**1. Introduction**

Let <sup>A</sup> denote the class of functions *<sup>f</sup>* which are analytic in the open unit disk <sup>D</sup> <sup>=</sup> {*<sup>z</sup>* : <sup>|</sup>*z*<sup>|</sup> <sup>&</sup>lt; <sup>1</sup>} of the form

$$f(z) = z + a\_2 z^2 + a\_3 z^3 + \cdots \quad (z \in \mathbb{D})\tag{1}$$

and let S denote the subclass of A consisting of univalent functions.

Suppose that P denotes the class of analytic functions *p* normalized by

$$p(z) = 1 + c\_1 z + c\_2 z^2 + c\_3 z^3 + \cdots$$

and satisfying the condition

$$
\Re(p(z)) > 0 \quad (z \in \mathbb{D}).
$$

We easily see that, if *p*(*z*) ∈ P, then a Schwarz function *ω*(*z*) exists with *ω*(0) = 0 and |*ω*(*z*)| < 1, such that (see [1])

$$p(z) = \frac{1 + w(z)}{1 - w(z)} \quad (z \in \mathbb{D})\,.$$

Very recently, Cho et al. [2] introduced the following function class S<sup>∗</sup> *<sup>s</sup>* , which are associated with sine function:

$$\mathcal{S}\_s^\* := \left\{ f \in \mathcal{A} : \frac{zf'(z)}{f(z)} \prec 1 + \sin z \; (z \in \mathbb{D}) \right\},\tag{2}$$

where "≺" stands for the subordination symbol (for details, see [3]) and also implies that the quantity *z f* (*z*) *<sup>f</sup>*(*z*) lies in an eight-shaped region in the right-half plane.

The *<sup>q</sup>th* Hankel determinant for *<sup>q</sup>* <sup>≥</sup> 1 and *<sup>n</sup>* <sup>≥</sup> 1 of functions *<sup>f</sup>* was stated by Noonan and Thomas [4] as

$$H\_{\emptyset}(n) = \begin{vmatrix} a\_n & a\_{n+1} & \cdots & a\_{n+q-1} \\ \\ a\_{n+1} & a\_{n+2} & \cdots & a\_{n+q} \\ \vdots & \vdots & & \vdots \\ a\_{n+q-1} & a\_{n+q} & \cdots & a\_{n+2q-2} \end{vmatrix} \quad (a\_1 = 1).$$

This determinant has been considered by several authors, for example, Noor [5] determined the rate of growth of *Hq*(*n*) as *n* → ∞ for functions *f*(*z*) given by Equation (1) with bounded boundary and Ehrenborg [6] studied the Hankel determinant of exponential polynomials.

In particular, we have

$$H\_3(1) = \left| \begin{array}{ccccc} a\_1 & a\_2 & a\_3 \\ & \\ a\_2 & a\_3 & a\_4 \\ & \\ a\_3 & a\_4 & a\_5 \end{array} \right| \quad (n=1, \ q=3).$$

Since *f* ∈ S, *a*<sup>1</sup> = 1,

$$H\mathfrak{z}(1) = a\mathfrak{z}(a\_2a\_4 - a\_3^2) - a\_4(a\_4 - a\_2a\_3) + a\mathfrak{z}(a\_3 - a\_2^2).$$

We note that <sup>|</sup>*H*2(1)<sup>|</sup> <sup>=</sup> <sup>|</sup>*a*<sup>3</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> <sup>2</sup>| is the well-known Fekete-Szego functional (see, for example, [7–9]).

On the other hand, Thomas and Halim [10] defined the symmetric Toeplitz determinant *Tq*(*n*) as follows:

$$T\_q(n) = \begin{vmatrix} a\_n & a\_{n+1} & \cdots & a\_{n+q-1} \\ \\ a\_{n+1} & a\_n & \cdots & a\_{n+q} \\ \vdots & \vdots & & \vdots \\ a\_{n+q-1} & a\_{n+q} & \cdots & a\_n \end{vmatrix} \quad (n \ge 1, \ q \ge 1).$$

The Toeplitz determinants are closely related to Hankel determinants. Hankel matrices have constant entries along the reverse diagonal, whereas Toeplitz matrices have constant entries along the diagonal. For a good summary of the applications of Toeplitz matrices to the wide range of areas of pure and applied mathematics, we can refer to [11].

As a special case, when *n* = 2 and *q* = 3, we have

$$T\_3(2) = \begin{vmatrix} a\_2 & a\_3 & a\_4 \\ \\ a\_3 & a\_2 & a\_3 \\ \\ a\_4 & a\_3 & a\_2 \end{vmatrix}.$$

In recent years, many authors studied the second-order Hankel determinant *H*2(2) and the third-order Hankel determinant *H*3(1) for various classes of functions (the interested readers can see, for instance, [12–25]). However, apart from the work in [10,21,26,27], there appears to be little literature dealing with Toeplitz determinants. Inspired by the aforementioned works, in this paper, we aim to investigate the third-order Hankel determinant *H*3(1) and Toeplitz determinant *T*3(2) for the above function class S<sup>∗</sup> *<sup>s</sup>* associated with sine function, and obtain the upper bounds of the above determinants.
