Abstract
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.
1. Introduction
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. [5] for the family of Bazilevic functions. For subfamilies of , more studies regarding can be seen in [6–12]. According to Thomas’ conjecture [13], if , then , but it was shown by Li and Srivastava in [14] that this conjecture is not true for . Also, Raducanu and Zaprawa [15] showed that it is false for . Rather, they showed that . As compared to estimation of is much more difficult. Babalola [16] 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 [26] improved the results of Babalola [16] 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. [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 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 [37].
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
Also,where
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
Now,
Consider
Since we have ,
Also,
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
Proof. As , using (34)–(37) in (20), we getRearranging the terms, we may writeAfter rearranging the terms, we getAfter using the triangular inequality and above-stated lemmas, we getThis reduces towhich completes the proof.
2.2. Bounds of for the Set
Let Rotation of a domain about the origin through an angle of containing onto itself is said to be -fold symmetric. An analytic function is -fold symmetric in ifholds for any . Denote as the set of -fold univalent functions which have the following Taylor series form:
Denote as the subfamily of of -fold symmetric convex functions. We can see that an analytic function of form (52) belongs to the family , if and only ifwhere . The family is defined as
Now, consider the following.
Theorem 3. Let be of form (52). Then,
Proof. Let of form (52). Consider the function asNow,The class which is associated with the sine functions can be written in the following form:By expanding and equating them, we get the following expression:This impliesBy using these coefficients, we can get and asBy using these values in , we getThe triangle inequality and the application of Lemma 1 lead us towhich completes the proof.
3. Conclusion
In this paper, we have found the upper bounds of fourth-order Hankel and Toeplitz determinants, followed by a review of such findings obtained so far for certain analytic functions. We have studied them for the convex functions associated with the function . A similar bound of the fourth-order Hankel determinant for 3-fold symmetric convex functions associated with has also been investigated.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Authors’ Contributions
All authors contributed equally to this study and approved the final manuscript.