*Article* **Third-Order Hankel and Toeplitz Determinants for Starlike Functions Connected with the Sine Function**

**Hai-Yan Zhang 1, Rekha Srivastava 2,\* and Huo Tang 1,\***


Received: 26 March 2019; Accepted: 25 April 2019; Published: 6 May 2019

**Abstract:** Let S<sup>∗</sup> *<sup>s</sup>* be the class of normalized functions *<sup>f</sup>* defined in the open unit disk <sup>D</sup> <sup>=</sup> {*<sup>z</sup>* : <sup>|</sup>*z*<sup>|</sup> <sup>&</sup>lt; <sup>1</sup>} such that the quantity *z f* (*z*) *<sup>f</sup>*(*z*) lies in an eight-shaped region in the right-half plane and satisfying the condition *z f* (*z*) *<sup>f</sup>*(*z*) <sup>≺</sup> <sup>1</sup> <sup>+</sup> sin *<sup>z</sup>* (*<sup>z</sup>* <sup>∈</sup> <sup>D</sup>). In this paper, we aim to investigate the third-order Hankel determinant *H*3(1) and Toeplitz determinant *T*3(2) for this function class S<sup>∗</sup> *<sup>s</sup>* associated with sine function and obtain the upper bounds of the determinants *H*3(1) and *T*3(2).

**Keywords:** starlike function; Toeplitz determinant; Hankel determinant; sine function; upper bound

**MSC:** 30C45; 30C50; 30C80
