Inverse and Logarithmic Coefficient Bounds of Concave Univalent Functions

Abstract

The concept of coefficient estimates on univalent functions is one of the interesting aspects explored recently by many researchers. Motivated by this direction, in this present work, we obtain the upper bounds of initial inverse coefficients and logarithmic coefficients and the upper bounds of differences between these successive coefficients related to concave univalent functions. Further, we also calculate the upper bounds of third-order Hankel, Toeplitz, and Vandermonde determinants in terms of specified coefficients connected to concave univalent functions.

1. Introduction

Define the class $\mathcal{S}$ consisting of functions expressed in the form

$$h\left(z\right)=z+\sum _{m=2}^{\infty }{a}_m{z}^m,$$

(1)

which are analytic and univalent within the open unit disk $\mathbb{D}=\left\{z:z\in \mathbb{C},|z|<1\right\}$ and normalized by the conditions $h\left(0\right)=0=h^{\prime }\left(0\right)-1$. Then, ${\mathcal{S}}^{*}$ is identified as the subclass of $\mathcal{S}$ that consists of starlike functions, i.e., the class of functions $h\in \mathcal{S}$ satisfying

$$Re\left[{\displaystyle \frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}}\right]>0.$$

The Koebe one-quarter theorem [1] asserts that the image of $\mathbb{D}$ under any function h in the class $\mathcal{S}$ is associated with an inverse function $h^{-1}$ that complies with the conditions

$$h^{-1}\left(h\left(z\right)\right)=z,z\in \mathbb{D}$$

and

$$h\left(h^{-1}\left(w\right)\right)=w,|w|<r_0\left(h\right);r_0\left(h\right)\ge 1/4,$$

where

$$h^{-1}\left(w\right)=g\left(w\right)=w+\sum _{m=2}^{\infty }{b}_m{w}^m.$$

By simple computation, one can get the first five inverse coefficients as

$$\begin{array}{ccc}\hfill b_1& =& 1,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill b_2& =& -a_2,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill b_3& =& 2a_2^2-a_3,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill b_4& =& 5a_2{a}_3-5a_2^3-a_4\hfill \end{array}$$

(5)

and

$$b_5=6a_2{a}_4+3a_3^2+14a_2^4-21a_2^2{a}_3-a_5.$$

(6)

Also, the logarithmic coefficients ${\gamma }_n$ of $h\in \mathcal{S}$ are defined with the following series expansion:

$$F_h\left(z\right)=log\left({\displaystyle \frac{h\left(z\right)}{z}}\right)=2\sum _{n=1}^{\infty }{\gamma }_n\left(h\right)z^n;z\in \mathbb{D}\setminus \left\{0\right\},log1=0.$$

These coefficients play an important role for various estimates in the theory of univalent functions. Note that we use ${\gamma }_n$ instead of ${\gamma }_n\left(h\right)$. The study of logarithmic coefficients is indeed interesting as it helped Kayumov [2] to solve Brannan’s conjecture for conformal mappings. The first three initial logarithmic coefficients are given by

$$\begin{array}{ccc}\hfill {\gamma }_1& =& {\displaystyle \frac{1}{2}}{a}_2,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\gamma }_2& =& {\displaystyle \frac{1}{2}}\left[a_3-{\displaystyle \frac{1}{2}}{a}_2^2\right]\hfill \end{array}$$

(8)

and

$${\gamma }_3={\displaystyle \frac{1}{2}}\left[a_4-a_2{a}_3+{\displaystyle \frac{1}{3}}{a}_2^3\right].$$

(9)

Several researchers have investigated about the logarithmic coefficients of univalent functions. For example, the rotation of Koebe function $k\left(z\right)=z{(1-e^{i\theta })}^{-2}$ for each real $\theta$ has logarithmic coefficients ${\gamma }_n=e^{i\theta n}/n,$$n\ge 1.$ If $h\in \mathcal{S},$ the sharp bounds for the logarithmic coefficients, when $n=1$ and $n=2$, are obtained as

$$|{\gamma }_1|\le 1,|{\gamma }_2|\le {\displaystyle \frac{1}{2}}(1+2e^{-2}).$$

The problem of determining the sharp bounds for $|{\gamma }_n|$ when $n\ge 3$ still remains unsolved. Recently, Lecko and Partyka [3] discussed the sharp inequalities for the difference of successive initial logarithmic coefficients of univalent functions.

Ponnusamy et al. [4] examined the logarithmic coefficients ${\mathrm{\Gamma }}_n$ of inverse univalent function $h^{-1}$ defined as

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

Upon expansion of the above series, it is easy to see that the first three logarithmic coefficients of inverse univalent function are given by

$$\begin{array}{ccc}\hfill {\mathrm{\Gamma }}_1& =& {\displaystyle \frac{1}{2}}\left[-a_2\right],\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill {\mathrm{\Gamma }}_2& =& {\displaystyle \frac{1}{2}}\left[-a_3+{\displaystyle \frac{3}{2}}{a}_2^2\right]\hfill \end{array}$$

(11)

and

$${\mathrm{\Gamma }}_3={\displaystyle \frac{1}{2}}\left[-a_4+4a_2{a}_3-{\displaystyle \frac{10}{3}}{a}_2^3\right].$$

(12)

Noonan and Thomas [5] introduced the concept of the rth Hankel determinant denoted by $H_{r,m}\left(h\right)$ for $r,m\in \mathbb{N}$. The expression for a function $h\in \mathcal{S}$, which is formed by its coefficients, is

$$H_{r,m}\left(h\right)=\left|\begin{array}{cccccc}{a}_m& a_{m+1}& .& .& .& a_{m+r-1}\\ a_{m+1}& a_{m+2}& .& .& .& a_{m+r}\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ a_{m+r-1}& a_{m+r}& .& .& .& a_{m+2r-2}\end{array}\right|.$$

Nowadays, Hankel matrices are used widely in the field of applied mathematics; see [6]. Hankel determinants are beneficial, for example, in determining whether certain coefficient functionals related to functions are bounded in $\mathbb{D}$ and if they achieve sharp bounds; see [7]. In the last decade, Altinkaya et al. [8] derived the upper bound of $H_{3,1}\left(h\right)$ for concave univalent functions. For a function $h\in \mathcal{S}$, the rth Hankel determinant $H_{r,m}\left(h^{-1}\right)$, where $r,m\in \mathbb{N}$ with entries as inverse coefficients specified by

$$H_{r,m}\left(h^{-1}\right)=\left|\begin{array}{cccccc}{b}_m& b_{m+1}& .& .& .& b_{m+r-1}\\ b_{m+1}& b_{m+2}& .& .& .& b_{m+r}\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ b_{m+r-1}& b_{m+r}& .& .& .& b_{m+2r-2}\end{array}\right|.$$

Let $h\in \mathcal{S}$, the rth Hankel determinant $H_{r,m}(F_h/2)$, where $r,m\in \mathbb{N}$ with entries as logarithmic coefficients specified by

$$H_{r,m}(F_h/2)=\left|\begin{array}{cccccc}{\gamma }_m& {\gamma }_{m+1}& .& .& .& {\gamma }_{m+r-1}\\ {\gamma }_{m+1}& {\gamma }_{m+2}& .& .& .& {\gamma }_{m+r}\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ {\gamma }_{m+r-1}& {\gamma }_{m+r}& .& .& .& {\gamma }_{m+2r-2}\end{array}\right|.$$

It is of interest to note that the expressions $H_{2,1}\left(h^{-1}\right)$ and $H_{2,1}(F_h/2)$ share a notable resemblance to $H_{2,1}\left(h\right)=a_1{a}_3-a_2^2$. Kowalczyk et al. [9] investigated the Hankel determinants formed using logarithmic coefficients. Additionally, Mundalia and Kumar [10] explored the problem of logarithmic coefficients for certain subclasses of close-to-convex functions. The sharp bound for the second Hankel determinant of logarithmic coefficients of convex and starlike functions of order $\alpha$ were found by Kowalczyk and Lecko [11]. More recently, Shi et al. [12] obtained the second Hankel determinant of inverse logarithmic coefficient bounds for starlike functions subordinated to the exponential functions. Also, Wang et al. [13] derived the sharp bounds on Hankel determinants of bounded turning functions through the hyperbolic tangent function.

In 2016, Ye and Lim [6] established that any $n\times n$ matrix over $\mathbb{C}$ can be represented as a product of certain Toeplitz or Hankel matrices. Toeplitz matrices, along with their determinants, play a crucial role in both applied and theoretical mathematics. A defining feature of Toeplitz matrices is that their elements remain constant along each diagonal. Giri and Kumar [14] explored the bounds on Toeplitz determinants for different subclasses of normalized univalent functions in higher-dimensional spaces. Motivated by all these works, for $h\in \mathcal{S}$ and $r,m\in \mathbb{N}$ we consider the Toeplitz determinant of the inverse coefficients as follows:

$$T_{r,m}\left(h^{-1}\right)=\left|\begin{array}{cccccc}{b}_m& b_{m+1}& .& .& .& b_{m+r-1}\\ b_{m+1}& b_m& .& .& .& b_{m+r-2}\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ b_{m+r-1}& b_{m+r-2}& .& .& .& b_m\end{array}\right|.$$

Also, Toeplitz determinant with logarithmic coefficients for $r\ge 1$ and $m\ge 1$ is described by

$$T_{r,m}(F_h/2)=\left|\begin{array}{cccccc}{\gamma }_m& {\gamma }_{m+1}& .& .& .& {\gamma }_{m+r-1}\\ {\gamma }_{m+1}& {\gamma }_m& .& .& .& {\gamma }_{m+r-2}\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ {\gamma }_{m+r-1}& {\gamma }_{m+r-2}& .& .& .& {\gamma }_m\end{array}\right|.$$

Similar to Hankel determinants, $T_{2,1}\left(h^{-1}\right)$ and $T_{2,1}(F_h/2)$ show the similarity of $T_{2,1}\left(h\right)=a_1^2-a_2^2$.

In 1958, Macon et al. [15] explicitly studied the Vandermonde matrices. The Vandermonde matrix is applied in Digital Signal Processing for Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT) computations. Vandermonde matrices play an important role in approximation problems as well. Recently, Vijayalakshmi et al. [16] worked on the Vandermonde determinant for a certain Sakaguchi-type function in the Limacon domain. For $h\in \mathcal{S}$, the Vandermonde determinants $V_{r,m}\left(h^{-1}\right)$ and $V_{r,m}(F_h/2)$ with $r,m\in \mathbb{N}$ related to the inverse coefficients and logarithmic coefficients, respectively, are described by

$$V_{r,m}\left(h^{-1}\right)=\left|\begin{array}{ccccccc}1& b_m& b_m^2& .& .& .& b_m^{r-1}\\ 1& b_{m+1}& b_{m+1}^2& .& .& .& b_{m+1}^{r-1}\\ .& .& .& .& .& .& .\\ .& .& .& .& .& .& .\\ .& .& .& .& .& .& .\\ 1& b_{m+r-1}& b_{m+r-1}^2& .& .& .& b_{m+r-1}^{r-1}\end{array}\right|$$

and

$$V_{r,m}(F_h/2)=\left|\begin{array}{ccccccc}1& {\gamma }_m& {\gamma }_m^2& .& .& .& {\gamma }_m^{r-1}\\ 1& {\gamma }_{m+1}& {\gamma }_{m+1}^2& .& .& .& {\gamma }_{m+1}^{r-1}\\ .& .& .& .& .& .& .\\ .& .& .& .& .& .& .\\ .& .& .& .& .& .& .\\ 1& {\gamma }_{m+r-1}& {\gamma }_{m+r-1}^2& .& .& .& {\gamma }_{m+r-1}^{r-1}\end{array}\right|.$$

In particular, $V_{2,1}\left(h^{-1}\right)$ and $V_{2,1}(F_h/2)$ resemble $V_{2,1}\left(h\right)=a_2-a_1$. The Vandermonde determinant has lot of applications in different fields.

