Second Hankel determinant of Logarithmic coefficients for Starlike and Convex functions associated with lune

Lemma 2.1. [17,18] If $p \in \mathcal{P}$ is of the form (2.1) with $c_1 \geq 0$ , then <span id="page-3-1"></span> $$(2.2) c_1 = 2\tau_1,$$ <span id="page-3-2"></span> $$(2.3) c_2 = 2\tau_1^2 + 2(1 - \tau_1^2)\tau_2$$ and $$(2.4) c_3 = 2\tau_1^3 + 4(1-\tau_1^2)\tau_1\tau_2 - 2(1-\tau_1^2)\tau_1\tau_2^2 + 2(1-\tau_1^2)(1-|\tau_2|^2)\tau_3$$ for some $\tau_1 \in [0,1]$ and $\tau_2, \tau_3 \in \overline{\mathbb{D}} := \{z \in \mathbb{C} : |z| \le 1\}.$ For $\tau_1 \in \mathbb{T} := \{z \in \mathbb{C} : |z| = 1\}$ , there is a unique function $p \in \mathcal{P}$ with $c_1$ as in (2.2), namely $$p(z) = \frac{1 + \tau_1 z}{1 - \tau_1 z}, \quad z \in \mathbb{D}.$$ For $\tau_1 \in \mathbb{D}$ and $\tau_2 \in \mathbb{T}$ , there is a unique function $p \in \mathcal{P}$ with $c_1$ and $c_2$ as in (2.2) and (2.3), namely $$p(z) = \frac{1 + (\overline{\tau_1}\tau_2 + \tau_1)z + \tau_2 z^2}{1 + (\overline{\tau_1}\tau_2 - \tau_1)z - \tau_2 z^2}, \quad z \in \mathbb{D}.$$ For $\tau_1, \tau_2 \in \mathbb{D}$ and $\tau_3 \in \mathbb{T}$ , there is a unique function $p \in \mathcal{P}$ with $c_1, c_2$ and $c_3$ as in (2.2)-(2.3), namely $$p(z) = \frac{1 + (\overline{\tau_2}\tau_3 + \overline{\tau_1}\tau_2 + \tau_1)z + (\overline{\tau_1}\tau_3 + \tau_1\overline{\tau_2}\tau_3 + \tau_2)z^2 + \tau_3z^3}{1 + (\overline{\tau_2}\tau_3 + \overline{\tau_1}\tau_2 - \tau_1)z + (\overline{\tau_1}\tau_3 - \tau_1\overline{\tau_2}\tau_3 - \tau_2)z^2 - \tau_3z^3}, \quad z \in \mathbb{D}.$$

Lemma 2.2. [5] Let A, B, C be real numbers and $$Y(A, B, C) := \max\{|A + Bz + Cz^2| + 1 - |z|^2 : z \in \overline{\mathbb{D}}\}.$$ (i) If AC > 0, then $$Y(A, B, C) = \begin{cases} |A| + |B| + |C|, & |B| \ge 2(1 - |C|), \\ 1 + |A| + \frac{B^2}{4(1 - |C|)}, & |B| < 2(1 - |C|). \end{cases}$$ (ii) If AC < 0, then (ii) If $$AC < 0$$ , then $$Y(A, B, C) = \begin{cases} 1 - |A| + \frac{B^2}{4(1 - |C|)}, & -4AC(C^{-2} - 1) \le B^2 \wedge |B| < 2(1 - |C|), \\ 1 + |A| + \frac{B^2}{4(1 + |C|)}, & B^2 < \min\{4(1 + |C|)^2, -4AC(C^{-2} - 1)\}, \\ R(A, B, C), & otherwise, \end{cases}$$ where $$R(A,B,C) := \begin{cases} |A| + |B| - |C|, & |C|(|B| + 4|A|) \le |AB|, \\ -|A| + |B| + |C|, & |AB| \le |C|(|B| - 4|A|), \\ (|C| + |A|)\sqrt{1 - \frac{B^2}{4AC}}, & otherwise. \end{cases}$$ For a better clarity in our presentation, we divide this into two section consisting of different families of functions from the class A and prove our main results for starlike functions and convex functions associated with lune.

