-\left(\right|\alpha |+1) \le \beta \le 1\right\},$$

this implies that the following sharp estimate holds:

$$\left|{\sigma }_3+\alpha {\sigma }_1{\sigma }_2+\beta {\sigma }_1^3\right| \le 1.$$

Using a consequence of the Schwarz–Pick Lemma, it is known that if $\omega \in \mathbb{B}$, then the following inequalities hold:

$$|{\sigma }_1| \le 1\mathrm{and}|{\sigma }_2| \le 1-|{\sigma }_1{| }^2.$$

This bound can be further refined (see, for instance, [4]). In fact, for any complex constant $\lambda$, the following improved estimate is valid:

$$|{\sigma }_2+\lambda {\sigma }_1^2| \le max\{1,\left|\lambda \right|\}.$$

(9)

Carlson presented an alternative extension of the classical Schwarz–Pick Lemma (see [5]). Below, we highlight only those inequalities that are relevant to our current investigation. For a comprehensive treatment and further results, the reader is referred to [6].

Lemma 2

([5]). Let $\omega \in \mathbb{B}$ of the form (8). Then, the following sharp coefficient estimates hold:

$$|{\sigma }_3| \le 1-|{{\sigma }_1|}^2-{\displaystyle \frac{|{\sigma }_2{|}^2}{1+|{\sigma }_1|}},|{\sigma }_4| \le 1-|{{\sigma }_1|}^2-|{{\sigma }_2|}^2,|{\sigma }_5| \le 1-|{{\sigma }_1|}^2-{\left|{\sigma }_2\right|}^2-{\displaystyle \frac{|{\sigma }_3{|}^2}{1+|{\sigma }_1|}}.$$

Lemma 3

([7]). Let $\omega \in \mathbb{B}$ of the form (8). Then,

$$|{\sigma }_1{\sigma }_3-{\sigma }_2^2| \le 1-|{\sigma }_1{|}^2.$$

Lemma 4

([4]). Let $\omega \in \mathbb{B}$ of the form (8) and $\lambda \in \mathbb{C}$. Then,

$$\left|{\sigma }_4+(1+\lambda ){\sigma }_1{\sigma }_3+{\sigma }_2^2+(1+2\lambda ){\sigma }_1^2{\sigma }_2+\lambda {\sigma }_1^4\right| \le max\{1,|\lambda \left|\right\},$$

(10)

and

$$\left|{\sigma }_4+2{\sigma }_1{\sigma }_3+\lambda {\sigma }_2^2+(1+2\lambda ){\sigma }_1^2{\sigma }_2+\lambda {\sigma }_1^4\right| \le max\{1,|\lambda \left|\right\}.$$

(11)

Lemma 5

([5,7]). Let $\omega \in \mathbb{B}$ of the form (8) and $\alpha \in \mathbb{C}$ with $\left|\alpha \right| \le 1$. Then,

$$\left|{\sigma }_5+(1+\alpha ){\sigma }_1{\sigma }_4+(1+\alpha ){\sigma }_2{\sigma }_3+3\alpha {\sigma }_1{\sigma }_2^2+(1+\alpha +{\alpha }^2){\sigma }_1^2{\sigma }_3+2\alpha (1+\alpha ){\sigma }_1^3{\sigma }_2+{\alpha }^2{\sigma }_1^5\right| \le 1.$$

(12)

In the subsequent statement, we present the Hankel determinant, a key concept underpinning our study. For any function

$$h\left(z\right)=\sum _{n=1}^{\infty }{h}_n{z}^n(z\in \mathbb{D}),$$

the qth Hankel determinant was formulated by Noonan and Thomas in [8], as follows:

$$H_{q,n}\left(h\right)=\left|\begin{array}{cccc}{h}_n& h_{n+1}& \dots & h_{n+q-1}\\ h_{n+1}& h_{n+2}& \dots & h_{n+q}\\ ⋮& ⋮& \ddots & ⋮\\ h_{n+q-1}& h_{n+q}& \dots & h_{n+2q-2}\end{array}\right|(q\ge 1).$$

The qth-order Hankel determinant of $f\in \mathbb{A}$ is given by $H_{q,n}\left(f\right):=det{\left\{a_{n+i+j-2}\right\}}_{i,j=1}^q,q\ge 1,$$i,j\ge 1.$

In special cases,

$$H_{2,2}\left(f\right)=a_2{a}_4-a_3^2,\mathrm{and}{H}_{2,1}\left({\displaystyle \frac{F_f}{2}}\right)={\gamma }_1{\gamma }_3-{\gamma }_2^2.$$

(13)

Similarly,

$$H_{2,2}\left(f^{-1}\right)=a_2{a}_4-a_3^2-a_2^2(a_3-a_2^2),$$

(14)

$$H_{2,3}\left(f^{-1}\right)=a_3{a}_5-a_4^2-3a_3^3,$$

(15)

$$H_{2,1}\left({\displaystyle \frac{F_{f^{-1}}}{2}}\right)={\displaystyle \frac{1}{48}}\left(13a_2^4-12a_2^2{a}_3-12a_3^2+12a_2{a}_4\right).$$

(16)

The forthcoming sections present results concerning coefficient problems, Hankel determinants, and Zalcman functionals. Most of these results yield sharp bounds, while a few remain without definitive sharpness. Moreover, we have not explored higher orders in this work, instead leaving them as a potential direction to inspire the reader.

