Results on Logarithmic Coefficients for the Class of Bounded Turning Functions

Theorem 3.1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{BT}_{\mathfrak{B}}$ and $\gamma_1, \gamma_2, \gamma_3, \gamma_4$ are given by (2.2). Then we have $$|\gamma_n| \le \frac{1}{4(n+1)}$$ for $n = 1, 2, 3, 4$ . The inequality is sharp for the following functions: <span id="page-7-1"></span>(3.1) $$f_1(z) = \int_0^z \left(\sqrt{1 + \tanh t}\right) dt = z + \frac{z^2}{4} - \frac{z^3}{24} - \frac{5z^4}{192} + \frac{17z^5}{1920} + \cdots$$  FIGURE 2. Pictorial representation of $f_1(z) = z + \frac{z^2}{4} - \frac{z^3}{24} - \frac{5z^4}{192} + \cdots$ <span id="page-7-2"></span>(3.2) $$f_2(z) = \int_0^z \left(\sqrt{1 + \tanh t^2}\right) dt = z + \frac{z^3}{6} - \frac{z^5}{40} - \frac{5z^7}{336} + \cdots$$ <span id="page-7-3"></span>(3.3) $$f_3(z) = \int_0^z \left(\sqrt{1 + \tanh t^3}\right) dt = z + \frac{z^4}{8} - \frac{z^7}{56} + \cdots$$ <span id="page-7-4"></span>(3.4) $$f_4(z) = \int_0^z \left(\sqrt{1 + \tanh t^4}\right) dt = z + \frac{z^5}{10} - \frac{z^9}{72} + \cdots$$

Theorem 3.2. Let $f \in \mathcal{BT}_{\mathfrak{B}}$ and has the series representation $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ , and $\gamma_1, \gamma_2, \gamma_3$ are given by (2.2). Then we have $$|H_{2,1}(F_f/2)| := |\gamma_1 \gamma_3 - \gamma_2^2| \le \frac{1}{144}.$$ The inequality is sharp for the function $f_2$ , which is defined in (3.2).

Theorem 4.1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{BT}_{\mathfrak{B}}$ and $\Gamma_1, \Gamma_2$ are given by (4.4). Then we have $$|\Gamma_1| \le \frac{1}{8}$$ and $|\Gamma_2| \le \frac{1}{12}$ . The inequalities are sharp for the functions $f_1$ and $f_2$ , defined in (3.1) and (3.2), respectively.

Theorem 4.2. Let $f \in \mathcal{BT}_{\mathfrak{B}}$ and has the series representation $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ , and $\Gamma_1, \Gamma_2, \Gamma_3$ are given by (4.4). Then we have $$|H_{2,1}(F_{f^{-1}}/2)| := |\gamma_1 \gamma_3 - \gamma_2^2| \le \frac{1}{144}.$$ The inequality is sharp for the function $f_2$ , defined in (3.2).

Theorem 5.1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{BT}_{\mathfrak{B}}$ . Then we have <span id="page-19-0"></span> $$|a_2 a_3 - a_4| \le \frac{1}{8}.$$ The inequality (5.1) is sharp for the function $f_3$ , which is defined in (3.3).

Theorem 6.2. Let $f \in \mathcal{BT}_{\mathfrak{B}}$ and has the series representation $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ , and $\Gamma_1, \Gamma_2$ are given by (4.4). Then we have $$-\frac{1}{2\sqrt{29}} \le |\Gamma_2| - |\Gamma_1| \le \frac{1}{12}.$$ Both inequalities are sharp.