Theorem 4.1. Let $f \in \mathcal{C}_{\mathbb{C}}$ . Then $$(4.1) |H_{2,1}(F_f/2)| \le \frac{23}{3264}$$ The inequality is sharp for the function $h \in \mathcal{C}_{\mathcal{C}}$ is given by $$h(z) = \int_0^z \frac{h_0(x)}{x} dx = z + \frac{\sqrt{69}}{12\sqrt{17}} z^3 + \frac{1}{20} \left( \frac{69}{136} + \frac{\sqrt{69}}{4\sqrt{17}} \right) z^5 + \cdots,$$ where $h_0(z)$ is given by (4.8).

Definition 1.1. Let f and g be two analytic functions. Then f is subordinated by g and written as $f(z) \prec g(z)$ , if there exists a self map w such that w(0) = 0 such that f(z) = g(w(z)). Moreover, if g is univalent and f(0) = g(0), then $f(\mathbb{D}) \subseteq g(\mathbb{D})$ . Raina and Sokol [24] introduced the class $\mathcal{S}_{(\!(}^*$ given by $$\mathcal{S}^*_{\mathbb{C}} := \left\{ f \in \mathcal{S} : \left| \left( \frac{zf'(z)}{f(z)} \right)^2 - 1 \right| \le 2 \frac{zf'(z)}{f(z)}, \ z \in \mathbb{D} \right\}.$$ Geometrically, a function $f \in \mathcal{S}_{\mathbb{Q}}$ is that, for any $z \in \mathbb{D}$ , the ratio $\frac{zf'(z)}{f(z)}$ contains the region which is bounded by the lune. It is given by the relation $\{w \in \mathbb{C} : |w^2 - 1| \le 2|w|\}$ . By using the definition of the subordination, the class $\mathcal{S}_{\mathbb{Q}}$ is defined as $$\mathcal{S}^*_{\mathbb{Q}} := \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} \prec z + \sqrt{1+z^2} = q(z), \ z \in \mathbb{D} \right\},$$ where branch of the square root is chosen to be q(0) = 1. The class $\mathcal{C}_{\mathbb{Q}}^*$ of convex function q is defined as $$\mathcal{C}^*_{\mathbb{Q}} := \left\{ f \in \mathcal{S} : 1 + \frac{zf''(z)}{f'(z)} \prec q(z), \ z \in \mathbb{D} \right\},$$ The class $\mathcal{S}^_{\mathbb{Q}}$ has been the subject of extensive investigation by several authors. The coefficient estimates of the class $\mathcal{S}^_{\mathbb{Q}}$ were investigated by Raina and Sokol [22, 23], whereas Gandhi and Ravichandran [10] examined the radius issues associated with the same class. Certain differential subordinations related to the class $\mathcal{S}^_{\mathbb{Q}}$ were studied by Sharma et~al. [27]. Raina et~al. [21] give integral representation and sufficient conditions for the functions in the class $\mathcal{S}^_{\mathbb{Q}}$ . A recent contribution by Cho et~al. [7] proposed a conjecture regarding the coefficients of this particular class. In geometric function theory, a lot of emphasis have been given to evaluate the bounds of Hankel determinants, whose elements are the coefficients of analytic functions f characterize in $\mathbb{D}$ of the form (1.2). Hankel matrices (and determinants) play a key role in several branches of mathematics and have various applications [31]. This study is dedicated to providing the sharp bound for the second Hankel determinant, whose entries are the logarithmic coefficients. We commence by presenting the definitions of Hankel determinants in the case where $f \in \mathcal{A}$ . The Hankel determinant $H_{q,n}(f)$ of Taylor's coefficients of functions $f \in \mathcal{A}$ represented by (1.1) is defined for $q, n \in \mathbb{N}$ as follows: $$H_{q,n}(f) := \begin{vmatrix} a_n & a_{n+1} & \cdots & a_{n+q-1} \\ a_{n+1} & a_{n+2} & \cdots & a_{n+q} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n+q-1} & a_{n+q} & \cdots & a_{n+2(q-1)} \end{vmatrix}.