Second Hankel Determinant for a Certain Subclass of Bi-Close to Convex Functions Defined by Kaplan
Abstract
:1. Introduction
1.1. Bi-Univalence
1.2. Subfamilies of and Related Bi-Univalent Functions
1.3. Hankel Determinant
1.4. Some Useful Bounds
2. Second Hankel Determinant in Class and
2.1. The Class
- (i)
- . In this case, , that is, is an increasing function. Therefore, for fixed the maximum of may occurs only at , and
- (ii)
- . As for , it is clear that so that . Therefore, similarly as in the case (i) the maximum of is attained for
2.2. The Class
3. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Lewin, M. On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 1967, 18, 63–68. [Google Scholar] [CrossRef]
- Brannan, D.A.; Clunie, J.G. Aspects of Contemporary Complex Analysis; Academic Press: London, UK, 1980. [Google Scholar]
- Netanyahu, E. The minimal distance of the Image boundary from the origin and the second coefficient of a univalent function in |z| < 1. Arch. Ration. Mech. Anal. 1969, 32, 100–112. [Google Scholar]
- Kaplan, W. Close-to-convex schlicht functions. Michigan Math. J. 1952, 1, 169–185. [Google Scholar] [CrossRef]
- Brannan, D.A.; Taha, Ṫ.S. On some classes of bi-univalent functions. Studia Univ. Babeş Bolyai Math. 1986, 31, 70–77. [Google Scholar]
- Ali, R.M.; Lee, S.K.; Ravichandran, V.; Supramaniam, S. Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions. Appl. Math. Lett. 2012, 25, 344–351. [Google Scholar] [CrossRef] [Green Version]
- Hayami, T.; Owa, Ṡ. Coefficient bounds for bi-univalent functions. PanAm. Math. J. 2012, 22, 15–26. [Google Scholar]
- Reade, M.O. The coefficients of close-to-convex functions. Duke Math. J. 1956, 23, 459–462. [Google Scholar] [CrossRef]
- Sivasubramanian, S.; Sivakumar, R.; Kanas, S.; Kim, S.A. Verification of Brannan and Clunie’s conjecture for certain subclasses of bi-univalent functions. Ann. Polon. Math. 2015, 113, 295–304. [Google Scholar] [CrossRef]
- Çağlar, M.; Deniz, E.; Srivastava, Ḣ.M. Second Hankel determinant for certain subclasses of bi-univalent functions. Turk. J. Math. 1986, 41, 694–706. [Google Scholar] [CrossRef]
- Pommerenke, C. On the coefficients and Hankel determinants of univalent functions. J. Lond. Math. Soc. 1966, 41, 111–122. [Google Scholar] [CrossRef]
- Pommerenke, C. On the Hankel determinants of univalent functions. Mathematika 1967, 14, 108–112. [Google Scholar] [CrossRef]
- Noonan, J.W.; Thomas, D.K. On the second Hankel determinant of areally mean p-valent functions. Trans. Am. Math. Soc. 1976, 223, 337–346. [Google Scholar]
- Dienes, P. The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable; Dover Publications, Inc.: New York, NY, USA, 1957. [Google Scholar]
- Edrei, A. Sur les déterminants récurrents et les singularitiés d’une fonction donée por son développement de Taylor. Compos. Math. 1940, 7, 20–88. [Google Scholar]
- Fekete, M.; Szegö, G. Eine bemerkung uber ungerade schlichten funktionene. J. Lond. Math. Soc. 1993, 8, 85–89. [Google Scholar]
- Wilson, R. Determinantal criteria for meromorphic functions. Proc. Lond. Math. Soc. 1954, 4, 357–374. [Google Scholar] [CrossRef]
- Vein, R.; Dale, P. Determinants and Their Applications in Mathematical Physics. In Applied Mathematical Sciences; Springer: New York, NY, USA, 1999; Volume 134. [Google Scholar]
- Kanas, S.; Adegani, E.A.; Zireh, Ȧ. An unified approach to second Hankel determinant of bi-subordinate functions. Mediterr. J. Math. 2017, 14, 12. [Google Scholar] [CrossRef]
- Keogh, K.R.; Merkes, E.P. A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 1969, 20, 8–12. [Google Scholar] [CrossRef]
- Lee, S.K.; Ravichandran, V.; Supramaniam, Ṡ. Bounds for the second Hankel determinant of certain univalent functions. J. Inequal. Appl. 2013, 2013, 17. [Google Scholar] [CrossRef] [Green Version]
- Motamednezhad, A.; Bulboacă, T.; Adegani, E.A.; Dibagar, N. Second Hankel determinant for a subclass of analytic bi-univalent functions defined by subordination. Turk. J. Math. 2018, 42, 2798–2808. [Google Scholar] [CrossRef]
- Deniz, E.; Çağlar, M.; Orhan, Ḣ. Second Hankel determinant for bi-starlike and bi-convex functions of order β. Appl. Math. Comput. 2015, 271, 301–307. [Google Scholar] [CrossRef] [Green Version]
- Orhan, H.; Magesh, N.; Yamini, J. Bounds for the second Hankel determinant of certain bi-univalent functions. Turk. J. Math. 2016, 40, 679–687. [Google Scholar] [CrossRef]
- Orhan, H.; Toklu, E.; Kadıoğlu, E. Second Hankel determinant problem for k-bi-starlike functions. Filomat 2017, 31, 3897–3904. [Google Scholar] [CrossRef]
- Pommerenke, C. Univalent Functions; Vandenhoeck & Ruprecht: Göttingen, Germany, 1975. [Google Scholar]
- Grenander, U.; Szegö, Ġ. Toeplitz forms and their applications. In California Monographs in Mathematical Sciences; University of California Press: Berkeley, CA, USA, 1958. [Google Scholar]
- Kanas, S. An unified approach to the Fekete-Szegö problem. Appl. Math. Comput. 2012, 218, 8453–8461. [Google Scholar] [CrossRef]
- Janteng, A.; Halim, S.; Darus, M. Hankel determinant for starlike and convex functions. Int. J. Math. Anal. (Ruse) 2007, 1, 619–625. [Google Scholar]
- Babalola, K.O. On H3(1) Hankel determinant for some classes of univalent functions. In Inequality Theory and Applications; Cho, Y.J., Kim, J.K., Dragomir, S.S., Eds.; Nova Science Publishers, Inc.: Hauppauge, NY, USA, 2003; Volume 2. [Google Scholar]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Kanas, S.; Sivasankari, P.V.; Karthiyayini, R.; Sivasubramanian, S. Second Hankel Determinant for a Certain Subclass of Bi-Close to Convex Functions Defined by Kaplan. Symmetry 2021, 13, 567. https://doi.org/10.3390/sym13040567
Kanas S, Sivasankari PV, Karthiyayini R, Sivasubramanian S. Second Hankel Determinant for a Certain Subclass of Bi-Close to Convex Functions Defined by Kaplan. Symmetry. 2021; 13(4):567. https://doi.org/10.3390/sym13040567
Chicago/Turabian StyleKanas, Stanislawa, Pesse V. Sivasankari, Roy Karthiyayini, and Srikandan Sivasubramanian. 2021. "Second Hankel Determinant for a Certain Subclass of Bi-Close to Convex Functions Defined by Kaplan" Symmetry 13, no. 4: 567. https://doi.org/10.3390/sym13040567