Replacing the univalent function h by $h^{-1}$ and the coefficients $\gamma$ by $\mathrm{\Gamma }$ in the above mentioned determinants $H_{r,m}(F_h/2)$, $T_{r,m}(F_h/2)$ and $V_{r,m}(F_h/2)$, we get the respective determinants in-terms of inverse logarithmic coefficients.

Now, we state the following definition of concave univalent functions, which has been discussed in a detailed way by Avkhadiev et al. [17] and Cruz et al. [18].

Definition 1. 

A function $h:\mathbb{D}\to \mathbb{C}$ is said to be in the family ${\mathcal{C}}_0\left(\alpha \right)$ of concave univalent functions if h satisfies the following conditions:

(i). h is analytic in $\mathbb{D}$ with the standard normalization $h\left(0\right)=0=h^{\prime }\left(0\right)-1$. In addition, $h\left(1\right)=\infty$.

(ii). h maps $\mathbb{D}$ conformally onto a set whose complement with respect to $\mathbb{C}$ is convex.

(iii). The opening angle of $h\left(D\right)$ at ∞ is less than or equal to $\pi \alpha$, $\alpha \in (1,2]$.

Bhowmik et al. [19] showed that an analytic function h maps $\mathbb{D}$ onto a concave domain of angle $\pi \alpha$ if and only if

$$Re\left\{{\displaystyle \frac{2}{\alpha -1}}\left[{\displaystyle \frac{\alpha +1}{2}}{\displaystyle \frac{1+z}{1-z}}-1-{\displaystyle \frac{z{h}^{\prime \prime }\left(z\right)}{h^{\prime }\left(z\right)}}\right]\right\}>0.$$

We need the following lemmas to prove our main results.

Lemma 1. 

[19] Let $\alpha \in (1,2]$. A function $h\in {\mathcal{C}}_0\left(\alpha \right)$ if and only if there exists a starlike function $\psi \in {\mathcal{S}}^{*}$ such that $h\left(z\right)={\mathrm{\Lambda }}_{\psi }\left(z\right)$, where

$${\mathrm{\Lambda }}_{\psi }\left(z\right)={\int }_0^z{\displaystyle \frac{1}{{(1-t)}^{\alpha +1}}}{\left[{\displaystyle \frac{t}{\psi \left(t\right)}}\right]}^{{\displaystyle \frac{\alpha -1}{2}}}dt.$$

Lemma 2. 

(de Branges’s theorem, [20]) If $h\in \mathcal{S}$, $h\left(z\right)=z+{\sum }_{m=2}^{\infty }{a}_m{z}^m,$ then $|a_m|\le m$ for $m\ge 2$.

Motivated by the aforementioned articles, in this present work, we obtain the upper bound of initial inverse coefficients and logarithmic coefficients and the upper bound of the difference between these successive coefficients related to concave univalent functions. Further, we also calculate the upper bound of third-order Hankel, Toeplitz, and Vandermonde determinants in terms of specified coefficients connected to concave univalent functions.

2. Bounds of Initial Coefficients and Bounds of Difference Between the Successive Coefficients for the Class ${\mathcal{C}}_{\mathbf{0}}\left(\mathbf{\alpha }\right)$ of Concave Univalent Functions

Theorem 1. 

Let the function $h\left(z\right)$ defined by (1) belong to the class ${\mathcal{C}}_0\left(\alpha \right),\alpha \in \left(1,2\right]$. Then, the initial upper bounds of the inverse coefficients of $"h"$ and $|b_i|$ for $2\le i\le 5$ are given by

$$\begin{array}{ccc}\hfill |b_2|& \le & \alpha ,\hfill \\ \hfill |b_3|& \le & {\displaystyle \frac{1}{3}}[2{\alpha }^2+6\alpha -5],\hfill \\ \hfill |b_4|& \le & \left\{\begin{array}{c}{\displaystyle \frac{1}{12}}[6{\alpha }^3+11{\alpha }^2+66\alpha -71],1<\alpha \le {\displaystyle \frac{16}{9}}\hfill \\ {\displaystyle \frac{1}{4}}[8{\alpha }^3-7{\alpha }^2+16\alpha -13],{\displaystyle \frac{16}{9}}<\alpha \le {\displaystyle \frac{13}{7}}\hfill \\ \alpha [2{\alpha }^2-1],{\displaystyle \frac{13}{7}}<\alpha \le 2\hfill \end{array}\right.\hfill \end{array}$$

and

$$|b_5|\le \left\{\begin{array}{c}{\displaystyle \frac{1}{60}}[24{\alpha }^4+50{\alpha }^3-91{\alpha }^2+1330\alpha -1253],1<\alpha \le {\displaystyle \frac{23+\sqrt{3121}}{48}}\hfill \\ {\displaystyle \frac{1}{60}}[168{\alpha }^4-88{\alpha }^3-397{\alpha }^2+1468\alpha -1091],{\displaystyle \frac{23+\sqrt{3121}}{48}}<\alpha \le {\displaystyle \frac{79}{46}}\hfill \\ {\displaystyle \frac{1}{60}}[168{\alpha }^4+188{\alpha }^3-871{\alpha }^2+1192\alpha -617],{\displaystyle \frac{79}{46}}<\alpha \le {\displaystyle \frac{19}{11}}\hfill \\ {\displaystyle \frac{1}{60}}[168{\alpha }^4+188{\alpha }^3-739{\alpha }^2+832\alpha -389],{\displaystyle \frac{19}{11}}<\alpha \le {\displaystyle \frac{13}{7}}\hfill \\ {\displaystyle \frac{1}{60}}[168{\alpha }^4+188{\alpha }^3-676{\alpha }^2+652\alpha -272],{\displaystyle \frac{13}{7}}<\alpha \le {\displaystyle \frac{23}{12}}\hfill \\ {\displaystyle \frac{1}{5}}[6{\alpha }^4+47{\alpha }^3-79{\alpha }^2+23\alpha +8],{\displaystyle \frac{23}{12}}<\alpha \le 2\hfill \end{array}\right..$$

Furthermore, the upper bounds of the logarithmic coefficients of the class ${\mathcal{C}}_0\left(\alpha \right)$ are given by

$$\begin{array}{ccc}\hfill |{\gamma }_1|& \le & {\displaystyle \frac{1}{2}}\alpha ,\hfill \\ \hfill |{\gamma }_2|& \le & {\displaystyle \frac{1}{12}}[{\alpha }^2+6\alpha -4]\hfill \end{array}$$

and

$$|{\gamma }_3|\le {\displaystyle \frac{1}{12}}[3{\alpha }^2+8\alpha -9].$$

Also, inverse logarithmic coefficients are given by

$$\begin{array}{ccc}\hfill |{\mathrm{\Gamma }}_1|& \le & {\displaystyle \frac{1}{2}}\alpha ,\hfill \\ \hfill |{\mathrm{\Gamma }}_2|& \le & {\displaystyle \frac{1}{4}}\hfill \end{array}$$

and

$$|{\mathrm{\Gamma }}_3|\le {\displaystyle \frac{1}{24}}[3{\alpha }^3+4{\alpha }^2+49\alpha -52].$$

Proof. 

We recall from Lemma 1 that $h\in {\mathcal{C}}_0\left(\alpha \right)$ if and only if there exists a function $\psi \in {\mathcal{S}}^{*}$ of the form

$$\psi \left(z\right)=z+\sum _{m=2}^{\infty }{\psi }_m{z}^m$$

such that

$$h^{\prime }\left(z\right)={\displaystyle \frac{1}{{(1-z)}^{\alpha +1}}}{\left({\displaystyle \frac{z}{\psi \left(z\right)}}\right)}^{{\displaystyle \frac{\alpha -1}{2}}}$$

(13)

where h has the form given by (1). Equating the coefficients on both sides of (13) yields

$$\begin{array}{ccc}\hfill a_2& =& {\displaystyle \frac{\alpha +1}{2}}-{\displaystyle \frac{\alpha -1}{4}}{\psi }_2,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill a_3& =& {\displaystyle \frac{(\alpha +1)(\alpha +2)}{6}}-{\displaystyle \frac{\alpha -1}{6}}{\psi }_3-{\displaystyle \frac{{\alpha }^2-1}{6}}{\psi }_2+{\displaystyle \frac{{\alpha }^2-1}{24}}{\psi }_2^2,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill a_4=& {\displaystyle \frac{(\alpha +1)(\alpha +2)(\alpha +3)}{24}}-{\displaystyle \frac{\alpha -1}{8}}{\psi }_4-{\displaystyle \frac{{\alpha }^2-1}{8}}{\psi }_3-{\displaystyle \frac{({\alpha }^2-1)(\alpha +2)}{16}}{\psi }_2\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)(\alpha +1)}{32}}{\psi }_2^2-{\displaystyle \frac{({\alpha }^2-1)(\alpha +3)}{192}}{\psi }_2^3+{\displaystyle \frac{{\alpha }^2-1}{16}}{\psi }_2{\psi }_3.\hfill \end{array}$$

(16)

and

$$\begin{array}{cc}\hfill a_5=& {\displaystyle \frac{(\alpha +1)(\alpha +2)(\alpha +3)(\alpha +4)}{120}}-{\displaystyle \frac{\alpha -1}{10}}{\psi }_5-{\displaystyle \frac{{\alpha }^2-1}{10}}{\psi }_4-{\displaystyle \frac{({\alpha }^2-1)(\alpha +2)}{20}}{\psi }_3\hfill \\ \hfill & +{\displaystyle \frac{{\alpha }^2-1}{40}}{\psi }_3^2-{\displaystyle \frac{({\alpha }^2-1)(\alpha +2)(\alpha +3)}{60}}{\psi }_2+{\displaystyle \frac{({\alpha }^2-1)(\alpha +1)(\alpha +2)}{80}}{\psi }_2^2\hfill \\ \hfill & -{\displaystyle \frac{({\alpha }^2-1)(\alpha +1)(\alpha +3)}{240}}{\psi }_2^3+{\displaystyle \frac{({\alpha }^2-1)(\alpha +3)(\alpha +5)}{1920}}{\psi }_2^4\hfill \\ & +{\displaystyle \frac{({\alpha }^2-1)(\alpha +1)}{20}}{\psi }_2{\psi }_3+{\displaystyle \frac{{\alpha }^2-1}{20}}{\psi }_2{\psi }_4-{\displaystyle \frac{({\alpha }^2-1)(\alpha +3)}{80}}{\psi }_2^2{\psi }_3.\hfill \end{array}$$