In recent years, considerable attention has been devoted to the study of coefficient problems and functionals such as the Hankel determinant, particularly for several well-established subclasses of analytic functions. Classical contributions by Hayman [9], Janteng et al. [10], and Babalola [11] laid the groundwork for bounding these determinants in classes such as $\mathcal{S}$, $\mathcal{K}$, and $\mathcal{C}$. These results were subsequently refined by Zaprawa and collaborators [6,12], who provided sharper bounds for the third-order Hankel determinant $H_{3,1}$ in various subclasses.

Further investigations explored determinant bounds for functions related to sigmoid-type mappings [13,14], lemniscate domains [15], and petal-shaped regions [1,16]. More recent advancements were achieved through the estimation of second- and third-order Hankel determinants for additional subclasses (see [17,18,19,20,21,22,23]), thus broadening the geometric scope of coefficient estimates in complex function theory.

Motivated by these advances and the structure of the symmetric class ${\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$ defined via subordination to the univalent function ${\phi }_H$, we aim to determine sharp upper bounds for the initial coefficients $|a_2|$, $|a_3|$, and $|a_4|$, as well as for the Hankel determinants $|H_{2,1}\left(f\right)|$, $|H_{2,2}\left(f\right)|$, and $|H_{3,1}\left(f\right)|$. Our approach blends classical analytic techniques with modern subordination theory and inequalities associated with Schwarz functions, offering new insights and refined bounds in the geometric theory of analytic functions.

2. Coefficient Bounds

In this section, we obtained estimates for the initial coefficients and the logarithmic coefficients.

Theorem 1.

If $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$, then

$$|a_2| \le {\displaystyle \frac{1}{2}},|a_3| \le {\displaystyle \frac{1}{2}},|a_4| \le {\displaystyle \frac{1}{4}},\left|a_5\right| \le {\displaystyle \frac{29}{112}}.$$

The bounds for $a_k$ ($k=2,3,4$) are sharp.

Proof. 

Since $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$, then there exists a Schwarz function $\omega \in \mathbb{B}$ of the form (8) such that

$${\displaystyle \frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)-f(-z)}}={\phi }_H\left(\omega \left(z\right)\right).$$

Thus, we have

$$\begin{array}{cc}\hfill 1+2a_{2}z+2a_3{z}^2+(-2a_2{a}_3+4a_4)z^3+(-2a_3^2+4a_5)z^4+(-4a_3{a}_4+2a_2{a}_3^2-2a_2{a}_5& \\ \hfill +6a_6)z^5+\dots =1+{\sigma }_{1}z+\left({\displaystyle \frac{{\sigma }_1^2}{3}}+{\sigma }_2\right)z^2+\left(-{\displaystyle \frac{{\sigma }_1^3}{9}}+{\displaystyle \frac{2{\sigma }_1{\sigma }_2}{3}}+{\sigma }_3\right)z^3+(-{\displaystyle \frac{1}{3}}{\sigma }_1^2{\sigma }_2& \\ \hfill +{\displaystyle \frac{1}{3}}({\sigma }_2^2+2{\sigma }_1{\sigma }_3)+{\sigma }_4)z^4+\left(-{\displaystyle \frac{1}{3}}{\sigma }_1({\sigma }_2^2+{\sigma }_1{\sigma }_3)+{\displaystyle \frac{1}{3}}(2{\sigma }_2{\sigma }_3+2{\sigma }_1{\sigma }_4)+{\sigma }_5\right)z^5+\dots .\end{array}$$

(17)

Comparing the coefficients yields

$$a_2={\displaystyle \frac{{\sigma }_1}{2}},$$

(18)

$$a_3={\displaystyle \frac{1}{6}}({\sigma }_1^2+3{\sigma }_2),$$

(19)

$$a_4={\displaystyle \frac{1}{72}}({\sigma }_1^3+21{\sigma }_1{\sigma }_2+18{\sigma }_3),$$

(20)

$$a_5={\displaystyle \frac{1}{72}}({\sigma }_1^4+15{\sigma }_2^2+12{\sigma }_1{\sigma }_3+18{\sigma }_4),$$

(21)

$$a_6={\displaystyle \frac{1}{1296}}(-{\sigma }_1^5+12{\sigma }_1^3{\sigma }_2+45{\sigma }_1{\sigma }_2^2+252{\sigma }_2{\sigma }_3+198{\sigma }_1{\sigma }_4+216{\sigma }_5).$$

(22)

Clearly,

$$|a_2| \le {\displaystyle \frac{1}{2}}\mathrm{and}\left|a_3\right| \le {\displaystyle \frac{1}{2}}.$$

Applying Lemma 1 with $\alpha ={\displaystyle \frac{7}{6}}$ and $\beta ={\displaystyle \frac{1}{18}}$, we have

$$|a_4| ={\displaystyle \frac{1}{4}}\left|{\sigma }_3+{\displaystyle \frac{7}{6}}{\sigma }_1{\sigma }_2+{\displaystyle \frac{1}{18}}{\sigma }_1^3\right| \le {\displaystyle \frac{1}{4}}.$$

