Abstract
In this paper, we derive a bound of the fourth Hankel determinant for the class of star-like functions. We also consider this problem for 2-fold and -fold symmetric star-like functions. In this case, we obtain sharp results.
1. Introduction and Definitions
Let denote the class of all analytic functions of the formwhich are analytic in the open unit disc . Let denote the subclass of all univalent functions in . Also, let , and denote the classes of star-like, convex, and close-to-convex functions, respectively, which are defined as follows:
It is easy to see that when , the class reduces to the class of bounded turning functions. Let denote the class of all analytic functions of the formwhose real parts are positive in The problem of finding coefficient bounds plays an important role in exploring the geometry of a complex-valued function. In particular, the second coefficient provides information about the growth and distortion theorems for functions in class . Similarly, the Hankel determinants are very useful in the investigations of the singularities and power series with integral coefficients.
The Hankel determinant for a function of form (1) was defined by Pommerenke [1, 2] (see also [3, 4]) asfor fixed integer and , and the growth of has been studied for different subclasses of univalent functions. The sharp bounds of for , , and were investigated by Janteng et al. [5–7]. They proved thatfor the class of Bazilevič functions, and the exact estimate of was obtained by Krishna et al. [8]. On the other hand, the sharp bound of this determinant for the class of close-to-convex functions is unknown (see [9]). For more results on , see [10–16].
The estimation of is much more difficult to obtain as compared to . In the first paper on , published in 2010, Babalola [17] obtained the upper bound of for , , and . Later on, some authors [18–24] derived the bounds of for different subclasses of analytic and univalent functions. In 2016, Zaprawa [25] improved the results of Babalola [17] and showed that
He also claimed that these bounds were still not sharp. Furthermore, he obtained the sharp bounds for -fold symmetric functions. Later on, Kwon et al. [26] proved thatfor . For some recent work on the bound of , refer [27–30]. Arif et al. and Arif et al. [31, 32] computed the bound on the fourth- and fifth-order Hankel determinants for the functions with bounded turning. Cho and Kumar [33] investigated the bound on for the class of functions associated with lune. Recently, Arif et al. [34] have determined the bound on the fourth Hankel determinant for the class of star-like functions related to lemniscate of Bernoulli.In this paper, we make a contribution to the subject by finding the fourth Hankel determinant for the class of star-like functions. We also study this determinant for the subclasses consisting of 2-fold and 3-fold symmetric functions.
2. Lemmas for Class
In the proofs of our results, we need the following sharp estimates for functions in the class .
Lemma 1. If and is of form (3), then the sharp estimateshold for each .
Inequalities (8), (9), and (10) are proved in [23, 35, 36], respectively. Inequality (11) is obvious.
Libera and Złotkiewicz proved the following result, see [37].
Lemma 2. Let be of form (3). Then, the expressionsare all bounded by 2.
Lemma 3. (See [37]). Let be given by (3) and . Then,for each complex numbers , such that , .
3. Bound of for the Set
From (4), we can write aswhere
We observe that is a polynomial of six successive coefficients , , , , , and of a function in a given class. Let be of form (1). Then, for given in (3), we have
Using (19), we get
From (20), we obtain that
By using the abovementioned formulae, we write (15), (16), (17), and (18) in the following way:
Now, we are ready to formulate our main result.
Theorem 1. If and is of form (1), then
Proof. Since , using (22), (23), (24), and (25) in (14), we getAfter rearranging the terms, we get (27) in the following form:LetApplying the triangle inequality and Lemma 1 and Lemma 2 in (28), we haveHence our stated bound for is obtained.
4. Bounds of for and
Let . A domain is said to be -fold symmetric if a rotation of about the origin through an angle carries onto itself. It is said that an analytic function is -fold symmetric in ifholds for any .
By , we denote the set of -fold univalent functions having the following Taylor series form:
The subfamily of is the set of -fold symmetric star-like functions. It is easy to see that an analytic function of form (32) belongs to the family if and only ifwhere the set is defined as follows:
Now, we are ready to prove the following theorem.
Theorem 2. Let be of form (32). Then, . The estimate is sharp.
Proof. Let be of the form , where , and let and have the following Taylor series expansions:For brevity, we write instead of .
From the equivalence , comparing coefficients inwe getConsequently, for , we haveFrom (20) and Lemma 3, we getTherefore, writing and also assuming in [0, 2] and , we havewhich takes the greatest value for , soSince the equality mentioned above holds for , one can deduce that if and, consequently, if , so the extremal star-like function is of the form .
Our next results are given in the following two theorems.
Theorem 3. If be of form (32); then,The estimate is sharp.
Proof. Let be given by (32). Since , there isUsing the same argument as given in the proof of Theorem 2, we obtainwhich is, by Lemma 1, less than or equal to (1/2).
The function is extremal in the estimate proven in Theorem 2 as well as in the estimate of (44). Hence, for this function, we have .
Theorem 4. If is of the form (32), thenThe estimate is sharp.
Proof. Let be of the form , where , and let and be of the formSince , from the equalitywe getConsequently, for , there is andFrom (48), (20), and Lemma 3, we getTherefore, writing , we haveThe equality mentioned above holds for and , so with . For this reason, the extremal function from the class is of the form , and the corresponding function is given by
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.