Methods in Dynamical Systems and Applications in EngineeringView this Special Issue
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 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.
Let the family of all functions be denoted by which are analytic in an open unit disc with Taylor series expansion:and represent a family of functions which are univalent in . Let , , and denote the families of starlike, convex, and close-to-convex functions, respectively, and they are defined as
Let denote the family of all analytic functions of the formwith the positive real parts in . As the coefficient for the functions belonging to the family is bounded by , this bound helps in the study of geometric properties of functions . Specifically, the second coefficient helps in finding the distortion and growth properties of a normalized univalent function. Likewise, the problems involving power series with integral coefficients and investigating the singularities are successfully handled by using Hankel determinants. Pommerenke [1, 2] introduced the idea of Hankel determinants, and he defined those for univalent functions of form (7) as follows:
In the theory of analytic functions, finding the upper bound of is one of the most studied problems. Several researchers found the above-mentioned bound for different subfamilies of univalent functions for fixed values of and . A few remarkable contributions in this regard are included here for reference. For the subfamilies , and (the class of functions with bounded turnings) of the set , the sharp bounds of were investigated by Janteng et al. [3, 4]. They proved the bounds as follows:
The accurate estimate of was obtained by Krishna et al.  for the family of Bazilevic functions. For subfamilies of , more studies regarding can be seen in [6–12]. According to Thomas’ conjecture , if , then , but it was shown by Li and Srivastava in  that this conjecture is not true for . Also, Raducanu and Zaprawa  showed that it is false for . Rather, they showed that . As compared to estimation of is much more difficult. Babalola  published the first paper on in 2010 in which he obtained the upper bound of for subfamilies of , and . After that, for different subfamilies of analytic and univalent functions, few other authors [17–25] also published their work regarding . Zaprawa  improved the results of Babalola  recently in 2017, by showing
He claimed that these bounds are not sharp. Furthermore, he considered the subfamilies of , and for sharpness, having functions with -fold symmetry, and obtained the sharp bounds. Arif et al. [27–30] made a remarkable contribution in studying the fourth- and fifth-order Hankel determinants and for certain subfamilies of univalent functions. Mashwani et al.  have studied the fourth-order Hankel determinant for starlike functions related to sigmoid functions, whereas Kaur et al.  studied the same problem for a subclass of bounded turning functions. Wang et al.  studied the problem for bounded turning functions related to the lemniscate of Bernoulli. Recently, Zhang and Tang  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 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 lies in the eight-shaped region in the right-half plane. For different subfamilies of univalent functions, growth of has been studied for fixed values of and . Particularly, we have
Also, Thomas and Halim defined the symmetric Toeplitz determinant as follows:
The 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 and , we have
In this paper, we intend to find the upper bound of and for the class of functions defined by (6). The following sharp results would be useful for investigating our main results.
Lemma 1. If and is of form (2), then for each the following sharp inequalities hold:
Inequalities (10)–(12) are proved in [26, 35, 36], respectively. Inequality (13) is obvious.
Libera and Złotkiewicz proved the following result .
Lemma 2. Let be of form (2). Then, the modulus of the expressionsare all bounded by 2.
2. Main Results
2.1. Bounds of and for the Set Connected with the Sine Function
Following (7), we can write , where and , aswhereand , and are determinants of order 3, given by
As, from (7), is a polynomial of six coefficients of function of the given class, these coefficients are taken as , and . However, there is a connection between these coefficients and the coefficients of function in the class in many problems. Consider that has form (1); then, there is a Schwartz function with and , such that
Since we have ,
On comparing coefficients between (25) and (28), we getand
By using these coefficients, we can write (16)–(19) in the following way:and
Similarly, in case of Toeplitz determinants,
By using the previous computations, we prove the following.
Theorem 1. If the function and is of form (1), then
Proof. As , then by using (30)–(33) in (15), we getAfter rearranging the terms, we getAfter using triangular inequalities and lemmas, we get the following expression:Hence,which completes the proof.
Theorem 2. If the function and is of form (1), then