The Second Hankel Determinant of Logarithmic Coefficients for Starlike and Convex Functions Involving Four-Leaf-Shaped Domain

Alshehry, Azzh Saad; Shah, Rasool; Bariq, Abdul

doi:https://doi.org/10.1155/2022/2621811

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Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Developments in Functions and Operators of Complex Variables 2022

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Research Article | Open Access

Volume 2022 | Article ID 2621811 | https://doi.org/10.1155/2022/2621811

The Second Hankel Determinant of Logarithmic Coefficients for Starlike and Convex Functions Involving Four-Leaf-Shaped Domain

Azzh Saad Alshehry,¹Rasool Shah,²and Abdul Bariq³

Academic Editor: Sarfraz Nawaz Malik

Received10 Aug 2022

Accepted02 Sept 2022

Published28 Sept 2022

Abstract

In this particular research article, we take an analytic function , which makes a four-leaf-shaped image domain. Using this specific function, two subclasses, and , of starlike and convex functions will be defined. For these classes, our aim is to find some sharp bounds of inequalities that consist of logarithmic coefficients. Among the inequalities to be studied here are Zalcman inequalities, the Fekete-Szegö inequality, and the second-order Hankel determinant.

1. Introduction and Definitions

To properly comprehend the findings provided in the paper, certain important literature on geometric function theory must first be discussed. In this regard, the letters and stand for the normalized univalent (or schlicht) functions class and the normalized holomorphic (or analytic) functions class, respectively. These primary notions are defined in the disc by where expresses holomorphic functions class, and

This class evolved as the foundational component of cutting-edge research in this area. In his paper [1], Koebe established the presence of a “covering constant” , demonstrating that if is holomorphic and Schlicht in with and then . Many mathematicians were intrigued by this beautiful result. Within a few years, the wonderful article by Bieberbach [2], which gave rise to the renowned coefficient hypothesis, was published.

The below expression provided the coefficients of logarithmic function for

The above coefficients have a considerable impact on the theory of Schlicht functions in many estimations. De Branges [3] achieved that in1985, and equality will be achieved if has the form for some It is obvious that this inequality provides the most general version of the well-known Bieberbach-Robertson-Milin conjectures concerning the Taylor coefficients of . We quote [4–6] for further information on the demonstration of de Brange’s conclusion. By taking into account, the logarithmic coefficients, in 2005, Kayumov [7] established Brennan’s conjecture for conformal mappings. The major contributions to study the bounds of logarithmic coefficients for various holomorphic univalent functions are due to Alimohammadi et al. [8], Obradović et al. [9], Ye [10], Deng [11], Girela [12], Roth [13], and Andreev and Duren [14].

For the prescribed functions , the relation of subordination between and is as follows (mathematically as ), if an holomorphic function comes in with the limitation and in a manner that satisfy. Consequently, the following relation applies if in : if and only if

By applying the notion of subordination, Ma and Minda [15] proposed a consolidated version of the set in 1992, and the following is a description of it: with the Schlicht function that satisfies

Various subclasses of the set have been examined in the past few years as particular choices for family . For instance, (i) (see [16]) and (see [17])(ii) (see [18, 19])(iii) (see [20, 21]) and (see [22])(iv) with and (see [23])(v) (see [24]) and (see [25, 26])

For given , and with the series representation (1), the Hankel determinant is expressed by and it was established by Pommerenke and Pommerenke [27, 28]. For several subcollections of Schlicht functions, the determinant has been examined. In specific, the sharp estimate of the functional for sets , and were determined in [29, 30]. However, for the class of close-to-convex functions, the exact bounds of this determinant remain open [31]. The researchers were inspired by the works of Babalola [32], Bansal, et al. [33], Zaprawa [34], Kwon et al. [35], Kowalczyk et al. [36], and Lecko et al. [37].

It is easy to deduce from equation (2) that, for , the logarithmic coefficients are computed by

Currently, Lecko and Kowalczyk and Kowalczyk and Lecko [38, 39] studied the following Hankel determinant of logarithmic coefficients

It has been noted that

By the virtue of the function , we define the following classes:

Alternatively, if and only if an analytic function occurs that satisfies in such that

By taking in (18), we achieve the following function, which serves as an extremal in many of the class problems.

The following Alexander-type connection-related two classes were mentioned above. The above two families are interlinked by the following Alexander-type relation

From (19) and (20), we easily obtain the following extremal functions in various problems of the class

Clearly, , and belong to the class . That is,

In the present paper, our core objective is to find the sharp coefficient type problems of logarithmic functions for the families and Among the inequalities to be studied here are Zalcman inequalities, the Fekete-Szegö inequality, and the second-order Hankel determinant .

