A study of sharp coefficient bounds for a new subfamily of starlike functions

In this article, by employing the hyperbolic tangent function tanhz, a subfamily Stanh∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{S}_{\tanh }^{\ast }$\end{document} of starlike functions in the open unit disk D⊂C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{D}\subset \mathbb{C}$\end{document}: D={z:z∈C and |z|<1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \mathbb{D}= \bigl\{ z:z\in \mathbb{C} \text{ and } \vert z \vert < 1 \bigr\} \end{aligned}$$ \end{document} is introduced and investigated. The main contribution of this article includes derivations of sharp inequalities involving the Taylor–Maclaurin coefficients for functions belonging to the class Stanh∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{S}_{\tanh }^{\ast } $\end{document} of starlike functions in D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{D}$\end{document}. In particular, the bounds of the first three Taylor–Maclaurin coefficients, the estimates of the Fekete–Szegö type functionals, and the estimates of the second- and third-order Hankel determinants are the main problems that are proposed to be studied here.

Though the subject of function theory was founded in 1851, the coefficient conjecture presented by Bieberbach [13] in 1916 led to the field's emergence as a promising area of new research. This conjecture was proved by de Branges [18] in 1985. Between 1916 and 1985, many of the finest scholars of the day sought to prove or disprove this Bieberbach conjecture. As a consequence, they discovered numerous sub-families of the class S of normalized univalent functions connected to distinct image domains. The families of starlike and convex functions, respectively, denoted by S * and K, are the most fundamental and significant subclasses of the set S. In 1992, Ma and Minda [36] considered the general form of the family as follows: where φ is a holomorphic function with φ (0) > 0 and has a positive real part in D. Also, the function φ maps D onto a star-shaped region with respect to φ(0) = 1 and is symmetric about the real axis. They addressed some specific results such as distortion, growth, and covering theorems. In recent years, several sub-families of the normalized analytic function class A were studied as a special case of the class S * (φ). For example, we have: which is described as the functions of the Janowski starlike class investigated in [22]. Furthermore, the class S * (ξ ) given by is the familiar starlike function family of order ξ with 0 ξ < 1. (ii) The following family: was studied in [49] by Sokól and Stankiewicz. The function φ(z) = √ 1 + z maps the region D onto the image domain which is bounded by |w 2 -1| < 1.
(iii) The class given by was examined by Sharma et al. [46]. It consists of functions f ∈ A in such a manner that is located in the region bounded by the cardioid given by 9x 2 + 9y 2 -18x + 5 2 -16 9x 2 + 9y 2 -6x + 1 = 0.
(iv) By selecting φ(z) = 1 + sin z, the class S * (φ(z)) leads to the family S * sin , which was investigated by Cho et al. [17]. On the other hand, the function class given by S * e ≡ S * e z was studied in [38] and, subsequently, in [48]. This function class was recently generalized by Srivastava et al. [56] in which the authors determined an upper bound of the Hankel determinant of the third order. (v) The following families: and S * cosh := S * (cosh z) were considered, respectively, by Raza and Bano [9] and Alotaibi et al. [2]. In both of these papers, the authors studied some interesting properties of the families which they studied. (vi) By choosing φ(z) = 1 + sin z, we obtain the following class: which was investigated in [17]. The authors in [17] addressed the radii problems for the defined class S * sin . (vii) By considering the function φ(z) = 1 + sinh -1 z, we get the recently-examined family given by which was introduced by Kumar and Arora [29]. They discussed relationships of this class with the already known classes. In 2021, Barukab et al. [12] derived sharp bounds for the Hankel determinant of the third order for the following function class: In the present paper, we consider the following hyperbolic function: Also, one can easily find that (ϕ 1 (z)) > 0.
Definition 1 ([59]) By using the above-defined hyperbolic function ϕ 1 (z), we define the following family of functions: In other words, a function f is in the class S * tanh if and only if there exists a holomorphic function q, fulfilling q(z) ≺ q 0 (z) := 1 + tanh z, such that By taking (3), we get the function that plays the role of the extremal function in many problems of the class S * tanh , given by for a function f ∈ S of the series form (1) was given by Pommerenke [40,41] as follows: .

