On Hankel and Inverse Hankel Determinants of Order Two for Some Subclasses of Analytic Functions

Let us denote by $\mathcal{A}$ the class of analytic functions f over the unit open disk $E=\{z\in \mathbb{C}:|z|<1\},$ which are normalized as $f\left(0\right)=0=f^{\prime }\left(z\right)-1,$ and hence of the form and the subclass of members of $\mathcal{A}$ which are univalent too, by $\mathcal{S}.$ The relation of subordination between analytic functions provides a convenient way to define various subclasses of $\mathcal{A}$ or $\mathcal{S}$. We say that an analytic function f in E is subordinate to an analytic function g in E written as $f\prec g,$ if there is some function $w\left(z\right)$ satisfying $\left|w\right(z\left)\right|<\left|z\right|$ and $w\left(0\right)=0$, known as the Schawarz function, such that $f\left(z\right)=g\left(w\right(z\left)\right)$. More specifically, if g is univalent then the relation $f\prec g$ holds if and only if $f\left(E\right)\subset g\left(E\right)$.

Further, for $f\in \mathcal{A}$ and of the form (1), we denote the Hankel determinant of order q by $H_{q,n}\left(f\right),$ where the integers q and n are such that $q\ge 2$ and $n\ge 1$ and is defined as follows:

This was from Pommerenke [1], who first worked on it. Hankel determinants have applications in the theory of singularities [2], and in the study of power series having integer coefficients [3,4,5]. Computing upper bounds of the absolute of Hankel determinants for various subclasses of analytic and univalent functions for larger values of q and n are tough to handle but at the same time interesting enough not to be left untouched. Hence, many authors worked in this direction and obtained upper bounds (sharp in many cases) of second- and third-order Hankel determinants for different subclasses of class $\mathcal{S}$; for example, see [6,7,8,9,10,11,12,13].

Expanding the second-order Hankel determinant $H_{2,2}\left(f\right)$ and third-order Hankel determinant $H_{3,1}\left(f\right)$ for f of the form (1), give and

While the Hankel determinant $H_{2,1}\left(f\right)=a_3-a_2^2$ is a special case of $a_3-\mu a_2^2,$ for $\mu =1,$ which is the famous Fekete–Szeg$\ddot{o}$ functional [6]. For work in this direction, see for example [14].

Further, researchers have also been started investigating the second Hankel determinant for $f^{-1},$ that is, inverse functions of the functions f in various subclasses of $\mathcal{S}.$ Since each $f\in \mathcal{S}$ is univalent, so if $w=f\left(z\right),$ then there must be some disk $\left|w\right|<r_0\left(f\right)$ for which $f^{-1}$ exists. More specifically, there will be at least the disk with $r_0=1/4,$ guaranteed by Koebe’s quarter theorem. Let where, $f\left(f^{-1}\left(w\right)\right)=w,$ so from (1) and (3) by equating coefficients, we have

Second and third coefficient estimates of inverse functions belonging to a certain subclass of Bazilevic functions have recently been obtained in [15].

Now, for the inverse function $f^{-1}$ of the form (3), the second Hankel determinant is given by

Before introducing our three subclasses, it is necessary to have definitions of the classes, $\mathcal{C}\left(\alpha \right),$${\mathcal{S}}^{*}\left(\alpha \right)$ and $\mathcal{G}\left(\alpha \right).$ For $0\le \alpha \le 1,$ the subclass $\mathcal{C}\left(\alpha \right)$ is known as the class of convex functions of order $\alpha$ and ${\mathcal{S}}^{*}\left(\alpha \right)$ the class of starlike functions of order $\alpha ,$ defined, respectively, as and