(17)

By applying (14)–(17) wherever required in (3)–(6), we get

$$\begin{array}{ccc}\hfill b_2& =& {\displaystyle \frac{1}{2}}\left[-(\alpha +1)+{\displaystyle \frac{\alpha -1}{2}}{\psi }_2\right],\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill b_3& =& {\displaystyle \frac{1}{3}}\left[{\displaystyle \frac{(\alpha +1)(2\alpha +1)}{2}}+{\displaystyle \frac{\alpha -1}{2}}{\psi }_3-({\alpha }^2-1){\psi }_2+{\displaystyle \frac{(\alpha -1)(\alpha -2)}{4}}{\psi }_2^2\right],\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill b_4=& {\displaystyle \frac{1}{8}}[-{\displaystyle \frac{(6{\alpha }^2+5\alpha +1)(\alpha +1)}{3}}+(\alpha -1){\psi }_4-{\displaystyle \frac{7({\alpha }^2-1)}{3}}{\psi }_3\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)(18\alpha +11)}{6}}{\psi }_2-{\displaystyle \frac{({\alpha }^2-1)(9\alpha -16)}{6}}{\psi }_2^2\hfill \\ \hfill & +{\displaystyle \frac{(3{\alpha }^2-13\alpha +14)(\alpha -1)}{12}}{\psi }_2^3+{\displaystyle \frac{(\alpha -1)(7\alpha -13)}{6}}{\psi }_2{\psi }_3].\hfill \end{array}$$

(20)

and

$$\begin{array}{cc}\hfill b_5=& {\displaystyle \frac{1}{10}}[{\displaystyle \frac{(24{\alpha }^3+26{\alpha }^2+9\alpha +1)(\alpha +1)}{12}}+(\alpha -1){\psi }_5-{\displaystyle \frac{11({\alpha }^2-1)}{4}}{\psi }_4\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)(23\alpha +16)}{6}}{\psi }_3+{\displaystyle \frac{(\alpha -1)(7\alpha -13)}{12}}{\psi }_3^2-{\displaystyle \frac{({\alpha }^2-1)(48{\alpha }^2+50\alpha +13)}{12}}{\psi }_2\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)(24{\alpha }^2-23\alpha -27)}{8}}{\psi }_2^2-{\displaystyle \frac{({\alpha }^2-1)(12{\alpha }^2-47\alpha +46)}{12}}{\psi }_2^3\hfill \\ \hfill & -{\displaystyle \frac{(-6{\alpha }^3+41{\alpha }^2-93\alpha +70)(\alpha -1)}{48}}{\psi }_2^4-{\displaystyle \frac{({\alpha }^2-1)(46\alpha -79)}{12}}{\psi }_2{\psi }_3\hfill \\ \hfill & +{\displaystyle \frac{(\alpha -1)(11\alpha -19)}{8}}{\psi }_2{\psi }_4+{\displaystyle \frac{(23{\alpha }^2-93\alpha +94)(\alpha -1)}{24}}{\psi }_2^2{\psi }_3].\hfill \end{array}$$

(21)

Applying the triangle inequality and Lemma 2 in (18)–(21), we get the desired bounds of the inverse coefficients till $|b_5|$.

Next, we find the upper bound of logarithmic and inverse logarithmic coefficients for the class ${\mathcal{C}}_0\left(\alpha \right)$:

By applying (14)–(16) wherever required in (7)–(9), we get

$$\begin{array}{ccc}\hfill {\gamma }_1& =& {\displaystyle \frac{1}{8}}[2(\alpha +1)-(\alpha -1){\psi }_2],\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\gamma }_2& =& {\displaystyle \frac{1}{12}}\left[{\displaystyle \frac{(\alpha +1)(\alpha +5)}{4}}-(\alpha -1){\psi }_3-{\displaystyle \frac{{\alpha }^2-1}{4}}{\psi }_2+{\displaystyle \frac{(\alpha -1)(\alpha +7)}{16}}{\psi }_2^2\right]\hfill \end{array}$$

(23)

and

$$\begin{array}{cc}\hfill {\gamma }_3=& {\displaystyle \frac{1}{48}}[(\alpha +1)(\alpha +3)-3(\alpha -1){\psi }_4-({\alpha }^2-1){\psi }_3-{\displaystyle \frac{{\alpha }^2-1}{2}}{\psi }_2+{\displaystyle \frac{{\alpha }^2-1}{2}}{\psi }_2^2\hfill \\ \hfill & -{\displaystyle \frac{(\alpha -1)(\alpha +3)}{4}}{\psi }_2^3+{\displaystyle \frac{(\alpha -1)(\alpha +5)}{2}}{\psi }_2{\psi }_3].\hfill \end{array}$$

(24)

Applying the triangle inequality and Lemma 2 in (22)–(24), we get the desired upper bounds of $|{\gamma }_1|$, $|{\gamma }_2|$ and $|{\gamma }_3|$.

By applying (14)–(16) wherever required in (10)–(12), we get

$$\begin{array}{ccc}\hfill {\mathrm{\Gamma }}_1& =& {\displaystyle \frac{1}{8}}[-2(\alpha +1)+(\alpha -1){\psi }_2],\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill {\mathrm{\Gamma }}_2& =& {\displaystyle \frac{1}{12}}\left[{\displaystyle \frac{(\alpha +1)(5\alpha +1)}{4}}+(\alpha -1){\psi }_3-{\displaystyle \frac{5({\alpha }^2-1)}{4}}{\psi }_2+{\displaystyle \frac{(\alpha -1)(5\alpha -13)}{16}}{\psi }_2^2\right]\hfill \end{array}$$

(26)

and

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_3=& {\displaystyle \frac{1}{48}}[-\alpha (\alpha +1)(3\alpha +1)+3(\alpha -1){\psi }_4-5({\alpha }^2-1){\psi }_3+{\displaystyle \frac{({\alpha }^2-1)(9\alpha +4)}{2}}{\psi }_2\hfill \\ \hfill & -{\displaystyle \frac{({\alpha }^2-1)(9\alpha -19)}{4}}{\psi }_2^2+{\displaystyle \frac{(3{\alpha }^2-16\alpha +21)(\alpha -1)}{8}}{\psi }_2^3\hfill \\ \hfill & +{\displaystyle \frac{(\alpha -1)(5\alpha -11)}{2}}{\psi }_2{\psi }_3].\hfill \end{array}$$

(27)

Applying the triangle inequality and Lemma 2 in (25)–(27), we get the desired bounds, which completes the proof of Theorem 1. □

Theorem 2. 

Let the function $h\left(z\right)$ defined by (1) belong to the class ${\mathcal{C}}_0\left(\alpha \right),\alpha \in \left(1,2\right]$. Then the upper bounds of successive difference between the inverse coefficients are given by

$$\begin{array}{ccc}\hfill |b_2-b_1|& \le & \alpha +1,\hfill \\ \hfill |b_3-b_2|& \le & {\displaystyle \frac{1}{3}}\left[(\alpha +5)(2\alpha -1)\right],\hfill \\ \hfill |b_4-b_3|& \le & \left\{\begin{array}{c}{\displaystyle \frac{1}{12}}[6{\alpha }^3+19{\alpha }^2+90\alpha -91],1<\alpha \le {\displaystyle \frac{1+\sqrt{97}}{6}}\hfill \\ {\displaystyle \frac{1}{12}}[24{\alpha }^3-5{\alpha }^2+48\alpha -43],{\displaystyle \frac{1+\sqrt{97}}{6}}<\alpha \le {\displaystyle \frac{13}{7}}\hfill \\ {\displaystyle \frac{1}{3}}[6{\alpha }^3+4{\alpha }^2-3\alpha -1],{\displaystyle \frac{13}{7}}<\alpha \le 2\hfill \end{array}\right.\hfill \end{array}$$

and

$$|b_5-b_4|\le \left\{\begin{array}{c}{\displaystyle \frac{1}{60}}[24{\alpha }^4+80{\alpha }^3-36{\alpha }^2+1660\alpha -1475],1<\alpha \le {\displaystyle \frac{2+\sqrt{326}}{12}}\hfill \\ {\displaystyle \frac{1}{60}}[168{\alpha }^4+32{\alpha }^3-502{\alpha }^2+1708\alpha -1153],{\displaystyle \frac{2+\sqrt{326}}{12}}<\alpha \le {\displaystyle \frac{19}{11}}\hfill \\ {\displaystyle \frac{1}{60}}[168{\alpha }^4+32{\alpha }^3-370{\alpha }^2+1348\alpha -925],{\displaystyle \frac{19}{11}}<\alpha \le {\displaystyle \frac{31+45\sqrt{41}}{184}}\hfill \\ {\displaystyle \frac{1}{60}}[168{\alpha }^4+308{\alpha }^3-739{\alpha }^2+772\alpha -256],{\displaystyle \frac{31+45\sqrt{41}}{184}}<\alpha \le {\displaystyle \frac{13}{7}}\hfill \\ {\displaystyle \frac{1}{60}}[168{\alpha }^4+308{\alpha }^3-676{\alpha }^2+592\alpha -139],{\displaystyle \frac{13}{7}}<\alpha \le {\displaystyle \frac{29+5\sqrt{1009}}{96}}\hfill \\ {\displaystyle \frac{1}{60}}[72{\alpha }^4+654{\alpha }^3-788{\alpha }^2-54\alpha +115],{\displaystyle \frac{29+5\sqrt{1009}}{96}}<\alpha \le 2\hfill \end{array}\right..$$

The upper bounds of successive difference between the logarithmic coefficients are given by

$$|{\gamma }_2-{\gamma }_1|\le {\displaystyle \frac{1}{24}}[-{\alpha }^2+18\alpha -11]$$

and

$$|{\gamma }_3-{\gamma }_2|\le {\displaystyle \frac{1}{24}}[{\alpha }^2+24\alpha -23].$$

The upper bound of successive difference between the inverse logarithmic coefficients are given by

$$|{\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_1|\le {\displaystyle \frac{1}{24}}[5{\alpha }^2+30\alpha -17]$$

and

$$|{\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_2|\le {\displaystyle \frac{1}{24}}[3{\alpha }^3+9{\alpha }^2+67\alpha -69].$$

Proof. 

By virtue of bound $|b_2|$ from Theorem 1 and (2), we readily obtain the upper bound of $|b_2-b_1|$.