From (21), we have

$$4|a_5| =\left|{\displaystyle \frac{1}{18}}{\sigma }_1^4+{\displaystyle \frac{5}{6}}{\sigma }_2^2+{\displaystyle \frac{2}{3}}{\sigma }_1{\sigma }_3+{\sigma }_4\right|,$$

(23)

so, we can rewrite (23) as follows:

$$4|a_5| =\left|{\displaystyle \frac{1}{2}}\left({\sigma }_4+{\displaystyle \frac{4}{3}}{\sigma }_1{\sigma }_3+{\sigma }_2^2+{\displaystyle \frac{5}{3}}{\sigma }_1^2{\sigma }_2+{\displaystyle \frac{1}{3}}{\sigma }_1^4\right)+{\displaystyle \frac{1}{2}}\left({\sigma }_4+{\displaystyle \frac{2}{3}}{\sigma }_2^2-{\displaystyle \frac{5}{3}}{\sigma }_1^2{\sigma }_2-{\displaystyle \frac{1}{9}}{\sigma }_1^4\right)\right|.$$

Applying Lemma 4 with $\lambda ={\displaystyle \frac{1}{3}}$, we obtain

$${\displaystyle \frac{1}{2}}\left|{\sigma }_4+{\displaystyle \frac{4}{3}}{\sigma }_1{\sigma }_3+{\sigma }_2^2+{\displaystyle \frac{5}{3}}{\sigma }_1^2{\sigma }_2+{\displaystyle \frac{1}{3}}{\sigma }_1^4\right| \le {\displaystyle \frac{1}{2}}.$$

Further, Lemma 2 leads to

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\left|{\sigma }_4+{\displaystyle \frac{2}{3}}{\sigma }_2^2-{\displaystyle \frac{5}{3}}{\sigma }_1^2{\sigma }_2-{\displaystyle \frac{1}{9}}{\sigma }_1^4\right|& \le {\displaystyle \frac{1}{2}}\left[1-|{{\sigma }_1|}^2-|{{\sigma }_2|}^2+{\displaystyle \frac{2}{3}}{|{\sigma }_2|}^2+{\displaystyle \frac{5}{3}}{|{\sigma }_1|}^2(1-|{{\sigma }_1|}^2)+{\displaystyle \frac{1}{9}}{\left|{\sigma }_1\right|}^4\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left[1+{\displaystyle \frac{2}{3}}{|{\sigma }_1|}^2-{\displaystyle \frac{1}{3}}{|{\sigma }_2|}^2-{\displaystyle \frac{14}{9}}{\left|{\sigma }_1\right|}^4\right]\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}{|{\sigma }_1|}^2-{\displaystyle \frac{7}{9}}{\left|{\sigma }_1\right|}^4:=\varrho \left(x\right),\hfill \end{array}$$

where $\varrho \left(x\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}{x}^2-{\displaystyle \frac{7}{9}}{x}^4$ and $0 \le x \le 1$. After several straightforward calculations, we find that the maximum value of the function $\varrho \left(x\right)$ is ${\displaystyle \frac{15}{28}}$. In conclusion, we obtain the bound

$$|a_5| \le {\displaystyle \frac{29}{112}}.$$

Moreover, when ${\sigma }_1=1$ and ${\sigma }_n=0$ for all $n\ne 1$, it follows that $|a_2| ={\displaystyle \frac{1}{2}}$. Also, if ${\sigma }_2=1$ and ${\sigma }_n=0$ for all $n\ne 2$, then $|a_3| ={\displaystyle \frac{1}{2}}$. Similarly, if ${\sigma }_3=1$ and ${\sigma }_n=0$ for all $n\ne 3$, it follows that $|a_4| ={\displaystyle \frac{1}{4}}$. □

Theorem 2.

If $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$, then

$$|{\gamma }_1| \le {\displaystyle \frac{1}{4}},|{\gamma }_2| \le {\displaystyle \frac{1}{4}},|{\gamma }_3| \le {\displaystyle \frac{1}{8}},|{\gamma }_4| \le {\displaystyle \frac{1}{8}},\left|{\gamma }_5\right| \le {\displaystyle \frac{34973}{311040}}.$$

The bounds for ${\gamma }_k$ (for $k=1,2,3,4$) are sharp.

Proof. 

