"Upper bounds of Toeplitz determinants for a subclass of alpha-close-to-convex functions"

Lemma 1.1. Let $p(z) = 1 + \sum_{n=1}^{\infty} c_n z^n \in \mathcal{P}$ with $c_1 \geq 0$ . Then for some complex valued z with $|z| \leq 1$ and some complex valued z with $|z| \leq 1$ , we have $$2c_2 = c_1^2 + x(4 - c_1^2),$$ $$4c_3 = c_1^3 + 2(4 - c_1^2)c_1x - c_1(4 - c_1^2)x^2 + 2(4 - c_1^2)(1 - |x|^2)z.$$

Theorem 2.1. Let f given by (1.1) be in the class $\tilde{\mathcal{R}}_{\alpha}$ . Then we have the bound $$|T_2(2)| \le \frac{5(1+\alpha)^2 + 9\alpha(2+3\alpha)}{9(1+\alpha)^2(1+2\alpha)^2}.$$ (2.4) The bound is sharp.

Theorem 2.2. Let f given by (1.1) be in the class $\tilde{\mathcal{R}}_{\alpha}$ . Then we have the bound $$|T_2(3)| \le \frac{4}{9(1+2\alpha)^2}. (2.7)$$ The bound is sharp.

Theorem 2.3. Let f given by (1.1) be in the class $\tilde{\mathcal{R}}_{\alpha}$ . Then we have the bound $$|T_3(1)| \le \frac{13 + 36\alpha + 36\alpha^2}{9(1 + 2\alpha)^2}. (2.10)$$ The bound is sharp.

Theorem 2.4. Let f given by (1.1) be in the class $\tilde{\mathcal{R}}_{\alpha}$ . Then we have the bound $$|T_{3}(2)| \leq \begin{cases} \frac{4(1+5\alpha)}{9(1+\alpha)(1+2\alpha)^{2}(1+3\alpha)} & \text{for } \alpha \leq \alpha_{0} \\ \frac{(1+5\alpha)(11+91\alpha+248\alpha^{2}+204\alpha^{3})}{36(1+\alpha)^{3}(1+2\alpha)^{2}(1+3\alpha)^{2}} & \text{for } \alpha > \alpha_{0}, \end{cases}$$ (2.14) where $\alpha_0 \approx 0.146157$ is the positive root of the equation $156\alpha^3 + 136\alpha^2 + 11\alpha - 5 = 0$ . For $\alpha > \alpha_0$ , equality attained by the function $g_{\alpha} \in \tilde{\mathcal{R}}_{\alpha}$ defined by (2.6).

Definition 1.1. ([4]) For $\alpha \geq 0$ , a function $f \in \mathcal{A}$ with $\frac{f(z)f'(z)}{z} \neq 0$ is said to be an alphaclose-to-convex function if for a starlike function $\phi(z)$ , it satisfies the condition $$\operatorname{Re}\left\{ (1-\alpha) \frac{zf'(z)}{\phi(z)} + \alpha \frac{(zf'(z))'}{\phi'(z)} \right\} > 0, \quad z \in \mathbb{U}.$$ We denote $C_{\alpha}$ by the class of all alpha-close-to-convex functions. This class was introduced and studied by Chichra [4]. For $\alpha = 0$ , $C_{\alpha} \equiv \mathcal{K}$ , the class of close-to-convex functions. We denote the subclass of $C_{\alpha}$ by $\tilde{\mathcal{R}}_{\alpha}$ for which $\phi(z) = z$ . Received: 17.01.2021. In revised form: 08.11.2021. Accepted: 15.11.2021 Corresponding author: Pravati Sahoo; pravatis@yahoo.co.in ²⁰¹⁰ Mathematics Subject Classification. 30C45. Key words and phrases. starlike function, convex function, close-to-convex function, Hankel determinant, Toeplitz

Definition 1.2. ([4]) Let $\tilde{\mathcal{R}}_{\alpha}$ be the class of all functions $f \in \mathcal{A}$ which satisfy $$\operatorname{Re}(f'(z) + \alpha z f''(z)) > 0$$ , for all $z \in \mathbb{U}$ . For $\alpha = 0$ , $\tilde{\mathcal{R}}_{\alpha} \equiv \mathcal{R}_0 \equiv \mathcal{R} = \{f(z) \in \mathcal{A} : \text{Re}(f'(z)) > 0, \text{ for all } z \in \mathbb{U}\}$ . These classes have been studied by many authors [4, 9, 10, 12, 13] in various viewpoints. Remark 1.1. Bieberbach's conjecture, which is also known as de Branges Theorem after its proof, has been one of the most popular problems in the theory of univalent functions. From the introduction of the Bieberbach conjecture in 1916, until its proof given by de Branges in 1985, a lot of methods and concepts have been developed. For instance, the problem of estimating bounds for successive coefficients was studied with an idea to solve the Bieberbach conjecture. The successive coefficient problem is still open for the whole class of univalent analytic functions as well as its many subclasses. Such investigations more often lead to new techniques and ideas to deal the main open problem or related problems. In the line of such investigation, later various types of coefficient problems appeared in function theory while considering various transformation of univalent functions such as the Fekete-Szegö coefficient problem which appears from the inversion transformation. Hankel and Toeplitz determinants of Taylor coefficients of analytic functions are also attracted to many researchers in the field of geometric function theory in the sense that those are somehow related to the above coefficient problems. In fact, Toeplitz determinants have many applications in operator theory, linear algebra, physics, etc. (see for instance [16] and references therein). The estimates of the first four Toeplitz determinants are obtained for the class $\mathcal{R}$ in [15]. In [13], Sahoo considered the class $\tilde{\mathcal{R}}_{\alpha}$ and obtained the bounds of first four Hankel determinants. We here consider the Toeplitz determinant for the class $\tilde{\mathcal{R}}_{\alpha}$ with an aim to provide generalizations with sharpness of the upper bounds of the results studied in [15] by Radhika et al. Here we recall Toeplitz symmetric matrices, that have constant entries along the diagonal.

Definition 1.3. The q-th Toeplitz determinant of f(z) for $q \ge 1$ and $n \ge 1$ is defined as $$T_{q}(n) = \begin{vmatrix} a_{n} & a_{n+1} & \dots & a_{n+q-1} \\ a_{n+1} & a_{n} & \dots & \dots \\ \vdots & \vdots & \vdots & \vdots \\ a_{n+q-1} & \dots & \dots & a_{n} \end{vmatrix}.$$ (1.3) So $$T_2(2) = \left| \begin{array}{ccc} a_2 & a_3 \\ a_3 & a_2 \end{array} \right|; T_2(3) = \left| \begin{array}{ccc} a_3 & a_4 \\ a_4 & a_3 \end{array} \right|; T_3(1) = \left| \begin{array}{cccc} 1 & a_2 & a_3 \\ a_2 & 1 & a_2 \\ a_3 & a_2 & 1 \end{array} \right|, T_3(2) = \left| \begin{array}{cccc} a_2 & a_3 & a_4 \\ a_3 & a_2 & a_3 \\ a_4 & a_3 & a_2 \end{array} \right|.$$ To prove our main results we need the following lemma.