Mathematical Problems in Engineering
Volume 2021 (2021), Article ID 6674010, 8 pages
https://doi.org/10.1155/2021/6674010

Research Article

Fourth Hankel Determinant for the Set of Star-Like Functions

Muhammad Arif,¹ Lubna Rani,¹ Mohsan Raza,² and Pawel Zaprawa³

¹Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
²Department of Mathematics, Government College University Faisalabad, Faisalabad, Pakistan
³Department of Mathematics, Faculty of Mechanical Engineering, Lublin University of Technology, Lublin, Poland

Correspondence should be addressed to Mohsan Raza; mohsan976@yahoo.com

Received 13 December 2020; Revised 5 February 2021; Accepted 19 April 2021; Published 3 May 2021

Academic Editor: Luigi Rodino

Copyright © 2021 Muhammad Arif et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 form

which 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 form

whose 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]) as

for 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 that

for 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 that

for . 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 estimates

hold 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 expressions

are 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 as

where

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 get

After rearranging the terms, we get (27) in the following form:

Let

Applying the triangle inequality and Lemma 1 and Lemma 2 in (28), we have

Hence 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 if

holds 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 if

where 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 in

we get

Consequently, for , we have

From (20) and Lemma 3, we get

Therefore, writing and also assuming in [0, 2] and , we have

which takes the greatest value for , so

Since 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 is

Using the same argument as given in the proof of Theorem 2, we obtain

which 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), then

The estimate is sharp.

Proof. Let be of the form , where , and let and be of the form

Since , from the equality

we get

Consequently, for , there is and

From (48), (20), and Lemma 3, we get

Therefore, writing , we have

The 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.

References

C. Pommerenke, “On the coefficients and Hankel determinants of univalent functions,” Journal of the London Mathematical Society, vol. s1-41, no. 1, pp. 111–122, 1966.

C. Pommerenke, “On the Hankel determinants of univalent functions,” Mathematika, vol. 14, no. 1, pp. 108–112, 1967.

M. Arif, K. I. Noor, and M. Raza, “Hankel determinant problem of a subclass of analytic functions,” Journal of Inequalities and Applications, vol. 2012, Article ID 22, 2012.

M. Arif, K. I. Noor, M. Raza, and H. WU, “Some properties of a generalized class of analytic functions related with Janowski functions,” Abstract and Applied Analaysis, vol. 2012, Article ID 279843, 11 pages, 2012.

A. Jangteng, S. A. Halim, and M. Darus, “Coefficient inequality for a function whose derivative has a positive real part,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 2, pp. 1–5, 2006.

A. Jangteng, S. A. Halim, and M. Darus, “Hankel determinant for starlike and convex functions,” International Journal of Mathematical Analysis, vol. 1, no. 13, pp. 619–625, 2007.

P. Zaprawa, “Second Hankel determinants for the class of typically real functions,” Abstract and Applied Analysis, vol. 2016, Article ID 3792367, 2016.

D. V. Krishna and T. RamReddy, “Second Hankel determinant for the class of Bazilevic functions,” Studia Universitatis Babes-Bolyai Mathematica, vol. 60, no. 3, pp. 413–420, 2015.

D. Rãducanu and P. Zaprawa, “Second Hankel determinant for close-to-convex functions,” Comptes Rendus Mathematique, vol. 355, pp. 1063–1071, 2017.

D. Bansal, “Upper bound of second Hankel determinant for a new class of analytic functions,” Applied Mathematics Letters, vol. 26, no. 1, pp. 103–107, 2013.

W. K. Hayman, “On the second Hankel determinant of mean univalent functions,” Proceedings of the London Mathematical Society, vol. s3-18, no. 1, pp. 77–94, 1968.

D. V. Krishna and T. RamReddy, “Hankel determinant for starlike and convex functions of order alpha,” Tbilisi Mathematical Journal, vol. 5, pp. 65–76, 2012.

S. K. Lee, V. Ravichandran, and S. Supramaniam, “Bounds for the second Hankel determinant of certain univalent functions,” Journal of Inequalities and Applications, vol. 2013, Article ID 281, 2013.

M. S. Liu, J. F. Xu, and M. Yang, “Upper bound of second Hankel determinant for certain subclasses of analytic functions,” Abstract and Applied Analysis, vol. 2014, Article ID 603180, 2014.

J. W. Noonan and D. K. Thomas, “On the second Hankel determinant of areally mean p-valent functions,” Transactions of the American Mathematical Society, vol. 223, no. 2, pp. 337–346, 1976.

H. Orhan, N. Magesh, and J. Yamini, “Bounds for the second Hankel determinant of certain bi-univalent functions,” Turkish Journal of Mathematics, vol. 40, pp. 679–687, 2016.