By utilizing (5) and the coefficient expressions from (18)–(22), we obtain:

$${\gamma }_1={\displaystyle \frac{1}{4}}{\sigma }_1,$$

(24)

$${\gamma }_2={\displaystyle \frac{1}{4}}\left({\sigma }_2+{\displaystyle \frac{1}{12}}{\sigma }_1^2\right),$$

(25)

$${\gamma }_3={\displaystyle \frac{1}{8}}\left({\sigma }_3+{\displaystyle \frac{1}{6}}{\sigma }_1{\sigma }_2-{\displaystyle \frac{1}{9}}{\sigma }_1^3\right),$$

(26)

$${\gamma }_4={\displaystyle \frac{1}{8}}\left({\sigma }_4+{\displaystyle \frac{1}{6}}{\sigma }_1{\sigma }_3+{\displaystyle \frac{1}{3}}{\sigma }_2^2-{\displaystyle \frac{5}{12}}{\sigma }_1^2{\sigma }_2+{\displaystyle \frac{11}{144}}{\sigma }_1^4\right),$$

(27)

$${\gamma }_5={\displaystyle \frac{1}{12}}\left({\sigma }_5+{\displaystyle \frac{1}{6}}{\sigma }_1{\sigma }_4+{\displaystyle \frac{5}{12}}{\sigma }_2{\sigma }_3+{\displaystyle \frac{41}{144}}{\sigma }_1^3{\sigma }_2-{\displaystyle \frac{3}{8}}{\sigma }_1^2{\sigma }_3-{\displaystyle \frac{13}{24}}{\sigma }_1{\sigma }_2^2-{\displaystyle \frac{47}{1080}}{\sigma }_1^5\right).$$

(28)

Obviously, we have

$$|{\gamma }_1| \le {\displaystyle \frac{1}{4}},\mathrm{and}\left|{\gamma }_2\right| \le {\displaystyle \frac{1}{4}}.$$

By taking $\alpha ={\displaystyle \frac{1}{6}}$ and $\beta =-{\displaystyle \frac{1}{9}}$ into Lemma 1, we obtain

$$|{\gamma }_3| \le {\displaystyle \frac{1}{8}}.$$

Note that

$$\begin{array}{cc}\hfill \left|{\sigma }_4+{\displaystyle \frac{1}{6}}{\sigma }_1{\sigma }_3+{\displaystyle \frac{1}{3}}{\sigma }_2^2-{\displaystyle \frac{5}{12}}{\sigma }_1^2{\sigma }_2+{\displaystyle \frac{11}{144}}{\sigma }_1^4\right|=|& {\displaystyle \frac{1}{2}}\left({\sigma }_4+{\displaystyle \frac{1}{3}}{\sigma }_1{\sigma }_3+{\sigma }_2^2-{\displaystyle \frac{1}{3}}{\sigma }_1^2{\sigma }_2-{\displaystyle \frac{2}{3}}{\sigma }_1^4\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left({\sigma }_4-{\displaystyle \frac{1}{3}}{\sigma }_2^2-{\displaystyle \frac{1}{2}}{\sigma }_1^2{\sigma }_2+{\displaystyle \frac{59}{72}}{\sigma }_1^4\right)|.\hfill \end{array}$$

Hence, applying Lemma 4 with $\lambda =-{\displaystyle \frac{2}{3}}$, we obtain

$${\displaystyle \frac{1}{2}}\left|{\sigma }_4+{\displaystyle \frac{1}{3}}{\sigma }_1{\sigma }_3+{\sigma }_2^2-{\displaystyle \frac{1}{3}}{\sigma }_1^2{\sigma }_2-{\displaystyle \frac{2}{3}}{\sigma }_1^4\right| \le {\displaystyle \frac{1}{2}}.$$

Also, Lemma 2 yields

$${\displaystyle \frac{1}{2}}\left|{\sigma }_4-{\displaystyle \frac{1}{3}}{\sigma }_2^2-{\displaystyle \frac{1}{2}}{\sigma }_1^2{\sigma }_2+{\displaystyle \frac{59}{72}}{\sigma }_1^4\right| \le {\displaystyle \frac{1}{2}}\left[1-|{{\sigma }_1|}^2-|{{\sigma }_2|}^2+{\displaystyle \frac{1}{3}}{|{\sigma }_2|}^2+{\displaystyle \frac{1}{2}}{|{\sigma }_1|}^2(1-|{{\sigma }_1|}^2)+{\displaystyle \frac{59}{72}}{\left|{\sigma }_1\right|}^4\right]$$

$$={\displaystyle \frac{1}{2}}\left[1-{\displaystyle \frac{1}{2}}{|{\sigma }_1|}^2-{\displaystyle \frac{2}{3}}{|{\sigma }_2|}^2+{\displaystyle \frac{23}{72}}{\left|{\sigma }_1\right|}^4\right] \le {\displaystyle \frac{1}{2}}.$$

Therefore, we conclude that

$$|{\gamma }_4| \le {\displaystyle \frac{1}{8}}.$$

From (28), we have

$$\begin{array}{cc}\hfill 12|{\gamma }_5|& =|{\sigma }_5+\frac{1}{6}{\sigma }_1{\sigma }_4+\frac{5}{12}{\sigma }_2{\sigma }_3+\frac{41}{144}{\sigma }_1^3{\sigma }_2-\frac{3}{8}{\sigma }_1^2{\sigma }_3-\frac{13}{24}{\sigma }_1{\sigma }_2^2-\frac{47}{1080}{\sigma }_1^5|\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}|\left({\sigma }_5+\frac{1}{3}{\sigma }_1{\sigma }_4+\frac{1}{3}{\sigma }_2{\sigma }_3-2{\sigma }_1{\sigma }_2^2+\frac{7}{9}{\sigma }_1^2{\sigma }_3-\frac{4}{9}{\sigma }_1^3{\sigma }_2+\frac{4}{9}{\sigma }_1^5\right)\hfill \\ \hfill & +\left({\sigma }_5+\frac{1}{2}{\sigma }_2{\sigma }_3+\frac{11}{12}{\sigma }_1{\sigma }_2^2-\frac{55}{36}{\sigma }_1^2{\sigma }_3+\frac{73}{72}{\sigma }_1^3{\sigma }_2-\frac{287}{540}{\sigma }_1^5\right)|.\hfill \end{array}$$