From (3) and (4), we get

$$b_3-b_2=2a_2^2+a_2-a_3.$$

(28)

Substituting (14) and (15) in (28), we receive

$$b_3-b_2={\displaystyle \frac{1}{3}}\left[(\alpha +1)(\alpha +2)+{\displaystyle \frac{\alpha -1}{2}}{\psi }_3-{\displaystyle \frac{(\alpha -1)(4\alpha +7)}{4}}{\psi }_2+{\displaystyle \frac{(\alpha -1)(\alpha -2)}{4}}{\psi }_2^2\right].$$

(29)

From (4) and (5), we obtain

$$b_4-b_3=5a_2{a}_3-5a_2^3-2a_2^2+a_3-a_4.$$

(30)

Employing (14)–(16) in (30), we receive

$$\begin{array}{cc}\hfill b_4-b_3=& {\displaystyle \frac{1}{8}}[-{\displaystyle \frac{(6{\alpha }^2+13\alpha +5)(\alpha +1)}{3}}+(\alpha -1){\psi }_4+{\displaystyle \frac{(\alpha -1)(7\alpha +11)}{3}}{\psi }_3\hfill \\ \hfill & +{\displaystyle \frac{3({\alpha }^2-1)(2\alpha +3)}{2}}{\psi }_2+{\displaystyle \frac{(-3{\alpha }^2+\alpha +8)(\alpha -1)}{2}}{\psi }_2^2\hfill \\ \hfill & +{\displaystyle \frac{(3{\alpha }^2-13\alpha +14)(\alpha -1)}{12}}{\psi }_2^3+{\displaystyle \frac{(\alpha -1)(7\alpha -13)}{6}}{\psi }_2{\psi }_3].\hfill \end{array}$$

(31)

From (5) and (6), we acquire

$$b_5-b_4=6a_2{a}_4-5a_2{a}_3-21a_2^2{a}_3+14a_2^4+5a_2^3+3a_3^2+a_4-a_5.$$

(32)

Applying (14)–(17) in (32), we receive

$$\begin{array}{cc}\hfill b_5-b_4=& {\displaystyle \frac{1}{10}}[{\displaystyle \frac{(12{\alpha }^3+28{\alpha }^2+17\alpha +3)(\alpha +1)}{6}}+(\alpha -1){\psi }_5-{\displaystyle \frac{(\alpha -1)(11\alpha +16)}{4}}{\psi }_4\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)(46\alpha +67)}{12}}{\psi }_3+{\displaystyle \frac{(\alpha -1)(7\alpha -13)}{12}}{\psi }_3^2\hfill \\ \hfill & -{\displaystyle \frac{({\alpha }^2-1)(96{\alpha }^2+190\alpha +81)}{24}}{\psi }_2+{\displaystyle \frac{({\alpha }^2-1)(72{\alpha }^2-24\alpha -161)}{24}}{\psi }_2^2\hfill \\ \hfill & -{\displaystyle \frac{(48{\alpha }^3-125{\alpha }^2-69\alpha +254)(\alpha -1)}{48}}{\psi }_2^3\hfill \\ \hfill & -{\displaystyle \frac{(-6{\alpha }^3+41{\alpha }^2-93\alpha +70)(\alpha -1)}{48}}{\psi }_2^4-{\displaystyle \frac{(92{\alpha }^2-31\alpha -223)(\alpha -1)}{24}}{\psi }_2{\psi }_3\hfill \\ \hfill & +{\displaystyle \frac{(\alpha -1)(11\alpha -19)}{8}}{\psi }_2{\psi }_4+{\displaystyle \frac{(23{\alpha }^2-93\alpha +94)(\alpha -1)}{24}}{\psi }_2^2{\psi }_3].\hfill \end{array}$$

(33)

Utilizing the triangle inequality and Lemma 2 in (29)–(33), we get the required upper bounds of successive difference between the inverse coefficients.

By referring (7) and (8), we obtain

$${\gamma }_2-{\gamma }_1={\displaystyle \frac{1}{2}}[a_3-a_2-{\displaystyle \frac{1}{2}}{a}_2^2].$$

(34)

Using (14) and (15) in (34), we get

$${\gamma }_2-{\gamma }_1={\displaystyle \frac{1}{12}}\left[{\displaystyle \frac{(\alpha +1)(\alpha -7)}{4}}-(\alpha -1){\psi }_3-{\displaystyle \frac{(\alpha -1)(\alpha -5)}{4}}{\psi }_2+{\displaystyle \frac{(\alpha -1)(\alpha +7)}{16}}{\psi }_2^2\right].$$

(35)

By referencing (8) and (9), we obtain

$${\gamma }_3-{\gamma }_2={\displaystyle \frac{1}{2}}[a_4-a_3-a_2{a}_3+{\displaystyle \frac{1}{2}}{a}_2^2+{\displaystyle \frac{1}{3}}{a}_2^3].$$

(36)

Employing (14)–(16) in (36), we obtain

$$\begin{array}{cc}\hfill {\gamma }_3-{\gamma }_2=& {\displaystyle \frac{1}{48}}[-2(\alpha +1)-3(\alpha -1){\psi }_4-(\alpha -1)(\alpha -3){\psi }_3+{\displaystyle \frac{{\alpha }^2-1}{2}}{\psi }_2+{\displaystyle \frac{(\alpha -1)(\alpha -5)}{4}}{\psi }_2^2\hfill \\ \hfill & -{\displaystyle \frac{(\alpha -1)(\alpha +3)}{4}}{\psi }_2^3+{\displaystyle \frac{(\alpha -1)(\alpha +5)}{2}}{\psi }_2{\psi }_3].\hfill \end{array}$$

(37)

Utilizing the triangle inequality and Lemma 2 in (35) and (37), we get the required upper bounds of successive difference between the logarithmic coefficients.

By referencing (10) and (11), we obtain

$${\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_1={\displaystyle \frac{1}{4}}[2a_2-2a_3+3a_2^2].$$

(38)

Substituting (14) and (15) in (38), we get

$${\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_1={\displaystyle \frac{1}{12}}\left[{\displaystyle \frac{(\alpha +1)(5\alpha +13)}{4}}+(\alpha -1){\psi }_3-{\displaystyle \frac{(\alpha -1)(5\alpha +11)}{4}}{\psi }_2+{\displaystyle \frac{(\alpha -1)(5\alpha -13)}{16}}{\psi }_2^2\right].$$

(39)

By considering (11) and (12), we have

$${\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_2={\displaystyle \frac{1}{12}}[6a_3-6a_4+24a_2{a}_3-9a_2^2-20a_2^3].$$

(40)

Utilizing (14)–(16) in (40), we get

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_2=& {\displaystyle \frac{1}{48}}[-(3{\alpha }^2+6\alpha +1)(\alpha +1)+3(\alpha -1){\psi }_4-(\alpha -1)(5\alpha +9){\psi }_3\hfill \\ & +{\displaystyle \frac{({\alpha }^2-1)(9\alpha +14)}{2}}{\psi }_2+{\displaystyle \frac{(-9{\alpha }^2+5\alpha +32)(\alpha -1)}{4}}{\psi }_2^2\hfill \\ \hfill & +{\displaystyle \frac{(3{\alpha }^2-16\alpha +21)(\alpha -1)}{8}}{\psi }_2^3+{\displaystyle \frac{(\alpha -1)(5\alpha -11)}{2}}{\psi }_2{\psi }_3].\hfill \end{array}$$

(41)

Utilizing the triangle inequality and Lemma 2 in (39) and (41) completes the proof of Theorem 2. □

Remark 1. 

$|b_2-b_1|$ and $|b_3-b_2|$ are called the bounds of the second-order Vandermonde determinants $|V_{2,1}\left(h^{-1}\right)|$ and $|V_{2,2}\left(h^{-1}\right)|$, respectively, in terms of inverse coefficients.

Remark 2. 

$|{\gamma }_2-{\gamma }_1|$ and $|{\gamma }_3-{\gamma }_2|$ are called the bounds of the second-order Vandermonde determinants $|V_{2,1}(F_h/2)|$ and $|V_{2,2}(F_h/2)|$, respectively, in terms of logarithmic coefficients. Similarly, replacing the function h by $h^{-1}$ and the coefficient γ by Γ gives $|V_{2,1}(F_{h^{-1}}/2)|$ and $|V_{2,2}(F_{h^{-1}}/2)|$.

3. Bounds of Hankel Determinants of Inverse and Logarithmic Coefficients for the Class ${\mathcal{C}}_{\mathbf{0}}\left(\mathbf{\alpha }\right)$ of Concave Univalent Functions

3.1. Bounds of Hankel Determinants of Inverse Coefficients

Theorem 3. 

Let the function $h\left(z\right)$ defined by (1) belong to the class ${\mathcal{C}}_0\left(\alpha \right),\alpha \in \left(1,2\right]$. Then

$$|b_2{b}_3-b_4|\le {\displaystyle \frac{1}{12}}\left[2{\alpha }^3+{\alpha }^2+46\alpha -49\right].$$

Proof. 

From (3)–(5), we observe that

$$b_2{b}_3-b_4=a_4-4a_2{a}_3+3a_2^3.$$

(42)

Substituting (14)–(16) in (42), we attain

$$\begin{array}{cc}\hfill b_2{b}_3-b_4=& {\displaystyle \frac{1}{8}}[{\displaystyle \frac{({\alpha }^2-1)(2\alpha +1)}{3}}-(\alpha -1){\psi }_4+{\displaystyle \frac{5({\alpha }^2-1)}{3}}{\psi }_3-{\displaystyle \frac{({\alpha }^2-1)(6\alpha +1)}{6}}{\psi }_2\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)(3\alpha -8)}{6}}{\psi }_2^2-{\displaystyle \frac{(\alpha -1)(\alpha -2)(\alpha -5)}{12}}{\psi }_2^3\hfill \\ \hfill & -{\displaystyle \frac{(\alpha -1)(5\alpha -11)}{6}}{\psi }_2{\psi }_3].\hfill \end{array}$$

(43)

Applying the triangle inequality and Lemma 2 in (43) ends the proof of Theorem 3. □

Theorem 4. 

Let the function $h\left(z\right)$ defined by (1) belong to the class ${\mathcal{C}}_0\left(\alpha \right),\alpha \in \left(1,2\right]$. Then

$$\begin{array}{ccc}\hfill |H_{2,1}\left(h^{-1}\right)|& \le & {\displaystyle \frac{1}{6}}\left[(\alpha +7)(\alpha -1)\right],\hfill \\ \hfill |H_{2,2}\left(h^{-1}\right)|& \le & \left\{\begin{array}{c}{\displaystyle \frac{1}{36}}[2{\alpha }^4-15{\alpha }^3+104{\alpha }^2-57\alpha -34],1<\alpha \le {\displaystyle \frac{7}{5}}\hfill \\ {\displaystyle \frac{1}{72}}[4{\alpha }^4-15{\alpha }^3+157{\alpha }^2-57\alpha -89],{\displaystyle \frac{7}{5}}<\alpha \le 2\hfill \end{array}\right.\hfill \end{array}$$

and

$${\displaystyle |H_{3,1}\left(h^{-1}\right)|\le }\left\{\begin{array}{c}{k}_1[404{\alpha }^6+1224{\alpha }^5+8691{\alpha }^4+16344{\alpha }^3+62514{\alpha }^2-196848\alpha +108211],1<\alpha \le {\displaystyle \frac{7}{5}}\hfill \\ k_1[404{\alpha }^6+1524{\alpha }^5+8571{\alpha }^4+13674{\alpha }^3+68064{\alpha }^2-200958\alpha +109261],{\displaystyle \frac{7}{5}}<\alpha \le {\alpha }_1\hfill \\ k_1[1268{\alpha }^6+5880{\alpha }^5-4281{\alpha }^4+9282{\alpha }^3+86856{\alpha }^2-200922\alpha +102457],{\alpha }_1<\alpha \le {\displaystyle \frac{79}{46}}\hfill \\ k_1[1268{\alpha }^6+7536{\alpha }^5+2811{\alpha }^4-21030{\alpha }^3+99672{\alpha }^2-172266\alpha +82549],{\displaystyle \frac{79}{46}}<\alpha \le {\displaystyle \frac{19}{11}}\hfill \\ k_1[1268{\alpha }^6+7536{\alpha }^5+3603{\alpha }^4-18438{\alpha }^3+82536{\alpha }^2-148938\alpha +72973],{\displaystyle \frac{19}{11}}<\alpha \le {\displaystyle \frac{16}{9}}\hfill \\ k_1[1808{\alpha }^6+6846{\alpha }^5+15003{\alpha }^4-53058{\alpha }^3+93576{\alpha }^2-113628\alpha +49453],{\displaystyle \frac{16}{9}}<\alpha \le {\displaystyle \frac{13}{7}}\hfill \\ k_2[904{\alpha }^6+3738{\alpha }^5+6948{\alpha }^4-18555{\alpha }^3+14451{\alpha }^2-15423\alpha +7937],{\displaystyle \frac{13}{7}}<\alpha \le {\displaystyle \frac{23}{12}}\hfill \\ k_2[616{\alpha }^6+3138{\alpha }^5+14916{\alpha }^4-32475{\alpha }^3+14499{\alpha }^2-903\alpha +209],{\displaystyle \frac{23}{12}}<\alpha \le 2\hfill \end{array}\right..$$

where $k_1={\displaystyle \frac{1}{2160}}$, $k_2={\displaystyle \frac{1}{1080}}$ and ${\alpha }_1={\displaystyle \frac{23+\sqrt{3121}}{48}}$.