2. A Set of Lemmas

We must first create the class in the below set-builder form in order to declare the Lemmas that are employed in our primary findings.

That is, if , then has the below series expansion

The following Lemma consists of the widely used formula [40], the formula [41], and the formula illustrated in [42].

Lemma 1. Let be given in the form (24), then for

Lemma 2. Let be of the form (24), then for

Lemma 3. Let and has the expansion (24). Then,

The inequalities (28)–(30) are taken from [40, 43] and [26, 44, 45], respectively.

Lemma 4 (see [40]). If has the representation (24), then

Lemma 5 [46]. Let and satisfy that and

If has the expansion (24), then

3. Coefficient Inequalities for the Class

We start by establishing out the class ’s initial coefficient bounds.

Theorem 6. Let be the series form (1) and if then These bounds are sharp.

Proof. Let Then, Schwarz function may therefore be used to express (16) as From the use of Schwarz function and if , we have and by simple computation, we get Using (1), we attain By some calculation and using the series expansion of (37), we get Now, by comparing (38) and (39) we get Utilizing (40) and (10), (11), (12), and (13), we have From (44), using triangle inequality and (29), we get Also, from (45), application (30), and triangle inequality, we get By rearranging (46), we have By Lemma 4 and triangle inequality, we obtain By rearranging (47), we have Comparing the equation of (52) right side with we get , and Thus, Lemma 5’s requirements are all met. Hence, These are sharp outcomes. Equality is determined by using (10)–(13) and

Theorem 7. If then The above stated inequality is best possible.

Proof. By utilizing (44) and (45), we have Implementation of (28) and triangle inequality, we get Equality is determined by using (10), (11), and

Corollary 8. If then This inequality is sharp and can be obtained by using (10), (11), and

Theorem 9. Let be the expansion (1) and if then The above stated result is the best possible.

Proof. From (44)–(46), we easily attain By using Lemma 4 and triangle inequality, we obtain Equality is determined by using (10), (11), (12), and

Theorem 10. Let be the expansion (1) and if then The last stated inequality is the finest.

Proof. From the use (45) and (47), we get Comparing the right side of (68) with we get , , and Thus, Lemma 5’s requirements are all met. Hence, Equality is determined by using (11), (13), and

Theorem 11. Let be the representation (1). Then, This result is sharp.

Proof. We can write the as From (44)–(46), we have Using (25) and (26) to express and in terms of and also with we obtain By changing and where and utilizing triangle inequality and pickings , so Differentiate with respect to , we have It is easy exercise to show that on so that Putting , we get As so is a decreasing function, so that it gives a maximum value at Equality is determined by using (10), (11), (12), and

4. Coefficient Inequalities for the Class

For the function of class , we start this portion by determining the absolute values of the first four initial logarithmic coefficients.

Theorem 12. Let be given by (1) and if then These bounds are sharp.

Proof. Let Then, (17) can be written in the form of Schwarz function as Using (1), we obtain Now, by comparing (84) and (39), we get Utilizing (85) and (10), (11), (12), and (13) we have From (86), using triangle inequality and (29), we get Also, from (87), application (30), and triangle inequality, we get By rearranging (88), we have By Lemma 4 and triangle inequality, we obtain By rearranging (89), we have Comparing the right side of (94) with we get , and . Thus, all the conditions of Lemma 5 are satisfied. Hence, we have These are sharp outcomes. Equality is determined by using (10), (11), (12), and (13) along with (22).

Theorem 13. Let be the series form (1). Then, This inequality is sharp.

Proof. By utilizing (86) and (87), we have Implementation of (28) and triangle inequality, we get Equality is determined by using (10), (11), and (22).

For we get the below corollary.

Corollary 14. Let , and it has the form (1). Then, This inequality is sharp and can be obtained by using (10), (11), and (22).

Theorem 15. Let be the form (1) and if then This result is sharp.

Proof. By using (86)–(88) we obtain By using Lemma 4 and triangle inequality, we obtain Equality is determined by using (10), (11), (12), and (22).

Theorem 16. Let be the form (1) and Then, This result is sharp.

Proof. By using (87) and (89) we obtain Comparing the right side of (68) with we get , and Thus, all the conditions of Lemma 5 are satisfied. Hence, we have Equality is determined by using (11), (13), and (22).

Theorem 17. Let be given the form (1) and Then, This result is sharp.

Proof. We can write the as; From (86)–(88), we have Using (25) and (26) to express and in terms of and also with we obtain By replacing and where and using triangle inequality and taking , so Differentiate with respect to , we have It is a simple exercise to show that on so that Putting gives As so is a decreasing function, so that it gives a maximum value at Equality is determined by using (10), (11), (12), and (22).

