Third Hankel Determinant Problem for Certain Subclasses of Analytic Functions Associated with Nephroid Domain
Abstract
In this research article we consider two well known subclasses of starlike and bounded turning functions associated with nephroid domain. Our aims to find third Hankel determinant for these classes.
References
L. Bieberbach, Uber die koeffizienten derjenigen Potenzreihen, weiche eine schlichte Abbildung des Einheitskreises vermittein, Sitzungsberichte Preussische Akademie der Wissenschaften 138 (1916), 940-955.
L. De Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137-152. https://doi.org/10.1007/BF02392821
W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23 (1970), 159-177. https://doi.org/10.4064/ap-23-2-159-177
J. Sokóƚ and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19 (1996), 101-105.
R.K. Raina and J. Sokóƚ, Some properties related to a certain class of starlike functions, C. R. Math. Acad. Sci. Paris 353(11) (2015), 973-978. https://doi.org/10.1016/j.crma.2015.09.011
R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38 (2015), 365-386. https://doi.org/10.1007/s40840-014-0026-8
S. Kumar and V. Ravichandran, A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math. 40 (2016), 199-212.
N.E. Cho, V. Kumar, S.S. Kumar and V. Ravichandran, Radius problems for starlike functions associated with the Sine function, Bull. Iran. Math. Soc. 45 (2019), 213-232. https://doi.org/10.1007/s41980-018-0127-5
R. Kargar, A. Ebadian and J. Sokóƚ, Radius problems for some subclasses of analytic functions, Complex Anal. Oper. Theory 11 (2017), 1639-1649. https://doi.org/10.1007/s11785-016-0584-x
R. Kargar, A. Ebadian and J. Sokóƚ, On Booth lemniscate and starlike functions, Anal. Math. Phys. 9 (2019), 143-154. https://doi.org/10.1007/s13324-017-0187-3
J. Dziok, R.K. Raina and J. Sokóƚ, On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers, Math. Comput. Model. 57 (2013), 1203-1211. https://doi.org/10.1016/j.mcm.2012.10.023
N.E. Cho, S. Kumar, V. Kumar, V. Ravichandran and H.M. Srivastava, Starlike functions related to the Bell numbers, Symmetry 11 (2019), 219. https://doi.org/10.3390/sym11020219
S. Mahmood, H.M. Srivastava and S.N. Malik, Some subclasses of uniformly univalent functions with respect to symmetric points, Symmetry 11 (2019), 287. https://doi.org/10.3390/sym11020287
Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. Lond. Math. Soc. 41 (1966), 111-122. https://doi.org/10.1112/jlms/s1-41.1.111
Ch. Pommerenke, On the Hankel determinants of univalent functions, Mathematika 14 (1967), 108-112. https://doi.org/10.1112/S002557930000807X
A. Jangteng, S.A. Halim and M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Ineq. Pure Appl. Math. 7 (2006), 1-5.
A. Jangteng, S.A. Halim and M. Darus, Coefficient inequality for starlike and convex functions, Int. J. Ineq. Math. Anal. 1 (2007), 619-625.
L.A. Wani and A. Swaminathan, Stralike and convex functions associated with a Nephroid domain having cusps on the real axis, 2019. arXiv.1912.05767v1
D. Răducanu and P. Zaprawa, Second Hankel determinant for close-to-convex functions, C. R. Math. Acad. Sci. Paris 355(10) (2017), 1063-1071. https://doi.org/10.1016/j.crma.2017.09.006
D. V. Krishna and T. RamReddy, Second Hankel determinant for the class of Bazilevic functions, Stud. Univ. Babes-Bolyai Math. 60 (2015), 413-420.
D. Bansal, Upper bound of second Hankel determinant for a new class of analytic functions, Appl. Math. Lett. 23 (2013), 103-107. https://doi.org/10.1016/j.aml.2012.04.002
S.K. Lee, V. Ravichandran and S. Supramaniam, Bounds for the second Hankel determinant of certain univalent functions, J. Inequal. Appl. 2013, 2013:281, 17 pp. https://doi.org/10.1186/1029-242X-2013-281
M. S. Liu, J. F. Xu and M. Yang, Upper bound of second Hankel determinant for certain subclasses of analytic functions, Abstr. Appl Anal. 2014, Art. ID 603180, 10 pp. https://doi.org/10.1155/2014/603180
J. W. Noonan and D. K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc. 223 (1976), 337-346. https://doi.org/10.2307/1997533
H. Orhan, N. Magesh and J. Yamini, Bounds for the second Hankel determinant of certain bi-univalent functions, Turk. J. Math. 40 (2016), 679-687. https://doi.org/10.3906/mat-1505-3
K. O. Babalola, On $H_{3}left( 1right)$ Hankel determinant for some classes of univalent functions, Inequal. Theory Appl. 6 (2010), 1-7.