By applying Lemma 5 with $\alpha =-{\displaystyle \frac{2}{3}}$, we obtain

$$12|{\gamma }_5| \le {\displaystyle \frac{1}{2}}(1+A),$$

where

$$A=\left|{\sigma }_5+{\displaystyle \frac{1}{2}}{\sigma }_2{\sigma }_3+{\displaystyle \frac{11}{12}}{\sigma }_1{\sigma }_2^2-{\displaystyle \frac{55}{36}}{\sigma }_1^2{\sigma }_3+{\displaystyle \frac{73}{72}}{\sigma }_1^3{\sigma }_2-{\displaystyle \frac{287}{540}}{\sigma }_1^5\right|.$$

By applying the triangle inequality and using Lemma 2, we have

$$A \le 1-|{{\sigma }_1|}^2-|{{\sigma }_2|}^2-{\displaystyle \frac{|{\sigma }_3{|}^2}{1+|{\sigma }_1|}}+{\displaystyle \frac{1}{2}}\left(\right|{\sigma }_2|+\frac{55}{18}{|{\sigma }_1|}^2\left)\right|{\sigma }_3|+\frac{73}{72}{|{\sigma }_1|}^3|{\sigma }_2|+\frac{11}{12}|{\sigma }_1\left|\right|{{\sigma }_2|}^2+\frac{287}{540}{\left|{\sigma }_1\right|}^5.$$

The maximum value of the above expression with respect to $\left|{\sigma }_3\right|$ is attained when

$$|{\sigma }_3| ={\displaystyle \frac{1}{4}}\left(\right|{\sigma }_2|+\frac{55}{18}{|{\sigma }_1|}^2\left)\right(1+|{\sigma }_1\left|\right),$$

that is,

$$A \le 1-|{{\sigma }_1|}^2+{\displaystyle \frac{3025}{5184}}{|{\sigma }_1|}^4+{\displaystyle \frac{28901}{25920}}{|{\sigma }_1|}^5+{\displaystyle \frac{55}{144}}{|{\sigma }_1|}^2|{\sigma }_2|+{\displaystyle \frac{67}{48}}{|{\sigma }_1|}^3|{\sigma }_2|-{\displaystyle \frac{15}{16}}{|{\sigma }_2|}^2+{\displaystyle \frac{47}{48}}|{\sigma }_1\left|\right|{{\sigma }_2|}^2.$$

Since $|{\sigma }_1{|}^3|{\sigma }_2| \le |{\sigma }_1{|}^2\left|{\sigma }_2\right|$, we obtain

$$A \le 1-|{{\sigma }_1|}^2+{\displaystyle \frac{3025}{5184}}{|{\sigma }_1|}^4+{\displaystyle \frac{28901}{25920}}{|{\sigma }_1|}^5+{\displaystyle \frac{16}{9}}{|{\sigma }_1|}^2|{\sigma }_2|-{\displaystyle \frac{15}{16}}{|{\sigma }_2|}^2+{\displaystyle \frac{47}{48}}|{\sigma }_1\left|\right|{{\sigma }_2|}^2.$$

Consider the domain $\Lambda =\left\{(\sigma ,\mathsf{d}):0 \le \sigma \le 1,0 \le \mathsf{d} \le 1-{\sigma }^2\right\},$ and define the real-valued function $h:\Lambda \to \mathbb{R}$ as follows:

$$h(\sigma ,\mathsf{d})=1-{\sigma }^2+{\displaystyle \frac{3025}{5184}}{\sigma }^4+{\displaystyle \frac{28901}{25920}}{\sigma }^5+{\displaystyle \frac{16}{9}}{\sigma }^2\mathsf{d}-{\displaystyle \frac{15}{16}}{\mathsf{d}}^2+{\displaystyle \frac{47}{48}}\sigma {\mathsf{d}}^2.$$

It follows that $A \le h(\sigma ,\mathsf{d})$, where the parameters $\sigma$ and $\mathsf{d}$ represent the moduli $|{\sigma }_1|$ and $|{\sigma }_2|$, respectively. After straightforward calculations, the maximum value of $h(\sigma ,\mathsf{d})$ is attained at $\sigma =1$ and $\mathsf{d}=0$, so

$$A \le h(1,0)={\displaystyle \frac{22013}{12960}}.$$

Therefore,

$$|{\gamma }_5| \le {\displaystyle \frac{34973}{311040}}\approx 0.11244.$$

Certainly, the function $\omega \left(z\right)=z^k$ attains the sharpness of the bounds ${\gamma }_k$ for $k=1,2,3,4$. □

3. Inequalities Involving Zalcman Functionals and Second Hankel Determinants

In this section, we establish upper bounds for the Zalcman functionals $a_{k+n-1}-a_k{a}_n,n,k\in \mathbb{N}=\{1,2,\dots \}$ and the second Hankel determinants.

