Abstract
In our present investigation, we obtain the improved third-order Hankel determinant for a class of starlike functions connected with modified sigmoid functions. Further, we investigate the fourth-order Hankel determinant, Zalcman conjecture, and also evaluate the fourth-order Hankel determinants for 2-fold, 3-fold, and 4-fold symmetric starlike functions.
1. Introduction and Motivation
Denoted by the class of functions which are analytic in and are of the form
Also, let be a subclass of class , containing all univalent functions in , and be normalized by the conditions
In 1916, working on the coefficients of class , Bieberbach conjectured that which was proved by De Branges in 1984 (see [1]). From 1916 to 1984, for some subclasses of , many researchers used different techniques and established a number of results. The classes , , and namely, the classes of convex, starlike, and bounded turning functions, respectively, are some major subclasses of the class . These classes are defined as follows:
Furthermore, we say that an analytic function is subordinated to in and is symbolically written as if there exists a Schwartz function with properties that such that
Moreover, if is in the class . Due to [2, 3], we get the following equivalence relation
Now, by using the principle of subordination, a generalized set of the classes and are given, respectively, as follows:
By changing the right-hand side in (10), several familiar classes can be obtained such as if we keep we get the class of starlike functions associated with the Janowski functions (see [4]). Moreover, if we take then, we have a class of starlike functions of order (see [5]). Also, for the choice of a corresponding class of starlike functions is obtained, which was introduced by Ronning (see [6]). Furthermore, if we take , the class of starlike functions related with the lemniscate of Bernoulli domain is resulted which was introduced and investigated by Jangteng et al. [7, 8]. Next, if we take the family of starlike functions connected with the sine function is obtained (see [9]). Mendiratta et al. [10] obtained a subclass of strongly starlike functions associated with exponential functions for the choice of . Sharma et al. [11] derived a class of starlike functions associated with a cardioid domain by taking
Moreover, several more subclasses of starlike functions have recently been presented in [12–15] through selecting specific functions for , like functions associated with conic domains, shell-like curves associated with Fibonacci numbers, and functions related with Bell numbers.
Lately, based on the techniques of Ma and Minda [16], Goel and Kumar in [17] defined the class , based on the subordination principle, as follows: and studied its various important geometric properties.
For a function of the form (2), Pommerenke [18, 19] defined Hankel determinant parameter with as follows:
The growth of for fixed integer and was evaluated for different subfamilies of univalent functions. Jangteng et al. [7, 20] investigated the sharp bound of the determinant for each of the classes and , while sharp estimation for the family of close-to-convex functions is still unknown (see [21]). On the other hand, Krishna et al. [22] proved the best estimate of for the class of Bazilevi functions. More detailed work on can be seen in [23–27] and also the references cited therein.
The determinant, is known as third-order Hankel determinant and the estimation of this determinant is more difficult and hence a potential attraction for a lot of researchers who focus on this field. In 1966–1967, Pommerenke defined this Hankel determinant but was an open problem till 2010. In 2010, Babalola [28] was the first researcher who worked on and successfully obtained the upper bound of for the functions belonging to the classes and A few mathematicians further expanded on this work to include other subclasses of holomorphic and univalent functions (see for example [29–35]). Zaprawa [36] enhanced their work in 2017 by demonstrating and asserted that these inequalities are still nonsharp. Additionally, for the sharpness, he thought about the subfamilies of and comprising of functions with -fold symmetry and acquired the sharp bounds. Recently, in 2018, Kowalczyk et al. [37] and Lecko et al. [38] got the sharp inequalities which are for the classes and , respectively, where the symbol indicates the family of starlike functions of order .
The main goal of this paper is to investigate the necessary and sufficient conditions for functions to get into the class in the form of coefficient inequality, convolution results, and the essential third-order Hankel determinant for this class in (6) and also for its 2-, 3-, and 4-fold symmetric functions
2. A Set of Lemmas
Let be the family of functions that are holomorphic in with and its series form is as follow:
Lemma 1. Ifand it is of the form ((19)), then,
Further results related to Lemma 1 can be found in [39, 40].
Lemma 2 (see [41]). Lethave the series expansion of the form ((19)). Then, for, ,
Lemma 3 (see [42]). Letsatisfy the inequalitiesandIfand has power series (19), then,
3. Improve Upper Bound for the Class
To prove Theorem 6, we need the following two lemmas (Lemma 4 and Lemma 5).
Lemma 4 (see [43]). Ifand is of the form (2), then,
Lemma 5 (see [43]). Letbe of the form (2), and then,
We now state and prove Theorem 6.