Definition 1.1. Let f and g be two analytic functions in the unit disk $\mathbb{D}$ . Then f is said to be subordinate to g, written as $f \prec g$ or $f(z) \prec g(z)$ , if there exists a function $\omega$ , analytic in $\mathbb{D}$ with w(0) = 0, |w(z)| < 1 such that f(z) = g(w(z)) for $z \in \mathbb{D}$ . Moreover, if g is univalent in $\mathbb{D}$ and f(0) = g(0), then $f(\mathbb{D}) \subseteq g(\mathbb{D})$ . File: Man-Aha-GFTP12-05-02-2025.tex, printed: 2025-2-6, 1.24 ²⁰²⁰ Mathematics Subject Classification. Primary 30C45; Secondary 30C50, 30C55. Key words and phrases. Univalent functions, Bounded Turning Functions, Hankel determinants, Logarithmic coefficients, Inverse functions, Zalcman functional, Schwarz functions. The most fundamental and significant subfamilies of the set S are the family $S^*$ of starlike functions and the family C of convex functions which are defined as follows: $$\mathcal{S}^* = \left\{ f \in \mathcal{A} : \frac{zf'(z)}{f(z)} \prec \psi(z), \ z \in \mathbb{D} \right\}$$ and $$C = \left\{ f \in \mathcal{A} : 1 + \frac{zf''(z)}{f'(z)} \prec \psi(z), \ z \in \mathbb{D} \right\},\,$$ with $$\psi(z) = 1 + 2\sum_{n=2}^{\infty} z^n := \frac{1+z}{1-z}, \ z \in \mathbb{D}.$$ A function $f \in \mathcal{A}$ is called starlike (resp. convex) if the image $f(\mathbb{D})$ is a starlike domain with respect to the origin (resp., convex). The classes of all starlike and convex functions that are univalent are denoted by $\mathcal{S}$ and $\mathcal{C}$ , respectively. It is well-known that a function f in $\mathcal{A}$ is starlike (resp. convex) if and only if Re(zf'(z)/f(z)) > 0 (resp. Re(1 + zf''(z)/f'(z))) > 0 for $z \in \mathbb{D}$ . By varying the function $\psi(z)$ in the above equations $\mathcal{S}$ and $\mathcal{C}$ , we get some subfamilies which have significant geometric sense. The function $\mathfrak{B}(z) = \sqrt{1 + \tanh z}$ represents a bean-shaped domain. Recently, Nandhini and Sruthakeerthi [28] defined a new subclass of bounded turning functions associated with a bean-shaped domain. There are several other subclasses that have been studied by researchers and each has significant geometrical properties.

Definition 1.2. [28] Let $f \in \mathcal{A}$ is in the class $\mathcal{BT}_{\mathfrak{B}}$ , if $$\mathcal{BT}_{\mathfrak{B}} = \{ f \in \mathcal{S} : f'(z) \prec \mathfrak{B}(z) \}.$$ Note that $$\mathfrak{B}(z) = \sqrt{1 + \tanh z} = \sqrt{\frac{2}{(1 + e^{-2z})}}$$ and $\mathfrak{B}(z)$ conformally maps $\mathbb{D}$ onto the region $$\Omega_{\mathfrak{B}} := \left\{ \omega \in \mathbb{C} : \left| \log \left( \frac{\omega^2}{2 - \omega^2} \right) \right| < 2 \right\}.$$ Geometrically, each $f \in \mathfrak{B}(z)$ maps to a bean-shaped region symmetric around the real axis, as shown in the following Fig. 1, left side is unit disk in z-plane and right side is $\omega$ -plane. Finding an upper bound for coefficients has been one of the central research topics in geometric function theory, as it reveals various properties of functions. The main challenge is to identify a suitable function from the class that effectively shown the sharpness of the bound. However, we point out that despite the extensive exploration of coefficient problems involving the Hankel determinant for the class $\mathcal{BT}_{\mathfrak{B}}$ , the corresponding determinant with the logarithmic coefficient for the class $\mathcal{BT}_{\mathfrak{B}}$ has not garnered as much attention from researchers. Furthermore, the sharpness of  FIGURE 1. The image $\mathfrak{B}(\mathbb{D})$ is a bean shaped domain by the function $w = \mathfrak{B}(z) = \sqrt{1 + \tanh z}$ . the logarithmic coefficients has not been explored for the class $\mathcal{BT}_{\mathfrak{B}}$ . This lack of attention serves as the primary motivation for the present paper and contribute to the understanding the several bounds of logarithmic coefficients for the class $\mathcal{BT}_{\mathfrak{B}}$ . In this article, we aim to determine the sharp bounds for various problems in geometric function theory. These problems include finding the sharp bounds for logarithmic coefficients as well as logarithmic inverse coefficients and the sharp bound for the Hankel determinant of logarithmic coefficients as well as logarithmic inverse coefficients. In the subsequent sections, we will discuss our findings and provide a background study on these topics. The organization of this paper is as follows: In Section 3, we establish the sharp bounds for $\gamma_1, \gamma_2, \gamma_3$ , and $\gamma_4$ and $|H_{2,1}(F_f/2)|$ for functions $f \in \mathcal{BT}_{\mathfrak{B}}$ . In Section 4, we establish the sharp bound of $\Gamma_1$ and $\Gamma_2$ and $|H_{2,1}(F_{f^{-1}}/2)|$ for functions in the class $\mathcal{BT}_{\mathfrak{B}}$ . In Section 5, we establish the sharp generalized Zalcman conjecture inequality for the class $\mathcal{BT}_{\mathfrak{B}}$ . In Section 6, we establish the sharp moduli differences of logarithmic coefficients for the class $\mathcal{BT}_{\mathfrak{B}}$ . The proofs of the results are discussed in detail in each respective section.