Proof. 

The second-order Hankel determinants of the inverse coefficients are given by

$$H_{2,1}\left(h^{-1}\right)=b_3-b_2^2$$

and

$$H_{2,2}\left(h^{-1}\right)=b_2{b}_4-b_3^2.$$

From (3) and (4), we observe

$$b_3-b_2^2=a_2^2-a_3.$$

(44)

Using (14) and (15) in (44) gives

$$b_3-b_2^2={\displaystyle \frac{1}{6}}\left[{\displaystyle \frac{{\alpha }^2-1}{2}}+(\alpha -1){\psi }_3-{\displaystyle \frac{{\alpha }^2-1}{2}}{\psi }_2+{\displaystyle \frac{(\alpha -1)(\alpha -5)}{8}}{\psi }_2^2\right].$$

(45)

Applying the triangle inequality and Lemma 2 in (45) results in the upper bound of $|H_{2,1}\left(h^{-1}\right)|$.

Now, we consider

$$H_{2,2}\left(h^{-1}\right)=b_2{b}_4-b_3^2.$$

From (3)–(5), we notice that

$$b_2{b}_4-b_3^2=a_2{a}_4-a_3^2-a_2^2{a}_3+a_2^4.$$

(46)

Substituting (14)–(16) in (46), we reach

$$\begin{array}{cc}\hfill b_2{b}_4-b_3^2=& {\displaystyle \frac{1}{4}}[{\displaystyle \frac{{(\alpha +1)}^2(\alpha -1)(2\alpha +1)}{36}}-{\displaystyle \frac{{\alpha }^2-1}{4}}{\psi }_4+{\displaystyle \frac{({\alpha }^2-1)(5\alpha +13)}{36}}{\psi }_3-{\displaystyle \frac{{(\alpha -1)}^2}{9}}{\psi }_3^2\hfill \\ \hfill & -{\displaystyle \frac{({\alpha }^2-1)(4{\alpha }^2+3\alpha +2)}{36}}{\psi }_2+{\displaystyle \frac{({\alpha }^2-1)(4{\alpha }^2-5\alpha -11)}{48}}{\psi }_2^2\hfill \\ \hfill & -{\displaystyle \frac{({\alpha }^2-1)(2{\alpha }^2-9\alpha +13)}{72}}{\psi }_2^3+{\displaystyle \frac{{(\alpha -1)}^2(\alpha -2)(\alpha -5)}{288}}{\psi }_2^4\hfill \\ \hfill & -{\displaystyle \frac{({\alpha }^2-1)(5\alpha -14)}{36}}{\psi }_2{\psi }_3+{\displaystyle \frac{{(\alpha -1)}^2}{8}}{\psi }_2{\psi }_4+{\displaystyle \frac{{(\alpha -1)}^2(5\alpha -7)}{144}}{\psi }_2^2{\psi }_3].\hfill \end{array}$$

(47)

Applying the triangle inequality and Lemma 2 in (47) results in the upper bound of $|H_{2,2}\left(h^{-1}\right)|$.

The third-order Hankel determinant of the inverse coefficients is given by

$$H_{3,1}\left(h^{-1}\right)=b_3(b_2{b}_4-b_3^2)-b_4(b_4-b_2{b}_3)+b_5(b_3-b_2^2).$$

$$|H_{3,1}\left(h^{-1}\right)|\le |b_3||b_2{b}_4-b_3^2|+|b_4||b_2{b}_3-b_4|+|b_5||b_3-b_2^2|.$$

(48)

By multiplying the $|b_3|$ bound from Theorem 1 and the result of $|H_{2,2}\left(h^{-1}\right)|$, we obtain

$$|b_3||b_2{b}_4-b_3^2|\le \left\{\begin{array}{c}{\displaystyle \frac{1}{108}}[4{\alpha }^6-18{\alpha }^5+108{\alpha }^4+585{\alpha }^3-930{\alpha }^2+81\alpha +170],1<\alpha \le {\displaystyle \frac{7}{5}}\hfill \\ {\displaystyle \frac{1}{216}}[8{\alpha }^6-6{\alpha }^5+204{\alpha }^4+903{\alpha }^3-1305{\alpha }^2-249\alpha +445],{\displaystyle \frac{7}{5}}<\alpha \le 2\hfill \end{array}\right..$$

(49)

By multiplying the $|b_4|$ bound from the result of Theorems 1 and 3, we get

$${\displaystyle |b_4||b_2{b}_3-b_4|\le }\left\{\begin{array}{c}{\displaystyle \frac{1}{144}}[12{\alpha }^6+28{\alpha }^5+419{\alpha }^4+136{\alpha }^3+2426{\alpha }^2-6500\alpha +3479],1<\alpha \le {\displaystyle \frac{16}{9}}\hfill \\ {\displaystyle \frac{1}{48}}[16{\alpha }^6-6{\alpha }^5+393{\alpha }^4-724{\alpha }^3+1066{\alpha }^2-1382\alpha +637],{\displaystyle \frac{16}{9}}<\alpha \le {\displaystyle \frac{13}{7}}\hfill \\ {\displaystyle \frac{1}{12}}[4{\alpha }^6+2{\alpha }^5+90{\alpha }^4-99{\alpha }^3-46{\alpha }^2+49\alpha ],{\displaystyle \frac{13}{7}}<\alpha \le 2\hfill \end{array}\right..$$

(50)

By multiplying the $|b_5|$ bound from Theorem 1 and the result of $|H_{2,1}\left(h^{-1}\right)|$, we attain

$${\displaystyle |b_5||b_3-b_2^2|\le }\left\{\begin{array}{c}{\displaystyle \frac{1}{360}}[24{\alpha }^6+194{\alpha }^5+41{\alpha }^4+434{\alpha }^3+7364{\alpha }^2-16828\alpha +8771],1<\alpha \le {\alpha }_1\hfill \\ {\displaystyle \frac{1}{360}}[168{\alpha }^6+920{\alpha }^5-2101{\alpha }^4-298{\alpha }^3+10496{\alpha }^2-16822\alpha +7637],{\alpha }_1<\alpha \le {\displaystyle \frac{79}{46}}\hfill \\ {\displaystyle \frac{1}{360}}[168{\alpha }^6+1196{\alpha }^5-919{\alpha }^4-5350{\alpha }^3+12632{\alpha }^2-12046\alpha +4319],{\displaystyle \frac{79}{46}}<\alpha \le {\displaystyle \frac{19}{11}}\hfill \\ {\displaystyle \frac{1}{360}}[168{\alpha }^6+1196{\alpha }^5-787{\alpha }^4-4918{\alpha }^3+9776{\alpha }^2-8158\alpha +2723],{\displaystyle \frac{19}{11}}<\alpha \le {\displaystyle \frac{13}{7}}\hfill \\ {\displaystyle \frac{1}{90}}[42{\alpha }^6+299{\alpha }^5-181{\alpha }^4-1180{\alpha }^3+2093{\alpha }^2-1549\alpha +476],{\displaystyle \frac{13}{7}}<\alpha \le {\displaystyle \frac{23}{12}}\hfill \\ {\displaystyle \frac{1}{30}}[6{\alpha }^6+83{\alpha }^5+161{\alpha }^4-780{\alpha }^3+699{\alpha }^2-113\alpha -56],{\displaystyle \frac{23}{12}}<\alpha \le 2\hfill \end{array}\right.$$

(51)

where ${\alpha }_1={\displaystyle \frac{23+\sqrt{3121}}{48}}$.

Substitution of (49)–(51) in (48) completes the proof of Theorem 4. □

3.2. Bounds of Hankel Determinants of Logarithmic Coefficients

Theorem 5. 

Let the function $h\left(z\right)$ defined by (1) belong to the class ${\mathcal{C}}_0\left(\alpha \right),\alpha \in \left(1,2\right]$. Then

$$|H_{2,1}(F_h/2)|\le {\displaystyle \frac{1}{576}}[-{\alpha }^4+12{\alpha }^3+218{\alpha }^2-156\alpha -61]$$

and

$$|H_{2,1}(F_{h^{-1}}/2)|\le \left\{\begin{array}{c}{\displaystyle \frac{1}{576}}[11{\alpha }^4-70{\alpha }^3+434{\alpha }^2-218\alpha -145],1<\alpha \le {\displaystyle \frac{7}{5}}\hfill \\ {\displaystyle \frac{1}{576}}[11{\alpha }^4-40{\alpha }^3+332{\alpha }^2-104\alpha -187],{\displaystyle \frac{7}{5}}<\alpha \le 2\hfill \end{array}\right..$$

Proof. 

The second-order Hankel determinant for logarithmic coefficients is given by

$$H_{2,1}(F_h/2)={\gamma }_1{\gamma }_3-{\gamma }_2^2.$$

From (7)–(9), we obtain

$${\gamma }_1{\gamma }_3-{\gamma }_2^2={\displaystyle \frac{1}{4}}\left[a_2{a}_4-a_3^2+{\displaystyle \frac{1}{12}}{a}_2^4\right].$$

(52)

By using (14)–(16) in (52), we obtain

$$\begin{array}{cc}\hfill {\gamma }_1{\gamma }_3-{\gamma }_2^2=& {\displaystyle \frac{1}{16}}[{\displaystyle \frac{{(\alpha +1)}^2(-{\alpha }^2+2\alpha +11)}{144}}-{\displaystyle \frac{{\alpha }^2-1}{4}}{\psi }_4-{\displaystyle \frac{({\alpha }^2-1)(\alpha -7)}{36}}{\psi }_3\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)({\alpha }^2-7)}{72}}{\psi }_2-{\displaystyle \frac{({\alpha }^2-1)({\alpha }^2-2\alpha +9)}{96}}{\psi }_2^2\hfill \\ \hfill & -{\displaystyle \frac{({\alpha }^2-1)(-{\alpha }^2+6\alpha +19)}{288}}{\psi }_2^3+{\displaystyle \frac{{(\alpha -1)}^2(-{\alpha }^2+10\alpha +23)}{2304}}{\psi }_2^4\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)(\alpha +8)}{36}}{\psi }_2{\psi }_3+{\displaystyle \frac{{(\alpha -1)}^2}{8}}{\psi }_2{\psi }_4-{\displaystyle \frac{(\alpha -1)({\alpha }^2-1)}{144}}{\psi }_2^2{\psi }_3\hfill \\ \hfill & -{\displaystyle \frac{{(\alpha -1)}^2}{9}}{\psi }_3^2].\hfill \end{array}$$

(53)

Employing the triangle inequality and Lemma 2 in (53) gives the upper bound of $|H_{2,1}(F_h/2)|$.

