Fourth-Order Hankel Determinants and Toeplitz Determinants for Convex Functions Connected with Sine Functions

This article deals with the upper bound of fourth-order Hankel and Toeplitz determinants for the convex functions which are defined by using the sine function. The main tools in this study are the coefficient inequalities for the class P of functions with positive real parts. Also, the investigation of the upper bound of the fourth-order Hankel determinant for 3-fold symmetric convex functions associated with the sine function is included.


Introduction
Let the family of all functions f be denoted by A which are analytic in an open unit disc D z ∈ C: |z| < 1 { } with Taylor series expansion: and S represent a family of functions f ∈ A which are univalent in D. Let S * , C, and K g denote the families of starlike, convex, and close-to-convex functions, respectively, and they are de ned as , K g f ∈ S: R zf ′ (z) g(z) > 0, for g ∈ S * , (z ∈ D) . (2) Let P denote the family of all analytic functions p of the form with the positive real parts in D. As the n th coe cient for the functions belonging to the family is bounded by n, this bound helps in the study of geometric properties of functions f ∈ S. Speci cally, the second coe cient a 2 helps in nding the distortion and growth properties of a normalized univalent function. Likewise, the problems involving power series with integral coe cients and investigating the singularities are successfully handled by using Hankel determinants. Pommerenke [1,2] introduced the idea of Hankel determinants, and he de ned those for univalent functions f ∈ S of form (7) as follows: He claimed that these bounds are not sharp. Furthermore, he considered the subfamilies of S * , C, and R for sharpness, having functions with m-fold symmetry, and obtained the sharp bounds. Arif et al. [27][28][29][30] made a remarkable contribution in studying the fourth-and fifthorder Hankel determinants H (4,1) (f) and H (5,1) (f) for certain subfamilies of univalent functions. Mashwani et al. [31] have studied the fourth-order Hankel determinant for starlike functions related to sigmoid functions, whereas Kaur et al. [32] studied the same problem for a subclass of bounded turning functions. Wang et al. [33] studied the problem for bounded turning functions related to the lemniscate of Bernoulli. Recently, Zhang and Tang [34] have studied the fourth-order Hankel determinant for the class of starlike functions related to sine functions. Motivated by the above-mentioned work, we intend to add some contributions to the fourth-order Hankel determinant for the class of convex functions associated with sine functions. Recently, the following class C s of convex functions was introduced, which is associated with the sine function: where ≺ is a subordination symbol and it also implies that the region defined by (zf ′ (D)) ′ /f ′ (D) lies in the eightshaped region in the right-half plane. For different subfamilies of univalent functions, growth of H q,n (f) has been studied for fixed values of q and n. Particularly, we have Also, omas and Halim defined the symmetric Toeplitz determinant T q (n) as follows: T q,n (f) � a n a n+1 . . . a n+q−1 a n+1 a n . . . a n+q : : . . . : a n+q−1 a n+q . . . a n (n ≥ 1, q ≥ 1).
e Toeplitz determinants are closely related to Hankel determinants. As Hankel matrices consist of constant entries along the reverse diagonal, the Toeplitz matrices consist of constant entries along the diagonal.
As a special case, when n � 1 and q � 4, we have In this paper, we intend to find the upper bound of |H 4,1 (f)| and |T 4,2 (f)| for the class of functions defined by (6). e following sharp results would be useful for investigating our main results. Lemma 1. If p ∈ P and p is of form (2), then for each n, k, m, l ∈ N � 1, 2, . . . { }, the following sharp inequalities hold: Inequalities (10)- (12) are proved in [26,35,36], respectively. Inequality (13) is obvious.

Lemma 2.
Let p ∈ P be of form (2). en, the modulus of the expressions are all bounded by 2.

Bounds of |H
where and R 1 , R 2 , and R 3 are determinants of order 3, given by Also, where As, from (7), H 4,1 (f) is a polynomial of six coefficients of function f of the given class, these coefficients are taken as a 2 , a 3, a 4 , a 5, a 6 , and a 7 . However, there is a connection between these coefficients and the coefficients of function p in the class P in many problems. Consider that f ∈ C s has form (1); then, there is a Schwartz function w(z) with w(0) � 0 and |w(z)| < 1, such that Now, Consider Since we have p ∈ P, Also, On comparing coefficients between (25) and (28), we get By using these coefficients, we can write (16)- (19) in the following way: (33) Similarly, in case of Toeplitz determinants, By using the previous computations, we prove the following.

Theorem 1. If the function f ∈ C s and is of form (1), then
Proof. As f ∈ C s , then by using (30)- (33) in (15), we get (39)

Journal of Mathematics
Hence,