Definition 2 The Hankel determinant
In particular, the following determinants are known as the first-, the second-, and the third-order Hankel determinants, respectively: and HD 3,1 (f ) = 1 a 2 a 3 a 2 a 3 a 4 a 3 a 4 a 5 = a 3 a 2 a 4a 2 3a 4 (a 4a 2 a 3 ) + a 5 a 3a 2 2 .
In the literature, there are just a few references to the Hankel determinant for functions belonging to the general family S. For the function f ∈ S, the best established sharp inequality is given by where λ is an absolute constant. This result is due to Hayman [21]. Further, for the same class S, it was derived in [39] as follows: and The challenge of finding the sharp bounds of Hankel determinants for a particular family of functions drew the attention of numerous researchers. For example, the sharp bounds of |HD 2,2 (f )| for the sub-families K, S * , and R (the family of bounded turning functions) of the class S were calculated by Janteng et al. [23,24]. These estimates are given by 4 9 (f ∈ R).
It is quite clear from the formulas given in (5), (6), and (7) that the calculation of the bound for |HD 3,1 (f )| is far more challenging in comparison with the finding of the bound for |HD 2,2 (f )|. In the year 2010, Babalola [8] investigated the bounds for the third-order Hankel determinant for the families of K, S * , and R. Subsequently, by using the same or analogous approach, several authors in [3,11,28,43,45] derived bounds for the thirdorder Hankel determinant |HD 3,1 (f )| for various sub-families of analytic and univalent functions. On the other hand, in the year 2017, Zaprawa [61] improved the findings of Babalola [8] by applying a new methodology to show that 41 60 (f ∈ R).
Zaprawa [61] remarked that such limits were indeed not the best ones. Later in the year 2018, Kwon et al. [31] strengthened Zaprawa's result for f ∈ S * and showed that |HD 3,1 (f )| 8 9 , and this bound was further improved by Zaprawa et al. [62] by showing in 2021 that In recent years, the following sharp bounds for the third-order Hankel determinant |HD 3,1 (f )| were given by Kowalczyk et al. [27] and Lecko et al. [32]: where S * ( 1 2 ) represents the family of starlike functions of order 1 2 in D. The interested readers may also refer to the research provided by Mahmood et al. [37] in which they calculated bounds for the third-order Hankel determinant for the basic (or q-) starlike functions in D.
In the present article, our aim is to calculate the sharp bounds of the coefficient inequalities, Fekete-Szegö type functional, and the Hankel determinants of order two and order three for the subclass S * tanh of starlike functions.

A set of lemmas
Definition 3 A function p is said to be in the class P if and only if it has the following series expansion: and satisfies the inequality given by Lemma 1 Let the function p ∈ P have the series form (8). Then, for x, δ, ρ ∈ D = D ∪ {1}, and Remark 2 In Lemma 1 and elsewhere in this paper, for the formula for c 2 , see [42]. The formula for c 3 is due to Libera and Złotkiewicz [34]. The formula for c 4 was proved in [30].
Lemma 3 If the function p ∈ P has the series form (8), then and |c n | 2 (n 1).

Coefficient inequalities for the function class S * tanh
The first two findings, Theorem 5 and Theorem 6, are special cases of the results established in the paper [1], and that is why we omitted both the proofs.

Theorem 5
Let the function f of the form (1) be in the class S * tanh . Then Each of these bounds is sharp.

Theorem 6
Let the function f of the form (1) be in the class S * tanh . Then This inequality is sharp.

Theorem 7
Let the function f of the form (1) be in the class S * tanh . Then This result is sharp.
After some calculation and by using the series expansion given by (16), we get Now, if we compare (17) and (18), we get By using (19), (20), and (21), we obtain which, in view of (9) and (10) Finally, upon taking G (c) = 0, we obtain c = 0, 1. Thus, clearly, G(c) has its maximum value at c = 0, so that in which the equality holds true for the extremal function given by This evidently completes our demonstration of Theorem 7.

Theorem 8
Let the function f of the form (1) be in the class S * tanh . Then This inequality is sharp.
Now, by using (9) and (10)  As G (c) 0, so G(c) is a decreasing function of c, so that it gives the maximum value at c = 0: Finally, the above bound for HD 2,2 (f ) is sharp and is achieved by the following extremal function: We have thus completed the proof of Theorem 8.

The third Hankel determinant
In this section, we determine the bounds of |HD 3,1 (f )| for the function f ∈ S * tanh .

Theorem 9
Let the function f of the form (1) be in the class S * tanh . Then This result is sharp.
Upon substituting these expressions into (25) and simplifying, we get Now, since t = (4c 2 ), we have Thus, upon setting |δ| = y and |x| = x, and by taking |ρ| 1, we obtain where Let the closed cuboid be of the following form: We need to find the points of maxima inside this closed cuboid , inside the six faces, and on the twelve edges in order to maximize the function H(c, x, y) given by (27). For this objective in view, we consider the following three cases. I. Let c, x, y ∈ (0, 2) × (0, 1) × (0, 1). In order to find the points of maxima inside , we take partial derivative of (27) with respect to y, so that we achieve and We now have to get the solutions which satisfy both of inequalities (29) and (30) for the existence of the critical points. Let us set Since h (x) < 0 for (0, 1), the function h(x) is decreasing in (0, 1). Hence c 2 > 7 4 , and a simple exercise shows that (29) does not hold true in this case for all values of x ∈ (0, 1) and there is no critical point of H(c, x, y) in (0, 2) × (0, 1) × (0, 1).

Concluding remarks and observations
In the present article, we have introduced and studied a new subfamily of starlike functions in the open unit disk D, which involves the hyperbolic function tanh z. For functions belonging to such a class of starlike functions, we have considered some interesting problems such as the bounds of the first three Taylor-Maclaurin coefficients, the estimates of the Fekete-Szegö type functional, and the estimates of the second-and third-order Hankel determinants. All of the bounds which we have investigated in this article have been shown to be sharp. A potential direction for further research based upon our present investigation would involve the use of the familiar quantum or basic (or q-) calculus as (for example) in the related recent works [37,44,50,53,54,56], [57], and [58]. However, as clearly pointed out in the survey-cum-expository review articles by Srivastava (see, for details, [50, p. 340]; see also [51, pp. 1511-1512]), any attempt to translate these suggested q-results in terms of the so-called trivial and inconsequential (p, q)-calculus would obviously lead to a shallow research, because the additional forced-in parameter p is obviously redundant or superfluous.