The second-order Hankel determinant for inverse logarithmic coefficients is given by

$$H_{2,1}(F_{h^{-1}}/2)={\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_2^2.$$

From (10), (11) and (12), we acquire

$${\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_2^2={\displaystyle \frac{1}{48}}[12a_2{a}_4-12a_3^2-12a_2^2{a}_3+13a_2^4].$$

(54)

By applying (14), (15) and (16) in (54), we get

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_2^2=& {\displaystyle \frac{1}{48}}[{\displaystyle \frac{{(\alpha +1)}^2(11{\alpha }^2+2\alpha -1)}{48}}-{\displaystyle \frac{3({\alpha }^2-1)}{4}}{\psi }_4+{\displaystyle \frac{({\alpha }^2-1)(5\alpha +13)}{12}}{\psi }_3\hfill \\ \hfill & -{\displaystyle \frac{{(\alpha -1)}^2}{3}}{\psi }_3^2-{\displaystyle \frac{({\alpha }^2-1)(11{\alpha }^2+2\alpha +7)}{24}}{\psi }_2-{\displaystyle \frac{({\alpha }^2-1)(-11{\alpha }^2+10\alpha +25)}{32}}{\psi }_2^2\hfill \\ \hfill & -{\displaystyle \frac{({\alpha }^2-1)(11{\alpha }^2-42\alpha +55)}{96}}{\psi }_2^3+{\displaystyle \frac{{(\alpha -1)}^2(11{\alpha }^2-62\alpha +83)}{768}}{\psi }_2^4\hfill \\ \hfill & -{\displaystyle \frac{({\alpha }^2-1)(5\alpha -14)}{12}}{\psi }_2{\psi }_3+{\displaystyle \frac{3{(\alpha -1)}^2}{8}}{\psi }_2{\psi }_4+{\displaystyle \frac{{(\alpha -1)}^2(5\alpha -7)}{48}}{\psi }_2^2{\psi }_3].\hfill \end{array}$$

(55)

Employing the triangle inequality and Lemma 2 in (55) concludes the proof of Theorem 5. □

4. Bounds of Toeplitz Determinants of Inverse and Logarithmic Coefficients for the Class ${\mathcal{C}}_{\mathbf{0}}\left(\mathbf{\alpha }\right)$ of Concave Univalent Functions

4.1. Bounds of Toeplitz Determinants of Inverse Coefficients

Theorem 6. 

Let the function $h\left(z\right)$ defined by (1) belong to the class ${\mathcal{C}}_0\left(\alpha \right),\alpha \in \left(1,2\right]$. Then

$$|b_3-2b_2^2|\le {\displaystyle \frac{1}{3}}\left[(\alpha +2)(2\alpha -1)\right].$$

Proof. 

From (3) and (4), we see that

$$b_3-2b_2^2=-a_3.$$

(56)

Substituting (15) and applying the triangle inequality as well as Lemma 2 in (56) concludes the proof of Theorem 6. □

Theorem 7. 

Let the function $h\left(z\right)$ defined by (1) belong to the class ${\mathcal{C}}_0\left(\alpha \right),\alpha \in \left(1,2\right]$. Then

$$\begin{array}{ccc}\hfill |T_{2,1}\left(h^{-1}\right)|& \le & 1+{\alpha }^2,\hfill \\ \hfill |T_{2,2}\left(h^{-1}\right)|& \le & \left\{\begin{array}{c}{\displaystyle \frac{1}{18}}[-8{\alpha }^4+48{\alpha }^3+55{\alpha }^2-138\alpha +43],1<\alpha \le {\alpha }_1\hfill \\ {\displaystyle \frac{1}{9}}[8{\alpha }^4+18{\alpha }^3-{\alpha }^2-54\alpha +29],{\alpha }_1<\alpha \le 2\hfill \end{array}\right.\hfill \end{array}$$

where ${\alpha }_1\approx 1.304540.$

and

$$|T_{3,1}\left(h^{-1}\right)|\le {\displaystyle \frac{1}{9}}\left[4{\alpha }^4+18{\alpha }^3+22{\alpha }^2-27\alpha +19\right].$$

Proof. 

The second-order Toeplitz determinants of the inverse coefficients are given by

$$T_{2,1}\left(h^{-1}\right)=b_1^2-b_2^2$$

and

$$T_{2,2}\left(h^{-1}\right)=b_2^2-b_3^2.$$

From (2) and (3), we have

$$b_1^2-b_2^2=1-a_2^2.$$

(57)

Substituting (14) and applying the triangle inequality as well as Lemma 2 in (57) yields the upper bound of $|T_{2,1}\left(h^{-1}\right)|$.

Now, we take

$$T_{2,2}\left(h^{-1}\right)=b_2^2-b_3^2.$$

From (3) and (4), we observe that

$$b_2^2-b_3^2=a_2^2-a_3^2+4a_2^2{a}_3-4a_2^4.$$

(58)

Substituting (14) and (15) in (58), we receive

$$\begin{array}{cc}\hfill b_2^2-b_3^2=& {\displaystyle \frac{1}{3}}[{\displaystyle \frac{-{(\alpha +1)}^2({\alpha }^2+\alpha -2)}{3}}-{\displaystyle \frac{({\alpha }^2-1)(2\alpha +1)}{6}}{\psi }_3-{\displaystyle \frac{{(\alpha -1)}^2}{12}}{\psi }_3^2\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)(8{\alpha }^2+12\alpha -5)}{12}}{\psi }_2+{\displaystyle \frac{(-8{\alpha }^3-4{\alpha }^2+15\alpha +5)(\alpha -1)}{16}}{\psi }_2^2\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)(\alpha -1)(\alpha -2)}{6}}{\psi }_2^3-{\displaystyle \frac{{(\alpha -1)}^2{(\alpha -2)}^2}{48}}{\psi }_2^4\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)(\alpha -1)}{3}}{\psi }_2{\psi }_3-{\displaystyle \frac{{(\alpha -1)}^2(\alpha -2)}{12}}{\psi }_2^2{\psi }_3].\hfill \end{array}$$

(59)

Applying the triangle inequality as well as Lemma 2 in (59) gives the upper bound of $|T_{2,2}\left(h^{-1}\right)|$.

The third-order Toeplitz determinant of the inverse coefficients is given by

$$T_{3,1}\left(h^{-1}\right)=1-2b_2^2-b_3(b_3-2b_2^2).$$

$$|T_{3,1}\left(h^{-1}\right)|\le 1+2|b_2{|}^2+|b_3||b_3-2b_2^2|.$$

(60)

Applying the $|b_2|$ and $|b_3|$ bounds from Theorems 1 and 6 in (60) finalizes the proof of Theorem 7. □

4.2. Bounds of Toeplitz Determinants of Logarithmic Coefficients

Theorem 8. 

Let the function $h\left(z\right)$ defined by (1) belong to the class ${\mathcal{C}}_0\left(\alpha \right),\alpha \in \left(1,2\right]$. Then

$$|T_{2,1}(F_h/2)|\le {\displaystyle \frac{1}{384}}[{\alpha }^4+20{\alpha }^3+102{\alpha }^2-44\alpha -7],$$

$$|T_{2,2}(F_h/2)|\le {\displaystyle \frac{1}{288}}[4{\alpha }^4+110{\alpha }^3+231{\alpha }^2-634\alpha +299]$$

and

$$|T_{3,1}(F_h/2)|\le {\displaystyle \frac{1}{6912}}[5{\alpha }^6+181{\alpha }^5+1699{\alpha }^4+2566{\alpha }^3-7085{\alpha }^2+3541\alpha -91].$$

Proof. 

The second-order Toeplitz determinants of the logarithmic coefficients are given by

$$T_{2,1}(F_h/2)={\gamma }_1^2-{\gamma }_2^2$$

and

$$T_{2,2}(F_h/2)={\gamma }_2^2-{\gamma }_3^2.$$

By referencing (7) and (8), we get

$${\gamma }_1^2-{\gamma }_2^2={\displaystyle \frac{1}{4}}\left[a_2^2-a_3^2+a_2^2{a}_3-{\displaystyle \frac{1}{4}}{a}_2^4\right].$$

(61)

By applying (14) and (15) in (61), we acquire

$$\begin{array}{cc}\hfill {\gamma }_1^2-{\gamma }_2^2=& {\displaystyle \frac{1}{48}}[{\displaystyle \frac{{(\alpha +1)}^2(-{\alpha }^2-10\alpha +119)}{48}}+{\displaystyle \frac{({\alpha }^2-1)(\alpha +5)}{6}}{\psi }_3-{\displaystyle \frac{{(\alpha -1)}^2}{3}}{\psi }_3^2\hfill \\ \hfill & -{\displaystyle \frac{({\alpha }^2-1)(-{\alpha }^2-6\alpha +67)}{24}}{\psi }_2-{\displaystyle \frac{({\alpha }^3+5{\alpha }^2-9\alpha +35)(\alpha -1)}{32}}{\psi }_2^2\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)(\alpha -1)(\alpha +7)}{96}}{\psi }_2^3-{\displaystyle \frac{{(\alpha -1)}^2({\alpha }^2+14\alpha +49)}{768}}{\psi }_2^4\hfill \\ \hfill & -{\displaystyle \frac{({\alpha }^2-1)(\alpha -1)}{6}}{\psi }_2{\psi }_3+{\displaystyle \frac{{(\alpha -1)}^2(\alpha +7)}{24}}{\psi }_2^2{\psi }_3].\hfill \end{array}$$

(62)

Applying the triangle inequality and Lemma 2 in (62) results in the upper bound of $|T_{2,1}(F_h/2)|$.

