Abstract
In this paper, upper bounds for the fourth-order Hankel determinant for the function class associated with the sine function are given.
1. Introduction
Let denote the class of functions which are analytic in the open unit disk of the form and let denote the subclass of consisting of univalent functions.
Suppose that is the class of analytic functions normalized by and satisfying the condition
Assume that and are two analytic functions in . Then, we say that the function is subordinate to the function , and we write if there exists a Schwarz function with and , such that (see [1])
In 2018, Cho et al. [2] introduced the following function class : which implies that the quantity lies in an eight-shaped region in the right-half plane.
In 1976, Noonan and Thomas [3] stated the Hankel determinant for and of functions as follows:
In particular, we have
Since thus
We note that is the well-known Fekete-Szegö functional (see [4–6]).
In recent years, many papers have been devoted to finding upper bounds for the second-order Hankel determinant and the third-order Hankel determinant , whose elements are various classes of analytic functions; it is worth mentioning that [7–20]. For instance, Murugusundaramoorthy and Bulboacă [21] defined a new subclass of analytic functions and got upper bounds for the Fekete-Szegö functional and the Hankel determinant of order two for Islam et al. [22] examined the -analog of starlike functions connected with a trigonometric sine function and discussed some interesting geometric properties, such as the well-known problems of Fekete-Szegö, the necessary and sufficient condition, the growth and distortion bound, closure theorem, and convolution results with partial sums for this class. Zaprawa et al. [23] obtained the bound of the third Hankel determinant for the univalent starlike functions. Very recently, Arif et al. [24] studied the problem of fourth Hankel determinant for the first time for the class of bounded turning functions and successfully obtained the bound of . Recently, Khan et al. [25] discussed some classes of functions with bounded turning which are connected to the sine functions and obtained upper bounds for the third- and fourth-order Hankel determinants related to such classes. Inspired by the aforementioned works, in this paper, we mainly investigate upper bounds for the fourth-order Hankel determinant for the function class associated with the sine function.
2. Main Results
By proving our desired results, we need the following lemmas.
Lemma 1 (see [26]). If, then exists somewith, such that
Lemma 2 (see [27]). Let, then
Lemma 3 (see [28]). Let, then we have
We now state and prove the main results of our present investigation.
Theorem 4. If the functionand of the form ((1)), then
Proof. Since , according to subordination relationship, thus there exists a Schwarz function with and , satisfying Here, Now, we define a function It is easy to see that and On the other hand, Comparing the coefficients of between equations (15) and (18), we obtain Applying Lemma 2, we easily get Let ; by using Lemma 3, we show also, let obviously, we find Setting , we have , and so, has a maximum value attained at also which is Let , according to Lemma 3, we obtain Putting we get Therefore, the function has a maximum value attained at also which is Let , in view of Lemma 3, we have that Taking we obtain Thus, is the root of the function and ; we are easy to see that the function has a maximum value attained at also which is Let , by virtue of Lemma 3, we have that Letting so we get Thus, the function has a maximum value attained at also which is Hence, the proof is complete.
Theorem 5. If the functionand of the form ((1)), then we have
Proof. Applying equation (21), we have Then, by applying Lemma 1, we get Suppose that Then, using the triangle inequality, we obtain Suppose then for any and , we get which means that is an increasing function on the closed interval [0,1] about . Therefore, the function can get the maximum value at , that is, So, obviously, Hence, the proof is complete.
Theorem 6. If the functionand of the form ((1)), then we have
Proof. From (21), we have Now, in view of Lemma 1, we get Let Then, using the triangle inequality, we deduce that Assume that Therefore, for any and we have that is, is an decreasing function on the closed interval [0,1] about . This implies that the maximum value of occurs at which is Define we clearly see that the function has a maximum value attained at also which is Hence, the proof is complete.
Theorem 7. If the functionand of the form ((1)), then we have
Proof. Let , then by equation (21), we get Now, in terms of Lemma 1, we obtain Let Then, using the triangle inequality, we get Setting then, for any and we have which implies that increases on the closed interval [0,1] about . That is, that has a maximum value at which is Putting then we have If then the root is Also, since so the function can take the maximum value at which is Hence, the proof is complete.
Theorem 8. If the functionand of the form ((1)), then we have
Proof. Let , then by using (21), we have Let , according to Lemma 3, we obtain Taking Then, we have which implies that increases on the closed interval [0,2] about . Namely, the maximum value of attains at also which is The proof of Theorem 8 is completed.
Theorem 9. If the functionand of the form ((1)), then we have
Proof. Assume that , then from (21), we obtain Next, by virtue of Lemma 3, we obtain Setting Then, we have Let , we get or and , which implies that the maximum value of attains at also which is Hence, the proof is complete.
Theorem 10. If the functionand of the form ((1)), then we have
Proof. Assume that , then from (21), we obtain Next, in terms of Lemma 3, we obtain Putting Then, for any we have , which means that the maximum value of arrives at also which is Hence, the proof is complete.
Theorem 11. If the functionand of the form ((1)), then we have
Proof. Because of then, by applying the triangle inequality, we get Next, substituting (13) and (39)–(78) into (85), we easily obtain the desired assertion (83).
3. Conclusion
In the present paper, we mainly get upper bounds of the fourth-order Hankel determinant of starlike functions connected with the sine function. However, the results obtained in this paper are not sharp. In the future, we will consider the sharpness of the results. Also, we can discuss the related research of the fifth-order Hankel determinant and fifth-order Toeplitz determinant for this function class.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The present investigation was partly supported by the National Natural Science Foundation of the People’s Republic of China under Grant 11561001, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant NJYT-18-A14, the Natural Science Foundation of Inner Mongolia of the People’s Republic of China under Grant 2018MS01026, and the Higher School Foundation of Inner Mongolia of the People’s Republic of China under Grant NJZY20200.