Since convex functions as well as starlike functions are univalent, both $\mathcal{C}\left(\alpha \right),$${\mathcal{S}}^{*}\left(\alpha \right)$ are subclasses of the class $\mathcal{S}$ and $\mathcal{C}\left(\alpha \right)\subset {\mathcal{S}}^{*}\left(\alpha \right).$ Also for $\alpha =0,$$\mathcal{C}\left(\alpha \right)$ and ${\mathcal{S}}^{*}\left(\alpha \right)$ become the well-known subclasses $\mathcal{C}$ and ${\mathcal{S}}^{*}$, respectively, of convex and starlike functions. For $f\in \mathcal{C}\left(\alpha \right),$ a non-sharp bound of $H_{2,2}\left(f\right)$ was given by V. Krishna et al. [7], which was further improved by Thomas et al. [8], and finally the best possible result was obtained by Y. J. Sim et al. [9]. While for $f\in S^{*}\left(\alpha \right)$, the sharp bound for $H_{2,2}\left(f\right)$ was obtained by N. E. Cho et al. [10]. In the paper [9], they are also successful in obtaining the sharp upper bounds of $H_{2,2}\left(f^{-1}\right)$ for the inverse function $f^{-1}$ when $f\in \mathcal{C}\left(\alpha \right)$, as well as when $f\in {\mathcal{S}}^{*}\left(\alpha \right),$ and as such the struggle for finding sharp upper bounds of $H_{2,1}\left(f\right)$ and $H_{2,2}\left(f^{-1}\right)$ for these two subclasses came to an end.

Further, the class $\mathcal{G}\left(\alpha \right),$ when $0<\alpha \le 1,$ is defined as

Ozaki [12] considered the class $\mathcal{G}\left(\alpha \right)$ for $\alpha =1,$ denoted by $\mathcal{G}$, and proved that $\mathcal{G}\subset \mathcal{S},$ the class of univalent functions. Later, a general form of this class was discussed by Umezawa [13]; in this paper he showed that each $f\in \mathcal{G}$ is convex in one direction. Apart from this, it has also been proven (see [16,17]) that $\mathcal{G}$ is a subclass of the class ${\mathcal{S}}^{*}$ of starlike functions, and so $\mathcal{G}\left(\alpha \right),$ for $0<\alpha \le 1$ is a subclass of the class of starlike functions ${\mathcal{S}}^{*}.$

Motivated by the work in [18] for the three classes $\mathcal{C}\left(\alpha \right),$${\mathcal{S}}^{*}\left(\alpha \right)$ and $\mathcal{G}\left(\alpha \right)$ and the way the authors in [19] defined the class $S{L}^{*}\left(\beta \right)$ as which for $\beta =0$ reduces to the class $S{L}^{*},$ defined by J. Sokól et al. in [20], we define the two subclasses $\mathcal{C}L\left(\beta \right)$ and $\mathcal{G}L\left(\beta \right)$ for $0\le \beta <1$ and $0<\beta <1$, respectively, by and

We need the following lemmas in the sequel.

First, we investigate upper bounds for second Hankel determinant, namely $H_{2,2}\left(f\right)$ as well as for $H_{2,1}\left(f\right)$, in all the above-mentioned three subclasses one by one.

Studying Hankel determinants of inverse functions provides valuable insights into different aspects of analytic functions and their geometry, including univalence and distortion, boundary behavior, geometric properties under inversion, analytic continuation and coefficient relationships of the analytic functions and its inverses. Researchers in Geometric Function Theory explore these properties to understand the intricate relationship between analytic functions, their inverses and the geometric implications within the complex plane.

Case-I: Here, $0\le \beta <\frac{3}{5}.$ For this interval of $\beta ,$$(3-5\beta )>0.$

Let $|c_1|=x,$ then by Lemma 2, $0\le x\le 1.$ Also, if

Then, the inequality $\left(26\right)$ becomes

Differentiating $g\left(x\right)$ twice with respect to x, we have and setting $g^{\prime }\left(x\right)=0,$ gives

Since for $x=0,$$g^{\prime \prime }\left(0\right)=2(3-5\beta )>0,$ and for $x^2=\frac{2(3-5\beta )}{31-7\beta -4{\beta }^2},$ we have $g^{\prime \prime }\left(x\right)=-4(3-5\beta )<0,$ therefore

Thus, substituting $|c_1|=\sqrt{\frac{2(3-5\beta )}{31-7\beta -4{\beta }^2}}$ into the inequality $\left(26\right),$ gives

Case-II: Here, $\frac{3}{5}\le \beta <1.$ For this range of $\beta ,$ we have $(3-5\beta )\le 0.$ Therefore, the right side of the inequality $\left(26\right),$ attains its maximum at $|c_1|=0,$ which gives

The last inequality implies the required result. This inequality is sharp for $w\left(z\right)=z^2$ in Equation $\left(7\right),$ or for $c_2=1$ and all other coefficients’ zeros in Equation $\left(25\right).$