Now,

$$T_{2,2}(F_h/2)={\gamma }_2^2-{\gamma }_3^2.$$

$$|T_{2,2}(F_h/2)|=|{\gamma }_2-{\gamma }_3||{\gamma }_2+{\gamma }_3|.$$

(63)

From (8) and (9), we have

$${\gamma }_2+{\gamma }_3={\displaystyle \frac{1}{2}}\left[a_3+a_4-a_2{a}_3-{\displaystyle \frac{1}{2}}{a}_2^2+{\displaystyle \frac{1}{3}}{a}_2^3\right].$$

(64)

Substituting (14)–(16) in (64), we get

$$\begin{array}{cc}\hfill {\gamma }_2+{\gamma }_3=& {\displaystyle \frac{1}{48}}[2(\alpha +1)(\alpha +4)-3(\alpha -1){\psi }_4-(\alpha -1)(\alpha +5){\psi }_3-{\displaystyle \frac{3({\alpha }^2-1)}{2}}{\psi }_2\hfill \\ \hfill & +{\displaystyle \frac{3(\alpha -1)(\alpha +3)}{4}}{\psi }_2^2-{\displaystyle \frac{(\alpha -1)(\alpha +3)}{4}}{\psi }_2^3+{\displaystyle \frac{(\alpha -1)(\alpha +5)}{2}}{\psi }_2{\psi }_3].\hfill \end{array}$$

(65)

Applying the triangle inequality and Lemma 2 in (65), we obtain

$$|{\gamma }_2+{\gamma }_3|\le {\displaystyle \frac{1}{12}}[4{\alpha }^2+14\alpha -13].$$

(66)

Using the result of $|V_{2,2}(F_h/2)|$ and (66) in (63) yields $|T_{2,2}(F_h/2)|$.

The third-order Toeplitz determinant of the logarithmic coefficients is given by

$$T_{3,1}(F_h/2)={\gamma }_1({\gamma }_1^2-{\gamma }_2^2)-{\gamma }_2^2({\gamma }_1-{\gamma }_3)+{\gamma }_3({\gamma }_2^2-{\gamma }_1{\gamma }_3).$$

$$|T_{3,1}(F_h/2)|\le |{\gamma }_1||{\gamma }_1^2-{\gamma }_2^2|+|{\gamma }_2{|}^2|{\gamma }_1-{\gamma }_3|+|{\gamma }_3||{\gamma }_1{\gamma }_3-{\gamma }_2^2|.$$

(67)

By multiplying the $|{\gamma }_1|$ bound from Theorem 1 and the result of $|T_{2,1}(F_h/2)|$, we obtain

$$|{\gamma }_1||{\gamma }_1^2-{\gamma }_2^2|\le {\displaystyle \frac{1}{768}}[{\alpha }^5+20{\alpha }^4+102{\alpha }^3-44{\alpha }^2-7\alpha ].$$

(68)

By referring to (7) and (9), we get

$${\gamma }_1-{\gamma }_3={\displaystyle \frac{1}{2}}\left[a_2+a_2{a}_3-a_4-{\displaystyle \frac{1}{3}}{a}_2^3\right].$$

(69)

Substituting (14)–(16) in (69), we obtain

$$\begin{array}{cc}\hfill {\gamma }_1-{\gamma }_3=& {\displaystyle \frac{1}{48}}[-(\alpha +1)(\alpha -9)+3(\alpha -1){\psi }_4+({\alpha }^2-1){\psi }_3+{\displaystyle \frac{(\alpha -1)(\alpha -11)}{2}}{\psi }_2\hfill \\ \hfill & -{\displaystyle \frac{({\alpha }^2-1)}{2}}{\psi }_2^2+{\displaystyle \frac{(\alpha -1)(\alpha +3)}{4}}{\psi }_2^3-{\displaystyle \frac{(\alpha -1)(\alpha +5)}{2}}{\psi }_2{\psi }_3].\hfill \end{array}$$

(70)

Applying the triangle inequality and Lemma 2 in (70), we get

$$|{\gamma }_1-{\gamma }_3|\le {\displaystyle \frac{1}{6}}[{\alpha }^2+6\alpha -5].$$

(71)

Using the $|{\gamma }_2|$ bound from Theorem 1 and (71), we obtain

$$|{\gamma }_2{|}^2|{\gamma }_1-{\gamma }_3|\le {\displaystyle \frac{1}{864}}[{\alpha }^6+18{\alpha }^5+95{\alpha }^4+60{\alpha }^3-412{\alpha }^2+336\alpha -80].$$

(72)

Multiplying the $|{\gamma }_3|$ bound from Theorem 1 and the $|H_{2,1}(F_h/2)|$ result, we get

$$|{\gamma }_3||{\gamma }_1{\gamma }_3-{\gamma }_2^2|\le {\displaystyle \frac{1}{6912}}[-3{\alpha }^6+28{\alpha }^5+759{\alpha }^4+1168{\alpha }^3-3393{\alpha }^2+916\alpha +549].$$

(73)

Utilizing (68), (72) and (73) in (67) finalizes the proof of Theorem 8. □

Theorem 9. 

Let the function $h\left(z\right)$ defined by (1) belong to the class ${\mathcal{C}}_0\left(\alpha \right),\alpha \in \left(1,2\right]$. Then

$${\displaystyle |T_{2,1}(F_{h^{-1}}/2)|\le }\left\{\begin{array}{c}{\displaystyle \frac{1}{1152}}[-175{\alpha }^4+180{\alpha }^3+830{\alpha }^2-1116\alpha +497],1<\alpha \le {\displaystyle \frac{-3+2\sqrt{19}}{5}}\hfill \\ {\displaystyle \frac{1}{384}}[-25{\alpha }^4+100{\alpha }^3+154{\alpha }^2-412\alpha +255],{\displaystyle \frac{-3+2\sqrt{19}}{5}}<\alpha \le \approx 1.490436\hfill \\ {\displaystyle \frac{1}{384}}[25{\alpha }^4+60{\alpha }^3+14{\alpha }^2-276\alpha +249],\approx 1.490436<\alpha \le 2\hfill \end{array}\right.,$$

$${\displaystyle |T_{2,2}(F_{h^{-1}}/2)|\le }\left\{\begin{array}{c}{\displaystyle \frac{1}{1152}}[-45{\alpha }^6+99{\alpha }^5-474{\alpha }^4+5742{\alpha }^3-9219{\alpha }^2+3799\alpha +138],1<\alpha \le {\displaystyle \frac{2+\sqrt{7}}{3}}\hfill \\ {\displaystyle \frac{1}{1152}}[-27{\alpha }^6+147{\alpha }^5-120{\alpha }^4+5098{\alpha }^3-9769{\alpha }^2+4355\alpha +276],{\displaystyle \frac{2+\sqrt{7}}{3}}<\alpha \le 2\hfill \end{array}\right.$$

and

$${\displaystyle |T_{3,1}(F_{h^{-1}}/2)|\le }k\left\{\begin{array}{c}33{\alpha }^7-166{\alpha }^6+511{\alpha }^5-1840{\alpha }^4+28579{\alpha }^3-40526{\alpha }^2+8941\alpha +6100,1<\alpha \le {\alpha }_a\hfill \\ 33{\alpha }^7-76{\alpha }^6+325{\alpha }^5-436{\alpha }^4+22351{\alpha }^3-29804{\alpha }^2+955\alpha +8284,{\alpha }_a<\alpha \le {\alpha }_b\hfill \\ 33{\alpha }^7-76{\alpha }^6+925{\alpha }^5+284{\alpha }^4+20143{\alpha }^3-30524{\alpha }^2+2563\alpha +8284,{\alpha }_b<\alpha \le {\alpha }_c\hfill \\ 33{\alpha }^7-76{\alpha }^6+1825{\alpha }^5-436{\alpha }^4+17623{\alpha }^3-28076{\alpha }^2+2455\alpha +8284,{\alpha }_c<\alpha \le {\alpha }_d\hfill \\ 33{\alpha }^7-76{\alpha }^6+1825{\alpha }^5-436{\alpha }^4+17731{\alpha }^3-27932{\alpha }^2+2059\alpha +7852,{\alpha }_d<\alpha \le 2\hfill \end{array}\right.$$

where, $k={\displaystyle \frac{1}{13824}}$, ${\alpha }_a={\displaystyle \frac{7}{5}}$, ${\alpha }_b={\displaystyle \frac{-3+2\sqrt{19}}{5}}$, ${\alpha }_c\approx 1.490436$ and ${\alpha }_d={\displaystyle \frac{-1+2\sqrt{145}}{6}}$.

Proof. 

The second-order Toeplitz determinants for the inverse logarithmic coefficients are given by

$$T_{2,1}(F_{h^{-1}}/2)={\mathrm{\Gamma }}_1^2-{\mathrm{\Gamma }}_2^2$$

and

$$T_{2,2}(F_{h^{-1}}/2)={\mathrm{\Gamma }}_2^2-{\mathrm{\Gamma }}_3^2.$$

By referencing (10) and (11), we obtain

$${\mathrm{\Gamma }}_1^2-{\mathrm{\Gamma }}_2^2={\displaystyle \frac{1}{16}}[4a_2^2-4a_3^2+12a_2^2{a}_3-9a_2^4].$$

(74)

By using (14) and (15) in (74), we receive

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_1^2-{\mathrm{\Gamma }}_2^2=& {\displaystyle \frac{1}{48}}[{\displaystyle \frac{{(\alpha +1)}^2(-25{\alpha }^2-10\alpha +143)}{48}}-{\displaystyle \frac{({\alpha }^2-1)(5\alpha +1)}{6}}{\psi }_3-{\displaystyle \frac{{(\alpha -1)}^2}{3}}{\psi }_3^2\hfill \\ \hfill & +{\displaystyle \frac{({\alpha }^2-1)(25{\alpha }^2+30\alpha -67)}{24}}{\psi }_2+{\displaystyle \frac{(-25{\alpha }^3-5{\alpha }^2+65\alpha -3)(\alpha -1)}{32}}{\psi }_2^2\hfill \\ \hfill & +{\displaystyle \frac{5({\alpha }^2-1)(\alpha -1)(5\alpha -13)}{96}}{\psi }_2^3-{\displaystyle \frac{{(\alpha -1)}^2(25{\alpha }^2-130\alpha +169)}{768}}{\psi }_2^4\hfill \\ \hfill & +{\displaystyle \frac{5({\alpha }^2-1)(\alpha -1)}{6}}{\psi }_2{\psi }_3-{\displaystyle \frac{{(\alpha -1)}^2(5\alpha -13)}{24}}{\psi }_2^2{\psi }_3].\hfill \end{array}$$

(75)

Applying the triangle inequality and Lemma 2 in (75) results in the upper bound of $|T_{2,1}(F_{h^{-1}}/2)|$.