Theorem 3.

If $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$, then the following sharp estimates hold:

$$|a_3-a_2^2| \le {\displaystyle \frac{1}{2}},\mathit{and}|a_4-a_2{a}_3| \le {\displaystyle \frac{1}{4}}.$$

Proof. 

It follows from Equations (9), (18) and (19) that the first inequality is satisfied. Also, using (18)–(20), we obtain

$$|a_4-a_2{a}_3| ={\displaystyle \frac{1}{4}}\left|{\sigma }_3+{\displaystyle \frac{1}{6}}{\sigma }_1{\sigma }_2-{\displaystyle \frac{5}{18}}{\sigma }_1^3\right|.$$

By setting $\alpha ={\displaystyle \frac{1}{6}}$ and $\beta =-{\displaystyle \frac{5}{18}}$ in Lemma 1, the second inequality follows. The sharpness of both inequalities is achieved by choosing $\omega \left(z\right)=z^3$ and $\omega \left(z\right)=z^4$, respectively. □

Theorem 4.

If $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$, then

$$|H_{2,2}\left(f\right)| \le {\displaystyle \frac{1}{4}}.$$

(29)

Proof. 

If $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$, then

$$|H_{2,2}\left(f\right)| =|a_2{a}_4-a_3^2| ={\displaystyle \frac{1}{4}}\left|{\sigma }_2^2-{\displaystyle \frac{1}{2}}{\sigma }_1{\sigma }_3+{\displaystyle \frac{1}{12}}{\sigma }_1^4+{\displaystyle \frac{1}{12}}{\sigma }_1^2{\sigma }_2\right|.$$

We can express this as

$$|H_{2,2}\left(f\right)| ={\displaystyle \frac{1}{4}}\left|{\displaystyle \frac{1}{2}}({\sigma }_2^2-{\sigma }_1{\sigma }_3)+{\displaystyle \frac{1}{2}}\left({\sigma }_2^2+{\displaystyle \frac{1}{6}}{\sigma }_1^4+{\displaystyle \frac{1}{6}}{\sigma }_1^2{\sigma }_2\right)\right| \le {\displaystyle \frac{1}{4}}\left({\displaystyle \frac{1}{2}}+A\right),$$

where the first component is bounded using Lemma 3. For the second component, we estimate:

$$A \le {\displaystyle \frac{1}{2}}\left[(1-|{{\sigma }_1|}^2{)}^2+{\displaystyle \frac{1}{6}}{|{\sigma }_1|}^4+{\displaystyle \frac{1}{6}}{|{\sigma }_1|}^2(1-|{\sigma }_1{|}^2)\right]={\displaystyle \frac{1}{2}}\left[1-{\displaystyle \frac{11}{6}}{|{\sigma }_1|}^2+{\left|{\sigma }_1\right|}^4\right] \le {\displaystyle \frac{1}{2}}.$$

Thus, the inequality (29) is satisfied. The sharpness is satisfied whenver $\omega \left(z\right)=z^2$. □

Theorem 5.

If $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$, then

$$|H_{2,1}(F_f/2)| \le {\displaystyle \frac{1}{16}}.$$

(30)

Proof. 

If $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$, then

$$\begin{array}{cc}\hfill |H_{2,1}(F_f/2)|& =|{\gamma }_1{\gamma }_3-{\gamma }_2^2|={\displaystyle \frac{1}{16}}\left|{\displaystyle \frac{1}{16}}{\sigma }_1^4+{\displaystyle \frac{1}{12}}{\sigma }_1^2{\sigma }_2+{\sigma }_2^2-{\displaystyle \frac{1}{2}}{\sigma }_1{\sigma }_3\right|\hfill \\ \hfill & ={\displaystyle \frac{1}{16}}\left|{\displaystyle \frac{1}{2}}({\sigma }_2^2-{\sigma }_1{\sigma }_3)+{\displaystyle \frac{1}{2}}\left({\sigma }_2^2+{\displaystyle \frac{1}{8}}{\sigma }_1^4+{\displaystyle \frac{1}{6}}{\sigma }_1^2{\sigma }_2\right)\right| \le {\displaystyle \frac{1}{16}}\left({\displaystyle \frac{1}{2}}+A\right).\hfill \end{array}$$

Lemma 3 has bounded the first term; therefore, it remains to show that $A \le {\displaystyle \frac{1}{2}}$, so

$$A \le {\displaystyle \frac{1}{2}}\left[(1-|{{\sigma }_1|}^2{)}^2+{\displaystyle \frac{1}{8}}{|{\sigma }_1|}^4+{\displaystyle \frac{1}{6}}{|{\sigma }_1|}^2(1-|{\sigma }_1{|}^2)\right]={\displaystyle \frac{1}{2}}\left[1-{\displaystyle \frac{11}{6}}{|{\sigma }_1|}^2+{\displaystyle \frac{23}{24}}{\left|{\sigma }_1\right|}^4\right] \le {\displaystyle \frac{1}{2}}.$$

When $\omega \left(z\right)=z^2$ is chosen, it becomes evident that the bound is sharp. □

4. Second Hankel Determinants for Inverse Functions

In this part, we present the sharp bounds of the second-order Hankel determinants for the inverses of functions belonging to the class ${\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$.