Data Availability

The numerical data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This research was supported by the Princess Nourah Bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R183), Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia.

References

P. Koebe, “Über die Uniformisierung der algebraischen Kurven. II,” Mathematische Annalen, vol. 69, no. 1, pp. 1–81, 1910.
View at: Publisher Site | Google Scholar
L. Bieberbach, “Über einige Extremalprobleme im Gebiete der konformen abbildung,” Mathematische Annalen, vol. 77, no. 2, pp. 153–172, 1916.
View at: Publisher Site | Google Scholar
L. De Branges, “A proof of the Bieberbach conjecture,” Acta Mathematica, vol. 154, no. 1-2, pp. 137–152, 1985.
View at: Publisher Site | Google Scholar
F. G. Avkhadiev and K. J. Wirths, Schwarz-Pick Type Inequalities, Springer Science & Business Media, 2009.
C. H. FitzGerald and C. Pommerenke, “The de Branges theorem on univalent functions,” Transactions of the American Mathematical Society, vol. 290, no. 2, pp. 683–690, 1985.
View at: Publisher Site | Google Scholar
C. H. FitzGerald and C. Pommerenke, “A theorem of de Branges on univalent functions,” Serdica, vol. 13, no. 1, pp. 21–25, 1987.
View at: Google Scholar
I. P. Kayumov, “On Brennan’s conjecture for a special class of functions,” Mathematical Notes, vol. 78, no. 3-4, pp. 498–502, 2005.
View at: Publisher Site | Google Scholar
D. Alimohammadi, E. Analouei Adegani, T. Bulboaca, and N. E. Cho, “Logarithmic coefficient bounds and coefficient conjectures for classes associated with convex functions,” Journal of Function Spaces, vol. 2021, Article ID 6690027, 7 pages, 2021.
View at: Publisher Site | Google Scholar
M. Obradović, S. Ponnusamy, and K. J. Wirths, “Logarithmic coefficients and a coefficient conjecture for univalent functions,” Monatshefte für Mathematik, vol. 185, no. 3, pp. 489–501, 2018.
View at: Publisher Site | Google Scholar
Z. Ye, “The logarithmic coefficients of close-to-convex functions,” Bulletin of the Institute of Mathematics, Academia Sinica, vol. 3, no. 3, pp. 445–452, 2008.
View at: Google Scholar
Q. Deng, “On the logarithmic coefficients of Bazilevic functions,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5889–5894, 2011.
View at: Publisher Site | Google Scholar
D. Girela, “Logarithmic coefficients of univalent functions,” Annales-Academiae Scientiarum Fennicae Series A1 Mathematica, vol. 25, no. 2, pp. 337–350, 2000.
View at: Google Scholar
O. Roth, “A sharp inequality for the logarithmic coefficients of univalent functions,” Proceedings of the American Mathematical Society, vol. 135, no. 7, pp. 2051–2054, 2007.
View at: Publisher Site | Google Scholar
V. V. Andreev and P. L. Duren, “Inequalities for logarithmic coefficients of univalent functions and their derivatives,” Indiana University Mathematics journal, vol. 37, no. 4, pp. 721–733, 1988.
View at: Publisher Site | Google Scholar
W. C. Ma and D. Minda, “A unified treatment of some special classesof univalent functions,” in Proceedings of the Conference on Complex Analysis, pp. 157–169, Tianjin, China, 1992.
View at: Google Scholar
K. Bano and M. Raza, “Starlike functions associated with cosine function,” Bulletin of Iranian Mathematical Society, vol. 47, no. 11, p. 20, 2021.
View at: Publisher Site | Google Scholar
A. Alotaibi, M. Arif, M. A. Alghamdi, and S. Hussain, “Starlikness associated with cosine hyperbolic function,” Mathematics, vol. 8, no. 7, 2020.
View at: Publisher Site | Google Scholar
K. Ullah, S. Zainab, M. Arif, M. Darus, and M. Shutaywi, “Radius problems for starlike functions associated with the tan hyperbolic function,” Journal of Function Spaces, vol. 2021, Article ID 9967640, 15 pages, 2021.
View at: Publisher Site | Google Scholar
K. Ullah, H. M. Srivastava, A. Rafiq, M. Arif, and S. Arjika, “A study of sharp coefficient bounds for a new subfamily of starlike functions,” Journal of Inequalities and Applications, vol. 2021, no. 1, 2021.
View at: Publisher Site | Google Scholar
R. Mendiratta, S. Nagpal, and V. Ravichandran, “On a subclass of strongly starlike functions associated with exponential function,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 38, no. 1, pp. 365–386, 2015.
View at: Publisher Site | Google Scholar
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.
View at: Publisher Site | Google Scholar
S. S. Kumar and K. Arora, “Starlike functions associated with a petal shaped domain,” 2020, https://arxiv.org/abs/2010.10072.