Theorem 6. Let be of the form (2), and then, The result is sharp for function
Proof. Since then, there exists a Schwarz function given in (7) such that Let Obviously, the function and This gives From (31) and (34), we have while
On equating coefficients of (35) and (36), we get
Now from (41), we have
By applying (20) and (21) to above we get
Now from (42), we have
By applying (20) and (21) to the above, we get
Using Lemma 2, we get
Let and along with triangle inequality, and we have
Differentiating (49) partially with respect to we have showing thatis an increasing function in interval so the maximum is attained atthat is,
Now since has root at and also so the maximum is attained at; therefore, we have
Hence,
For the third Hankel determinant, we need the following result.
Lemma 7 (see [17]). Letbe of the form (2). Then,
Theorem 8. Let be of the form (2). Then,
Proof. Since from (16), we have by applying triangle inequality, we obtain Now, using Lemmas 4–7 and Theorem 6 in conjunction with (59), we can get the required result.
4. Bounds of for the Class
In recent years, researchers has started to evaluate the fourth-order Hankel determinant for different subclasses of analytic functions. The trend of finding the fourth-order Hankel determinant in geometric function theory started in 2018, when Arif et al. [44] studied and obtained the upper bound for the class of bounded turning functions. Recently Zhang and Tang [31] studied the fourth-order Hankel determinant for a subclass of starlike functions associated with the sine function. Inspired from the recent research going on and from the above works, we discuss here the fourth-order Hankel determinant for the class
From (15), we can write as where
Theorem 9. Let be of the form (2), and then,
Proof. From equations (37), (38), (39), and (40) we get Now, making use of (20), (21), and (22) in conjunction with (65), we can get the required result.
Theorem 10. Let be of the form (2), and then,
Proof. From equations (37), (39), and (40) we get Applying Lemma 3 to the last term, we get the required result.
Theorem 11. Let be of the form (2), and then,
Proof. From equations (37), (39), and (40), we get Now, making use of (20), (21), and (22) in conjunction with (69), we can get the required result.
Theorem 12. Let be of the form (2), and then,
Proof. From (15), we have where and are defined in (61), (62), and (63), respectively. Now, using triangle inequalities, we have since By applying Lemmas 4 and 7 and Theorems 6, 9, and 10, we get And also, Using Lemmas 5 and 7 and Theorems 6, 10, and 11 we get Also, again, Using Lemmas 5 and 7 and Theorems 6, 9, and 11, we get Now, using the values of (74), (76), and (78) along with Theorem 6 and Lemma 7 to (72), we get the desired the estimate.
5. Zalcman Conjecture for Class
One of the main conjectures in the geometric function theory, suggested by Lawrence Zalcman in 1960, is that the coefficients of class satisfy the inequality
Only the well-known Koebe function and its rotations have equality in the above form. For the popular Fekete-Szego inequality, when , the equality holds. Recently, Khan et al. [43] evaluated the Zalcman conjecture for the class of starlike functions with respect to symmetric points associated with the sine function. Many researchers have studied the Zalcman function in the literature [45–47].
Theorem 13. Let be of the form (2), and then, The result is sharp for function
Proof. From equations (37) and (40), we get Using Lemma 3 and equation (82), we can get the required result.
6. Bounds of for 2-Fold, 3-Fold, and 4-Fold Symmetric Functions
Let It is called -fold symmetric if a rotation of domain about the origin through an angle carries itself on the domain . It is obvious that in , an analytic function is -fold symmetric if
The set of -fold symmetric univalent functions with the following series:
is referred to as .
The subclass is a collection of -fold symmetric starlike functions associated with the modified sigmoid function. More precisely, an analytic function of the form (84) belongs to class if and only if where the set is defined by
Theorem 14. If and be of the form (84), then,
Proof. Since therefore, there exists a function such that Using the series forms (84) and (86), when in the above relation, we have Now, using (89), (90), and (91), we get Now, using (20) and (21) to the above, we get the required result.
Theorem 15. If and be of the form (84), then,
Proof. Using (89) and (90), we have Using (21) to the above, we get the required result.
Theorem 16. If and be of the form (84), then,
Proof. Since therefore, and we have Then, Using (87) and (93), we get the required result.
Theorem 17. If and be of the form (84), then,
Proof. Since therefore, there exists a function such that Using the series forms (84) and (86), when in the above relation, we have Now, Therefore, Using (20) and (21), we get the desired result.
Theorem 18. If and be of the form (84), then,
Proof. Since therefore, there exists a function such that Using the series forms (84) and (86), when in the above relation, we have Since therefore, and we have Now, Using (20) to the above, we get the required result.
7. Conclusion
In our present investigation, we have obtained the improved third-order Hankel determinant for a class of starlike functions connected with modified sigmoid functions. Furthermore, we have investigated the fourth-order Hankel determinant and Zalcman conjecture and also evaluated fourth-order Hankel determinants for 2-fold, 3-fold, and 4-fold symmetric starlike functions.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors equally contributed to this manuscript and approved the final version.