Theorem 6.

Let $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$. Then

$$|H_{2,2}\left(f^{-1}\right)| \le {\displaystyle \frac{1}{4}}.$$

(31)

Proof. 

For functions belonging to the class ${\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$, we examine the second-order Hankel determinant of the inverse function, and find:

$$|H_{2,2}\left(f^{-1}\right)| =\left|a_2{a}_4-a_3^2-a_2^2(a_3-a_2^2)\right|={\displaystyle \frac{1}{4}}\left|-{\displaystyle \frac{7}{12}}{\sigma }_1^2{\sigma }_2-{\sigma }_2^2+{\displaystyle \frac{1}{2}}{\sigma }_1{\sigma }_3\right|.$$

To proceed, we rewrite this expression as follows:

$${\displaystyle \frac{1}{4}}\left|{\displaystyle \frac{1}{2}}({\sigma }_2^2-{\sigma }_1{\sigma }_3)+{\displaystyle \frac{1}{2}}\left({\sigma }_2^2+{\displaystyle \frac{7}{6}}{\sigma }_1^2{\sigma }_2\right)\right|,$$

leading to

$$|H_{2,2}\left(f^{-1}\right)| \le {\displaystyle \frac{1}{4}}\left({\displaystyle \frac{1}{2}}+A\right),$$

where the term ${\displaystyle \frac{1}{2}}({\sigma }_2^2-{\sigma }_1{\sigma }_3)$ is bounded using the Lemma 3. For the remaining component, an application of classical coefficient bounds yields the following:

$$A \le {\displaystyle \frac{1}{2}}\left[(1-|{{\sigma }_1|}^2{)}^2+{\displaystyle \frac{7}{6}}{|{\sigma }_1|}^2(1-|{\sigma }_1{|}^2)\right]={\displaystyle \frac{1}{2}}\left[1-{\displaystyle \frac{5}{6}}{|{\sigma }_1|}^2-{\displaystyle \frac{1}{6}}{\left|{\sigma }_1\right|}^4\right] \le {\displaystyle \frac{1}{2}}.$$

Thus, the inequality (31) is satisfied, and its sharpness is also attained when $\omega \left(z\right)=z^2$. □

Theorem 7.

Let $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$. Then,

$$\left|H_{2,1}\left({\displaystyle \frac{F_{f^{-1}}}{2}}\right)\right| \le {\displaystyle \frac{1}{16}}.$$

(32)

Proof. 

Consider $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$. From Equations (16) and (18)–(20), we derive

$$\begin{array}{cc}\hfill \left|H_{2,1}\left({\displaystyle \frac{F_{f^{-1}}}{2}}\right)\right|& ={\displaystyle \frac{1}{16}}\left|{\sigma }_2^2-{\displaystyle \frac{1}{2}}{\sigma }_1{\sigma }_3-{\displaystyle \frac{1}{48}}{\sigma }_1^4+{\displaystyle \frac{7}{12}}{\sigma }_1^2{\sigma }_2\right|\hfill \\ \hfill & ={\displaystyle \frac{1}{16}}\left|{\displaystyle \frac{1}{2}}({\sigma }_2^2-{\sigma }_1{\sigma }_3)+{\displaystyle \frac{1}{2}}\left({\sigma }_2^2-{\displaystyle \frac{1}{24}}{\sigma }_1^4+{\displaystyle \frac{7}{6}}{\sigma }_1^2{\sigma }_2\right)\right|.\hfill \end{array}$$

It follows

$$\left|H_{2,1}\left({\displaystyle \frac{F_{f^{-1}}}{2}}\right)\right| \le {\displaystyle \frac{1}{16}}\left({\displaystyle \frac{1}{2}}+A\right),$$

where the first term ${\displaystyle \frac{1}{2}}({\sigma }_2^2-{\sigma }_1{\sigma }_3)$ is controlled by Lemma 3, and the second term satisfies the following:

$$A \le {\displaystyle \frac{1}{2}}\left[(1-|{{\sigma }_1|}^2{)}^2+{\displaystyle \frac{1}{24}}{|{\sigma }_1|}^4+{\displaystyle \frac{7}{6}}{|{\sigma }_1|}^2(1-|{\sigma }_1{|}^2)\right]={\displaystyle \frac{1}{2}}\left[1-{\displaystyle \frac{5}{6}}{|{\sigma }_1|}^2-{\displaystyle \frac{1}{8}}{\left|{\sigma }_1\right|}^4\right] \le {\displaystyle \frac{1}{2}}.$$

Thus, the inequality (32) is satisfied, and its sharpness is also attained when $\omega \left(z\right)=z^2$. □

5. Third Hankel Determinant for ${\mathcal{S}}_{\mathbf{sym}}^{\mathbf{*}}$

In this final section, we present a single result provides the best possible bound for the third-order Hankel determinant for the class ${\mathcal{S}}_{\mathrm{sym}}^{*}$. However, since the result is not sharp, we also establish a conjecture below it.