H. M. Srivastava, S. Altinkaya and S. Yalçin, Hankel determinant for a subclass of bi-valent functions defined by using a symmetric q-derivative operator, Filomat 32 (2018), 503-516. https://doi.org/10.2298/FIL1802503S
H. M. Srivastava, Q. Z. Ahmad, M. Darus, N. Khan, B. Khan, N. Zaman and H. H. Shah, Upper bound of the third Hankel determinant for a subclass of close-to-convex functions associated with Lemniscate of Bernouli, Mathematics 7(9) (2019), 848. https://doi.org/10.3390/math7090848
H. M. Srivastava, Q. Z. Ahmad, N. Khan, N. Khan and B. Khan, Hankel and Toeplitz determinant for a subclass of q-starlike functions associated with a general conic domain, Mathematics 7(2) (2019), 181. https://doi.org/10.3390/math7020181
M. Shafiq, H. M. Srivastava, N. Khan, Q. Z. Ahmad, M. Darus and S. Kiran, An upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with k-Fibonacci numbers, Symmetry 12(6) (2020), 1043. https://doi.org/10.3390/sym12061043
L. Shi, M. G. Khan and B. Ahmad, Some geometric properties of a family of analytic functions involving a generalized q-operator, Symmetry 12(2) (2020), 291. https://doi.org/10.3390/sym12020291
M.G. Khan, B. Ahmad, G. Muraugusundaramoorthy, R. Chinram and W. K. Mashwani, Applications of modified sigmoid functions to a class of starlike functions, J. Funct. Spaces 2020, Art. ID 8844814, 8 pp. https://doi.org/10.1155/2020/8844814
M.G. Khan, B. Ahmad, J. Sokol and Z. Muhammad, Coefficient problems in a class of functions with bounded turning associated with Sine function, European Journal of Pure and Applied Mathematics 14 (2021), 53-64. https://doi.org/10.29020/nybg.ejpam.v14i1.3902
S. Altinkaya and S. Yalçin, Third Hankel determinant for Bazilevič functions, Advances in Math. 5 (2016), 91-96.
D. Bansal, S. Maharana and J. K. Prajapat, Third order Hankel determinant for certain univalent functions, J. Korean Math. Soc. 52 (2015), 1139-1148. https://doi.org/10.4134/JKMS.2015.52.6.1139
N.E. Cho, B. Kowalczyk, O.S. Kwon, A. Lecko and J. Sim, Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha, J. Math. Inequal. 11 (2017), 429-439. https://doi.org/10.7153/jmi-2017-11-36
D.V. Krishna, B. Venkateswarlua and T. RamReddy, Third Hankel determinant for bounded turning functions of order alpha, J. Niger. Math. Soc. 34 (2015), 121-127. https://doi.org/10.1016/j.jnnms.2015.03.001
G. Shanmugam, B.A. Stephen and K.O. Babalola, Third Hankel determinant for -starlike functions, Gulf J. Math. 2 (2014), 107-113.
P. Zaprawa, Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math. 14(1) (2017), Paper No. 19, 10 pp. https://doi.org/10.1007/s00009-016-0829-y
B. Kowalczyk, A. Lecko and Y.J. Sim, The sharp bound of the Hankel determinant of the third kind for convex functions, Bull. Australian Math. Soc. 97 (2018), 435-445. https://doi.org/10.1017/S0004972717001125
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 Anal. Oper. Theory 13(5) (2019), 2231-2238. https://doi.org/10.1007/s11785-018-0819-0
O.S. Kwon, A. Lecko and Y.J. Sim, The bound of the Hankel determinant of the third kind for starlike functions, Bull. Malays. Math. Sci. Soc. 42 (2018), 1-14. https://doi.org/10.1007/s40840-018-0683-0
C. Pommerenke, Univalent Functions, Math. Lehrbucher, Band XXV, Vandenhoeck & Ruprecht, Gottingen, 1975.
F. Keogh and E. Merkes, A coefficient inequality for certain subclasses of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8-12. https://doi.org/10.1090/S0002-9939-1969-0232926-9
V. Ravichandran and S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris 353(6) (2015), 505-510. https://doi.org/10.1016/j.crma.2015.03.003
W.C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157-169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA, 1994.
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 functions, Open Mathematics 17 (2019), 1615-1630. https://doi.org/10.1515/math-2019-0132
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