Abstract
In this present paper, a new generalized class from the family of function with bounded turning was introduced by using - derivative operator. Our aim for this class is to find out the upper bound of third- and fourth-order Hankel determinant. Moreover, the upper bounds for two-fold and three-fold symmetric functions for this class are also obtained.
1. Introduction and Motivation
In order to better explain the terminology included in our key observations, some of the essential relevant literatures on geometric function theory need to be provided and discussed here. We start with symbol which represents the class of holomorphic functions in the region of open unit disc and satisfy the relationship for . That is; if , then it has the following Taylor series form:
Also, let represent all univalent functions in . Next, we are going to define the most useful class of geometric function theory known as the class of Caratheodory functions and is defines as; a holomorphic function belongs to if it satisfies along with the series expansion:
Using the Caratheodory functions family, we consider the following basics subclasses of as
The investigation of -calculus ( stands for quantum) fascinated and inspired many scholars due its use in various areas of the quantitative sciences. Jackson [1, 2] was among the key contributors of all the scientists who introduced and developed the -calculus theory. Just like -calculus was used in other mathematical sciences, the formulations of this idea are commonly used to examine the existence of various structures of function theory. Though the link between certain geometric nature of the analytic function and the -derivative operator was established by the authors in [3], but, for the usage of -calculus in function theory, a solid and comprehensive foundation is given in [4] by Srivastava. After this development, many researchers introduced and studied some useful operators in -analog with the applications of convolution concepts. For example; Kanas and Răducanu [5] established the -differential operator and then examined the behavior of this operator in function theory. This operator was generalized further for multivalent analytic functions by Arif et al. [6]. Analogous to -differential operator, Arif et al. and Khan et al. contributed the integral operators for analytic and multivalent functions in [7, 8], respectively. Similarly, in the article [9], the authors developed and analyzed the operator in -analog for meromorphic functions. Also, see the survey type article [10] on quantum calculus and their applications. With the use of these operators, many researchers were contributed some good articles in this direction in the field of geometric function theory, see [11, 12]. The substitution of by in -calculus has given us -calculus which is the extension of -calculus. Chakrabarti and Jagannathan [13] considered -integer.
For and , the -derivative operator of the function is defined as
From the use (1) and (4), the following formulae are easily obtained:and for a constant ,
Here, we also note that when we take and .
Further by using (1) and (4), we havewith
Let us define the class , which consists all the functions , satisfying
The Hankel determinant with and for a function of the series form (1) was given by Pommerenke [14, 15], as
In particular, the following determinants are known as the first-, second-, and third-order Hankel determinants, respectively.
There are comparatively few observations in literature in relation to the Hankel determinant for the function which belongs to the general family . For the function , the best established sharp inequality is , where is the absolute constant, which is due to Hayman [16]. Further for the same class , it was obtained in [17] that
In a given family of functions, the problem of calculating the bounds, probably sharp, of Hankel determinants attracted the minds of several mathematicians. For example, the sharp bound of , for the subfamilies , , and (family of bounded turning functions) of the set , was calculated by Janteng et al. [18, 19]. These estimates are
For the following two families, of star-like functions of order and for of strongly star-like functions of order , the authors computed in [20, 21] that is bounded by and , respectively. The exact bound for the family of Ma-Minda star-like functions was measured in [22], see also [23]. For more work on , see references [24–28].
It is quite clear from the formulae given in (12) that the calculation of is far more challenging compared with finding the bound of . Babalola was the first mathematician who investigated the bounds of third=order Hankel determinant for the families of , , and in an article [29] published in 2010. Using the same approach, later, several authors [30–34] published their articles regarding for certain subfamilies of analytic and univalent functions. After this study, Zaprawa [35] improved the findings of Babalola in 2017 by applying a new methodology. He obtained the following bounds:
He argued that such limits are indeed not the best. Later in 2018, Kwon et al. [36] strengthened the Zaprawa’s result for and showed that and this bound was further improved by Zaprawa et al. [37] in 2021. They got for . Recently in 2018, Kowalczyk et al. [38] and Lecko et al. [39] succeeded in finding the sharp bounds of for the families and , respectively, where indicate the star-like functions family of order . These results are given as
The estimation of fourth Hankel determinant for the bounded turning functions has been obtained by Arif et al. [40] and they proved the following bounds for :
After that, Kaur and Singh [41] proved fourth Hankel determinant for bounded turning function of order . For more contributions, see [42–46]. Recently, Srivastava et al. [47] consider a family of normalized analytic functions with bounded turnings in the open unit disk which are connected with the cardioid domains and they obtained the estimates of fourth Hankel determinant.
2. A Set of Results
In order to investigate , we need the following results.
Lemma 1. (see [48]). If , having the form (2), then
Theorem 1. (see [40]). Let and for real ,
3. Bounds of Third Hankel Determinant
The third Hankel determinant is a polynomial of four variables, as:
In order to solve , we need to know the correspondence between and .
Thus,
By simplifying, we yield
Now, using the above coefficients in (22), we obtain
Rearranging the above terms,
Using triangular inequality and the results (18) and (19) of lemma of Section 2, we get
Remark 1. As we know by (7), if and , then the above result (28) coincides with Zaprawa [35].
4. Bounds of Fourth Hankel Determinant
Firstly, is the fourth Hankel determinant of the form (11) with six coefficients which can be written in the form,where , and are third-order determinants given as
Theorem 2. If , thenwhere
Proof. By using definition (10), we have . Thus, .Therefore,Substituting (36) in (30) and (31) and in (32) yields thatNow, rewrite the above equations as follows:Using the triangular inequality with the inequalities (18) and (19) of lemma on the above equations, we obtainNow, using the values (39)–(41) and (28) along with the inequality in (29), we get our desired result.
5. Bounds of for Two-Fold and Three-Fold Symmetric Functions
-Fold symmetric function consists all those functions which satisfy the following condition:where . The set of univalent functions with -fold symmetry (that is ) has the expansion of the form,
Furthermore, univalent function belongs to if and only ifwhereasand function if .
Now, if , then ; hence, . In the same way, if , then ; clearly, we can see that two-fold symmetric functions are odd. So, .
Theorem 3. If is three-fold symmetric bounded turning function, that is, , then
Proof. Firstly, consider that , then is a function of the form such that . Since and with the use of (43) for , it follows thatIdentifying the coefficients, we getWe already know , then function of the form such that . Thus,Therefore,Now, by rearranging (48) and (50), we getSince , this impliesWith the use of Theorem 1 in Section 2, where for , we get our theorem proved.
Theorem 4. If is two-fold symmetric bounded turning function, that is, , then
Proof. By the definition of two-fold symmetric function, the Hankel determinant can be written asSince , then is a function such that ; then, the expansion of (43) and (45) for yields thatNow, with the help of lemma in Section 2, we get our desired result as asserted by the statement.
Remark 2. For results of (29), (45) and (43) will coincide with results derived in [40].
Data Availability
The required data are included in this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.