K. O. Babalola, “On H₃(1) Hankel determinant for some classes of univalent functions,” Inequality Theory and Applications, vol. 6, pp. 1–7, 2007.

S. Altinkaya and S. Yalçin, “Third Hankel determinant for bazilevi č functions,” Advances in Mathematics, vol. 5, no. 2, pp. 91–96, 2016.

D. Bansal, S. Maharana, and J. K. Prajapat, “Third order Hankel Determinant for certain univalent functions,” Journal of the Korean Mathematical Society, vol. 52, no. 6, pp. 1139–1148, 2015.

N. E. Cho, B. Kowalczyk, O. S. Kwon, A. Lecko, and Y. J. Sim, “Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha,” Journal of Mathematical Inequalities, vol. 11, no. 2, pp. 429–439, 2017.

D. V. Krishna, B. Venkateswarlua, and T. RamReddy, “Third Hankel determinant for bounded turning functions of order alpha,” Journal of the Nigerian Mathematical Society, vol. 34, pp. 121–127, 2015.

M. Raza and S. N. Malik, “Upper bound of third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli,” Journal of Inequalities and Applications, vol. 2013, Article ID 412, 2013.

G. Shanmugam, B. A. Stephen, and K. O. Babalola, “Third Hankel determinant for α-starlike functions,” Gulf Journal of Mathematics, vol. 2, no. 2, pp. 107–113, 2014.

D. V. Krishna, B. Venkateswarlua, and T. RamReddy, “Third Hankel determinant for certain subclass of p-valent functions,” Complex Variables and Elliptic Equations, vol. 60, no. 9, pp. 1301–1307, 2015.

P. Zaprawa, “Third Hankel determinants for subclasses of univalent functions,” Mediterranean Journal of Mathematics, vol. 14, Article ID 19, 2017.

O. S. Kwon, A. Lecko, and Y. J. Sim, “The bound of the Hankel determinant of the third kind for starlike functions,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 42, no. 2, pp. 767–780, 2019.

B. Kowalczyk, A. Lecko, and Y. J. Sim, “The sharp bound for the Hankel determinant of the third kind for convex functions,” Bulletin of the Australian Mathematical Society, vol. 97, no. 3, pp. 435–445, 2018.

A. Lecko, Y. J. Sim, and B. Śmiarowska, “The sharp bound of the Hankel determinant of the third kind for starlike functions of order 1/2,” Complex Analysis and Operator Theory, vol. 13, no. 5, pp. 2231–2238, 2019.

S. Mahmood, H. Srivastava, N. Khan, Q. Ahmad, B. Khan, and I. Ali, “Upper bound of the third Hankel determinant for a subclass of q-starlike functions,” Symmetry, vol. 11, no. 3, p. 347, 2019.

L. Shi, H. M. Srivastava, M. Arif, S. Hussain, and H. Khan, “An investigation of the third Hankel determinant problem for certain subfamilies of univalent functions involving the exponential function,” Symmetry, vol. 11, no. 5, p. 598, 2019.

M. Arif, R. Rani, M. Raza, and P. Zaprawa, “Fourth Hankel determinant for a family of functions with bounded turning,” Bullettin of Korean Mathematical Society, vol. 55, no. 6, pp. 1703–1711, 2018.

M. Arif, I. Ullah, M. Raza, and P. Zaprawa, “Investigation of the fifth Hankel determinant for a family of functions with bounded turnings,” Mathematica Slovaca, vol. 70, no. 2, pp. 319–328, 2020.

N. E. Cho and V. Kumar, “Initial coefficients and fourth Hankel determinant for certain analytic functions,” Miskolc Mathematical Notes, vol. 21, no. 2, pp. 763–779, 2020.

M. Arif, S. Umer, M. Raza, T. Bulboaca, M. U. Farooq, and H. Khan, “On fourth Hankel determinant for functions associated with Bernoulli’s lemniscate,” Hacettepe Journal of Mathematics and Statistics, vol. 49, pp. 1777–1780, 2020.

C. Carathéodory, “Über den variabilitätsbereich der fourier'schen konstanten von positiven harmonischen funktionen,” Rendiconti del Circolo Matematico di Palermo, vol. 32, no. 1, pp. 193–217, 1911.

A. E. Livingston, “The coefficients of multivalent close-to-convex functions,” Proceedings of the American Mathematical Society, vol. 21, no. 3, p. 545, 1969.

R. J. Libera and E. J. Złotkiewicz, “Early coefficients of the inverse of a regular convex function,” Proceedings of the American Mathematical Society, vol. 85, no. 2, p. 225, 1982.