$$ Kowalczyk and Lecko [13] recently proposed a Hankel determinant whose elements are the logarithmic coefficients of $f \in \mathcal{S}$ , realizing the extensive use of these coefficients. This determinant is expressed as follows: $$H_{q,n}(F_f/2) = \begin{vmatrix} \gamma_n & \gamma_{n+1} & \cdots & \gamma_{n+q-1} \\ \gamma_{n+1} & \gamma_{n+2} & \cdots & \gamma_{n+q} \\ \vdots & \vdots & \vdots & \vdots \\ \gamma_{n+q-1} & \gamma_{n+q} & \cdots & \gamma_{n+2(q-1)} \end{vmatrix}.$$ The study of Hankel determinants for starlike, convex, or many other functions has been done extensively (see [13, 15, 20, 25, 28]), their sharp bounds have been established. Recently, the Hankel determinants with logarithmic coefficients have been examined for certain subclasses of starlike, convex, univalent, strongly starlike and strongly convex functions (see [3,13,14] and references therein). However, a little is known about sharp bounds of Hankel determinants of logarithmic coefficients and need to explore them for many classes of functions. Differentiating (1.3) and using (1.2), a simple computation shows that $$\begin{cases} \gamma_1 = \frac{1}{2}a_2, \\ \gamma_2 = \frac{1}{2}\left(a_3 - \frac{1}{2}a_2^2\right), \\ \gamma_3 = \frac{1}{2}\left(a_4 - a_2a_3 + \frac{1}{3}a_2^3\right), \\ \gamma_4 = \frac{1}{2}\left(a_5 - a_2a_4 + a_2^2a_3 - \frac{1}{2}a_3^2 - \frac{1}{4}a_2^4\right), \\ \gamma_5 = \frac{1}{2}\left(a_6 - a_2a_5 - a_3a_4 + a_2a_3^2 + a_2^2a_4 - a_2^3a_3 + \frac{1}{5}a_2^5\right). \end{cases}$$ the great importance of logarithmic coefficients in the recent te and interesting to compute the Hankel determinant whose Due to the great importance of logarithmic coefficients in the recent years, it is appropriate and interesting to compute the Hankel determinant whose entries are logarithmic coefficients. In particular, the second Hankel determinant of $F_f/2$ is defined as <span id="page-2-0"></span>(1.4) $$H_{2,1}(F_f/2) = \gamma_1 \gamma_3 - \gamma_2^2 = \frac{1}{48} \left( a_2^4 - 12a_3^2 + 12a_2 a_4 \right).$$ In this paper, we aim to explore by examining the sharp bound of the Hankel determinant $H_{2,1}(F_f/2)$ for two class of functions, namely, starlike and convex functions associated with lune. It is known that for the Koebe function $f(z) = z/(1-z)^2$ , the logarithmic coefficients are $\gamma_n = 1/n$ , for each positive integer n. Since the Koebe function appears as an extremal function in many problems of geometric theory of analytic functions, one could expect that $\gamma_n = 1/n$ holds for functions in $\mathcal{S}$ . But this is not true in general, even in order of magnitude. The problem of computing the bound of the logarithmic coefficients are studied recently by several authors in different contexts, for instance see [1, 2, 6, 20, 29]. As usual, instead of the whole class S, one can take into account their subclasses for which the problem of finding sharp estimates of Hankel determinant of logarithmic coefficients can be studied. The problem of computing the sharp bounds of $H_{2,1}(F_f/2)$ was considered in [13] for starlike and convex functions. It is now appropriate to remark that $H_{2,1}(F_f/2)$ is invariant under rotation since for $f_{\theta}(z) := e^{-i\theta} f(e^{i\theta}z), \theta \in \mathbb{R}$ when $f \in S$ we have $$H_{2,1}(F_{f_{\theta}}/2) = \frac{e^{4i\theta}}{48} \left( a_2^4 - 12a_3^2 + 12a_2a_4 \right) = e^{4i\theta} H_{2,1}(F_f/2).$$