Now,

$$T_{2,2}(F_{h^{-1}}/2)={\mathrm{\Gamma }}_2^2-{\mathrm{\Gamma }}_3^2.$$

$$|T_{2,2}(F_{h^{-1}}/2)|=|{\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_3||{\mathrm{\Gamma }}_2+{\mathrm{\Gamma }}_3|.$$

(76)

From (11) and (12), we have

$${\mathrm{\Gamma }}_2+{\mathrm{\Gamma }}_3={\displaystyle \frac{1}{2}}\left[4a_2{a}_3-a_3-a_4+{\displaystyle \frac{3}{2}}{a}_2^2-{\displaystyle \frac{10}{3}}{a}_2^3\right].$$

(77)

Substituting (14), (15) and (16) in (77), we get

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_2+{\mathrm{\Gamma }}_3=& {\displaystyle \frac{1}{48}}[(-3{\alpha }^2+4\alpha +1)(\alpha +1)+3(\alpha -1){\psi }_4-(\alpha -1)(5\alpha +1){\psi }_3\hfill \\ & +{\displaystyle \frac{3({\alpha }^2-1)(3\alpha -2)}{2}}{\psi }_2+{\displaystyle \frac{(-9{\alpha }^2+15\alpha +6)(\alpha -1)}{4}}{\psi }_2^2\hfill \\ \hfill & -{\displaystyle \frac{(3{\alpha }^2-16\alpha +21)(\alpha -1)}{8}}{\psi }_2^3+{\displaystyle \frac{(\alpha -1)(5\alpha -11)}{2}}{\psi }_2{\psi }_3].\hfill \end{array}$$

(78)

Using the triangle inequality and Lemma 2 in (78), we obtain

$$|{\mathrm{\Gamma }}_2+{\mathrm{\Gamma }}_3|\le \left\{\begin{array}{c}{\displaystyle \frac{1}{48}}[-15{\alpha }^3+78{\alpha }^2-57\alpha -2],1<\alpha \le {\displaystyle \frac{2+\sqrt{7}}{3}}\hfill \\ {\displaystyle \frac{1}{48}}[-9{\alpha }^3+76{\alpha }^2-67\alpha -4],{\displaystyle \frac{2+\sqrt{7}}{3}}<\alpha \le 2\hfill \end{array}\right..$$

(79)

By using the result of $|V_{2,2}(F_{h^{-1}}/2)|$ and (79) in (76) yields $|T_{2,2}(F_{h^{-1}}/2)|$.

The third-order Toeplitz determinant of logarithmic coefficients is given by

$$T_{3,1}(F_{h^{-1}}/2)={\gamma }_1({\gamma }_1^2-{\gamma }_2^2)-{\gamma }_2^2({\gamma }_1-{\gamma }_3)+{\gamma }_3({\gamma }_2^2-{\gamma }_1{\gamma }_3).$$

$$|T_{3,1}(F_{h^{-1}}/2)|\le |{\mathrm{\Gamma }}_1||{\mathrm{\Gamma }}_1^2-{\mathrm{\Gamma }}_2^2|+|{\mathrm{\Gamma }}_2{|}^2|{\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_3|+|{\mathrm{\Gamma }}_3||{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_2^2|.$$

(80)

By multiplying the $|{\mathrm{\Gamma }}_1|$ bound from Theorem 1 and the result of $|T_{2,1}(F_{h^{-1}}/2)|$, we obtain

$${\displaystyle |{\mathrm{\Gamma }}_1||{\mathrm{\Gamma }}_1^2-{\mathrm{\Gamma }}_2^2|\le }\left\{\begin{array}{c}{\displaystyle \frac{1}{2304}}[-175{\alpha }^5+180{\alpha }^4+830{\alpha }^3-1116{\alpha }^2+497\alpha ],1<\alpha \le {\displaystyle \frac{-3+2\sqrt{19}}{5}}\hfill \\ {\displaystyle \frac{1}{768}}[-25{\alpha }^5+100{\alpha }^4+154{\alpha }^3-412{\alpha }^2+255\alpha ],{\displaystyle \frac{-3+2\sqrt{19}}{5}}<\alpha \le \approx 1.490436\hfill \\ {\displaystyle \frac{1}{768}}[25{\alpha }^5+60{\alpha }^4+14{\alpha }^3-276{\alpha }^2+249\alpha ],\approx 1.490436<\alpha \le 2\hfill \end{array}\right..$$

(81)

By referring (10) and (12), we get

$${\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_3={\displaystyle \frac{1}{6}}[3a_4-3a_2-12a_2{a}_3+10a_2^3].$$

(82)

Substituting (14)–(16) in (82), we obtain

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_3=& {\displaystyle \frac{1}{48}}[(3{\alpha }^2+\alpha -12)(\alpha +1)-3(\alpha -1){\psi }_4+5({\alpha }^2-1){\psi }_3\hfill \\ \hfill & -{\displaystyle \frac{(9{\alpha }^2+13\alpha -8)(\alpha -1)}{2}}{\psi }_2+{\displaystyle \frac{({\alpha }^2-1)(9\alpha -19)}{4}}{\psi }_2^2\hfill \\ \hfill & -{\displaystyle \frac{(3{\alpha }^2-16\alpha +21)(\alpha -1)}{8}}{\psi }_2^3-{\displaystyle \frac{(\alpha -1)(5\alpha -11)}{2}}{\psi }_2{\psi }_3].\hfill \end{array}$$

(83)

Applying the triangle inequality and Lemma 2 in (83), we get

$$|{\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_3|\le \left\{\begin{array}{c}{\displaystyle \frac{1}{3}}[6\alpha -5],1<\alpha \le {\displaystyle \frac{-1+\sqrt{145}}{6}}\hfill \\ {\displaystyle \frac{1}{24}}[3{\alpha }^3+4{\alpha }^2+37\alpha -52],{\displaystyle \frac{-1+\sqrt{145}}{6}}<\alpha \le 2\hfill \end{array}\right..$$

(84)

Using the $|{\mathrm{\Gamma }}_2|$ bound from Theorem 1 and (84), we obtain

$$|{\mathrm{\Gamma }}_2{|}^2|{\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_3|\le \left\{\begin{array}{c}{\displaystyle \frac{1}{48}}[6\alpha -5],1<\alpha \le {\displaystyle \frac{-1+\sqrt{145}}{6}}\hfill \\ {\displaystyle \frac{1}{384}}[3{\alpha }^3+4{\alpha }^2+37\alpha -52],{\displaystyle \frac{-1+\sqrt{145}}{6}}<\alpha \le 2\hfill \end{array}\right..$$

(85)

Multiplying the $|{\mathrm{\Gamma }}_3|$ bound from Theorem 1 and the $|H_{2,1}(F_{h^{-1}}/2)|$ result, we have

$${\displaystyle |{\mathrm{\Gamma }}_3||{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_2^2|\le }k\left\{\begin{array}{c}33{\alpha }^7-166{\alpha }^6+1561{\alpha }^5-2920{\alpha }^4+23599{\alpha }^3-33830{\alpha }^2+4231\alpha +7540,1<\alpha \le {\displaystyle \frac{7}{5}}\hfill \\ 33{\alpha }^7-76{\alpha }^6+1375{\alpha }^5-1516{\alpha }^4+17371{\alpha }^3-23108{\alpha }^2-3755\alpha +9724,{\displaystyle \frac{7}{5}}<\alpha \le 2\hfill \end{array}\right.$$

(86)

where $k={\displaystyle \frac{1}{13824}}$.

Utilizing (81), (85), and (86) in (80) finalizes the proof of Theorem 9. □

5. Bounds of Vandermonde Determinants of Inverse and Logarithmic Coefficients for the Class ${\mathcal{C}}_{\mathbf{0}}\left(\mathbf{\alpha }\right)$ of Concave Univalent Functions

5.1. Bounds of Vandermonde Determinant of Inverse Coefficients

Theorem 10. 

Let the function $h\left(z\right)$ defined by (1) belong to the class ${\mathcal{C}}_0\left(\alpha \right),\alpha \in \left(1,2\right]$. Then

$$|V_{3,1}\left(h^{-1}\right)|\le {\displaystyle \frac{2}{9}}[2{\alpha }^5+17{\alpha }^4+35{\alpha }^3-4{\alpha }^2-19\alpha +5].$$

Proof. 

The third-order Vandermonde determinant of inverse coefficients is given by

$$V_{3,1}\left(h^{-1}\right)=(b_3-b_2)(b_3-b_1)(b_2-b1).$$

$$|V_{3,1}\left(h^{-1}\right)|=|b_3-b_2||b_3-b_1||b_2-b1|.$$

(87)

By considering (2) and the $|b_3|$ bound from Theorem 1, we have

$$|b_3-b_1|\le {\displaystyle \frac{2}{3}}\left[{\alpha }^2+3\alpha -1\right].$$

(88)

Applying the $|V_{2,1}\left(h^{-1}\right)|$ and $|V_{2,2}\left(h^{-1}\right)|$ bounds from Theorem 2 and (88) in (87) completes the proof of Theorem 10. □

5.2. Bounds of Vandermonde Determinants of Logarithmic Coefficients

Theorem 11. 

Let the function $h\left(z\right)$ defined by (1) belong to the class ${\mathcal{C}}_0\left(\alpha \right),\alpha \in \left(1,2\right]$. Then

$$|V_{3,1}(F_h/2)|\le {\displaystyle \frac{1}{3456}}[-{\alpha }^6-12{\alpha }^5+413{\alpha }^4+2016{\alpha }^3-6035{\alpha }^2+4908\alpha -1265].$$

and

$${\displaystyle |V_{3,1}(F_{h^{-1}}/2)|\le }\left\{\begin{array}{c}{\displaystyle \frac{1}{1728}}[90{\alpha }^6+735{\alpha }^5+2649{\alpha }^4+6302{\alpha }^3-26814{\alpha }^2+23083\alpha -5865],1<\alpha \le {\displaystyle \frac{-1+\sqrt{145}}{6}}\hfill \\ \begin{array}{c}\hfill {\displaystyle \frac{1}{13824}}[45{\alpha }^8+465{\alpha }^7+2757{\alpha }^6+10967{\alpha }^5+9899{\alpha }^4+17819{\alpha }^3\\ \hfill -192665{\alpha }^2+210269\alpha -60996],\end{array}{\displaystyle \frac{-1+\sqrt{145}}{6}}<\alpha \le 2\hfill \end{array}\right..$$

Proof. 

The third-order Vandermonde determinant for logarithmic coefficients is given by

$$V_{3,1}(F_h/2)=({\gamma }_3-{\gamma }_2)({\gamma }_3-{\gamma }_1)({\gamma }_2-{\gamma }_1).$$

$$|V_{3,1}(F_h/2)|=|{\gamma }_3-{\gamma }_2||{\gamma }_3-{\gamma }_1||{\gamma }_2-{\gamma }_1|.$$

(89)

Applying the $|V_{2,1}(F_h/2)|$ and $|V_{2,2}(F_h/2)|$ results from Theorem 2 and (71) in (89) results in the upper bound of $|V_{3,1}(F_h/2)|$.

The third-order Vandermonde determinant for inverse logarithmic coefficients is given by

$$V_{3,1}(F_{h^{-1}}/2)=({\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_2)({\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_1)({\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_1).$$

$$|V_{3,1}(F_{h^{-1}}/2)|=|{\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_2||{\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_1||{\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_1|.$$

(90)

Applying the $|V_{2,1}(F_{h^{-1}}/2)|$ and $|V_{2,2}(F_{h^{-1}}/2)|$ results from Theorem 2 and (84) in (90) ends the proof of Theorem 11. □

6. Concluding Remarks and Observations

In this article, we found the bounds of inverse coefficients, logarithmic coefficients, inverse logarithmic coefficients, and differences between these successive coefficients for a certain subclass of univalent functions. Also, we considered the Hankel, Toeplitz, and Vandermonde determinants by taking inverse coefficients, logarithmic coefficients, and inverse logarithmic coefficients for concave univalent functions. We obtained the upper bounds for the mentioned different types of determinants related to the given subclass of univalent functions.

Given the importance of the inverse and logarithmic coefficients of univalent functions, our work provides a new basis for the study of the Hankel, Toeplitz, and Vandermonde determinants. It could also inspire further similar results investigating other subfamilies of univalent functions or taking the bounds of those specified higher-order determinants.