Theorem 8.

If $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$, then

$$|H_{3,1}\left(f\right)| \le {\displaystyle \frac{239+41\sqrt{41}}{4800}}\approx 0.1045\dots .$$

(33)

Proof. 

If $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$, then

$$\begin{array}{cc}\hfill |H_{3,1}\left(f\right)|& =\left|2a_2{a}_3{a}_4-a_3^3-a_4^2-a_3{a}_5-a_2^2{a}_5\right|\hfill \\ \hfill & ={\displaystyle \frac{1}{16}}|2{\sigma }_2{\sigma }_4-{\sigma }_3^2+\left({\sigma }_2+{\displaystyle \frac{1}{3}}{\sigma }_1^2\right){\sigma }_1{\sigma }_3-{\displaystyle \frac{47}{36}}{\sigma }_1^2{\sigma }_2^2+{\displaystyle \frac{11}{54}}{\sigma }_1^4{\sigma }_2-{\displaystyle \frac{19}{324}}{\sigma }_1^6-{\displaystyle \frac{1}{3}}{\sigma }_2^3-{\displaystyle \frac{1}{3}}{\sigma }_1^2{\sigma }_4|.\hfill \end{array}$$

By applying Lemma 1 with $\alpha =-1$ and $\beta =-{\displaystyle \frac{1}{3}}$, we obtain

$$\left|-{\sigma }_3^2+\left({\sigma }_2+{\displaystyle \frac{1}{3}}{\sigma }_1^2\right){\sigma }_1{\sigma }_3\right|=\left|{\sigma }_3\left({\sigma }_3-{\sigma }_1{\sigma }_2-{\displaystyle \frac{1}{3}}{\sigma }_1^3\right)\right| \le \left|{\sigma }_3\right|.$$

Consequently, it follows that

$$|H_{3,1}\left(f\right)| \le {\displaystyle \frac{1}{16}}\left[2|{\sigma }_2\left|\right|{\sigma }_4|+|{\sigma }_3|+{\displaystyle \frac{47}{36}}{|{\sigma }_1|}^2{|{\sigma }_2|}^2+{\displaystyle \frac{11}{54}}{|{\sigma }_1|}^4|{\sigma }_2|+{\displaystyle \frac{19}{324}}{|{\sigma }_1|}^6+{\displaystyle \frac{1}{3}}{|{\sigma }_2|}^3+{\displaystyle \frac{1}{3}}{|{\sigma }_1|}^2\left|{\sigma }_4\right|\right].$$

It follows from Lemma 2 that

$$|{\sigma }_3| \le 1 - |{{\sigma }_1|}^2-{\displaystyle \frac{1}{2}}{|{\sigma }_2|}^2,\mathrm{and}|{\sigma }_4| \le 1 - |{{\sigma }_1|}^2-{\left|{\sigma }_2\right|}^2.$$

Hence,

$$|H_{3,1}\left(f\right)| \le h(\sigma ,\mathsf{d}),$$

where $0 \le \sigma =|{\sigma }_1| \le 1$, $0 \le \mathsf{d}=|{\sigma }_2| \le 1-{\sigma }^2$, and

$$h(\sigma ,\mathsf{d})={\displaystyle \frac{1}{16}}\left[1+{\displaystyle \frac{19}{324}}{\sigma }^6+2\mathsf{d}-{\displaystyle \frac{1}{2}}{\mathsf{d}}^2-{\displaystyle \frac{5}{3}}{\mathsf{d}}^3+{\displaystyle \frac{1}{54}}{\sigma }^4(-18+11\mathsf{d})+{\sigma }^2\left(-{\displaystyle \frac{2}{3}}-\mathsf{d}+{\displaystyle \frac{35}{36}}{\mathsf{d}}^2\right)\right].$$

Computations using a computer leads to the maximum value of $h(\sigma ,\mathsf{d})$ as

$$h\left(0,{\displaystyle \frac{-1+\sqrt{41}}{10}}\right)={\displaystyle \frac{239+41\sqrt{41}}{4800}}\approx 0.1045\dots .$$

Thus, the proof of inequality (33) is complete. □

Conjecture 1. If $f\in {\mathcal{S}}_{\mathrm{sym}}^{*}\left({\phi }_H\right)$, then

$$|H_{3,1}\left(f\right)| \le {\displaystyle \frac{1}{16}}.$$

(34)

6. Conclusions

In conclusion, we introduced a new subclass of starlike functions with respect to symmetric points within a recently established domain. Sharp bounds for both the coefficients and logarithmic coefficients were derived. Furthermore, employing a novel and concise proof technique, we obtained exact bounds for the second-order Hankel determinants. The best known estimate for the third-order Hankel determinant was also established, accompanied by a conjecture for its sharp bound to stimulate further research. This work opens several avenues for future investigations, including the study of higher-order Hankel determinants, proving the stated conjecture, or exploring analogous function classes within the same domain. We believe these results provide valuable contributions and serve as a foundational reference for subsequent studies related to this domain.