View at: Google Scholar
D. A. Brannan and W. E. Kirwan, “On some classes of bounded univalent functions,” Journal of the London Mathematical Society, vol. 2, no. 1, pp. 431–443, 1969.
View at: Publisher Site | Google Scholar
J. Sokół and J. Stankiewicz, “Radius of convexity of some subclasses of strongly starlike functions,” Zeszyty naukowe Politechniki Rzeszowskiej Matematyka, vol. 19, pp. 101–105, 1996.
View at: Google Scholar
K. Sharma, N. K. Jain, and V. Ravichandran, “Starlike functions associated with a cardioid,” Afrika Matematika, vol. 27, no. 5-6, pp. 923–939, 2016.
View at: Publisher Site | Google Scholar
L. Shi, I. Ali, M. Arif, N. E. Cho, S. Hussain, and H. Khan, “A study of third Hankel determinant problem for certain subfamilies of analytic functions involving cardioid domain,” Mathematics, vol. 7, no. 5, p. 418, 2019.
View at: Publisher Site | Google Scholar
C. Pommerenke, “On the coefficients and Hankel determinants of univalent functions,” Journal of the London Mathematical Society, vol. 1, no. 1, pp. 111–122, 1966.
View at: Publisher Site | Google Scholar
C. Pommerenke, “On the Hankel determinants of univalent functions,” Mathematika, vol. 14, no. 1, pp. 108–112, 1967.
View at: Publisher Site | Google Scholar
A. Janteng, 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.
View at: Google Scholar
A. Janteng, 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.
View at: Google Scholar
D. Raducanu and P. Zaprawa, “Second Hankel determinant for close-to-convex functions,” Comptes Rendus Mathematique, vol. 355, no. 10, pp. 1063–1071, 2017.
View at: Publisher Site | Google Scholar
K. O. Babalola, “On Hankel determinant for some classes of univalent functions,” Inequality Theory and Applications, vol. 6, pp. 1–7, 2010.
View at: Google Scholar
D. Bansal, S. Maharana, and J. K. Prajapat, “Third order Hankel Determinant for certain univalent functions,” Journal of Korean Mathematical Society, vol. 52, no. 6, pp. 1139–1148, 2015.
View at: Publisher Site | Google Scholar
P. Zaprawa, “Third Hankel determinants for subclasses of univalent functions,” Mediterranean Journal of Mathematics, vol. 141, no. 19, p. 10, 2017.
View at: Publisher Site | Google Scholar
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.
View at: Publisher Site | Google Scholar
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.
View at: Publisher Site | Google Scholar
A. Lecko, Y. J. Sim, and B. Smiarowska, “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.
View at: Publisher Site | Google Scholar
B. Kowalczyk and A. Lecko, “Second Hankel determinant of logarithmic coefficients of convex and starlike functions,” Bulletin of the Australian Mathematical Society, vol. 105, no. 3, pp. 458–467, 2022.
View at: Publisher Site | Google Scholar
B. Kowalczyk and A. Lecko, “Second Hankel determinant of logarithmic coefficients of convex and starlike functions of order alpha,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 45, pp. 727–740, 2022.
View at: Publisher Site | Google Scholar
C. Pommerenke, Univalent Function, Göttingen, Germany, Vanderhoeck & Ruprecht, 1975.
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, pp. 225–230, 1982.
View at: Publisher Site | Google Scholar
O. S. Kwon, A. Lecko, and Y. J. Sim, “On the fourth coefficient of functions in the Carathéodory class,” Computational Methods and Function Theory, vol. 18, no. 2, pp. 307–314, 2018.
View at: Publisher Site | Google Scholar
F. Keough and E. Merkes, “A coefficient inequality for certain classes of analytic functions,” Proceedings of the American Mathematical Society, vol. 20, no. 1, pp. 8–12, 1969.
View at: Publisher Site | Google Scholar
M. Arif, M. Raza, H. Tang, S. Hussain, and H. Khan, “Hankel determinant of order three for familiar subsets of analytic functions related with sine function,” Open Mathematics, vol. 17, no. 1, pp. 1615–1630, 2019.
View at: Publisher Site | Google Scholar
M. Arif, S. Umar, 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, no. 5, pp. 1–11, 2020.
View at: Publisher Site | Google Scholar
V. Ravichandran and S. Verma, “Borne pour le cinquieme coefficient des fonctions etoilees,” Comptes Rendus Mathematique, vol. 353, no. 6, pp. 505–510, 2015.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Azzh Saad Alshehry 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.

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