Upper Bound of the Third Hankel Determinant for a Subclass of Analytic Functions Connected with Sine Functions
Abstract
turning and multivalent starlike functions that are subordinate with sine function in the open unit disk E = fz : jzj < 1g : For these families our aim is to nd the bounds of Hankel determinant of order three. Further, the estimate of third order Hankel determinant for the family SL sin in this work improve the bounds which was investigated recently. Moreover, the same bounds have been investigated for 2-fold symmetric and 3-fold symmetric functions. Also we discuss H (3) determinant for the above mentioned families. 2 sin
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Ali, R.M.; Cho, N.E.; Ravichandran, V.; Kumar, S.S. First order di⁄erential
subordination for functions associated with the lemniscate of Bernoulli. Taiwanese
J. Math. 2012; 16; 1017 1026:
Altinkaya, ‚S.; Yalçin, S. Third Hankel determinant for Bazilevi
µ
c functions. Adv.
Math. 2016; 5; 91 96:
Alt‹nkaya, ‚S.; Yalç‹n, S. Upper bound of second Hankel determinant for biBazilevic
functions.
Mediterr. J. Math. 2016; 13; 4081 4090:
Arif, M.; Su¢ ciency criteria for a class of p-valent analytic functions of complex
order, Abstract and Applied Analysis, Volume 2013, Article ID 517296, 4 pages.
Arif, M.; Ayaz. M.; Iqbal, J.; Haq, W. Su¢ cient conditions for functions to be
in a class of p-valent analytic functions, Journal of Computational Analysis and
Applications, 2013, 16(1); 159 164.
Arif, M.; Dziok, J.; Raza, M.; Sokó÷, J. On products of multivalent close-to-star
functions, Journal of Inequality and Applications, Vol. 2015; 2015 : 5; 14 Pages.
Arif, M.; Noor, K.I,; Raza, M. Hankel determinant problem of a subclass of
analytic functions. J. Ineq. Appl. 2012(1); Art. 22, 7 pages.
Arif, M.; Sokó÷, J.; Ayaz, M.; Su¢ cient condition for functions to be in a class
of meromorphic multivalent Sakaguchi type spiral-like functions, Acta Mathematica
Scientia, 2014; 34B(2); 1 4.
Babalola, K.O. On H
(1) Hankel determinant for some classes of univalent
functions. Inequal. Theory Appl. 2010; 6; 1 7:
Bansal, D. Upper bound of second Hankel determinant for a new class of analytic
functions. Appl. Math. Lett. 2013; 23; 103 107:
Bansal, D.; Maharana, S.; Prajapat, J.K. Third order Hankel Determinant for
certain univalent functions. J. Korean Math. Soc. 2015; 32; 1139 1148:
˙aglar, M.; Deniz, E.; Srivastava, H.M. Second Hankel determinant for certain
subclasses of bi-univalent functions. Turk. J. Math. 2017; 41; 694 706:
Caratheodory, C. Uber den variabilitatsbereich der fourierschen konstanten von
positiven harmonischen funktionen. Rend. Circ. Mat. Palermo. 1911; 32; 193
:
Cho, N.E.; Kowalczyk, B.; Kwon, O.S.; Lecko, A.; Sim, Y.J. Some coe¢ cient
inequalities related to the Hankel determinant for strongly starlike functions of
order alpha. J. Math. Inequal. 2017; 11; 429 439:
Grenander, U.; Sezego. G.Toeplitz Forms and Their Applications. University of
California Press, Berkeley, 1958:
Hayman, W.K. On second Hankel determinant of mean univalent functions.
Proc. Lond. Math. Soc. 1968; 3; 77 94:
Jangteng, A.; Halim, S.A.; Darus, M. Coe¢ cient inequality for a function whose
derivative has a positive real part. J. Ineq. Pure Appl. Math. 2006; 7; 1 5:
Jangteng, A.; Halim, S.A.; Darus, M. Coe¢ cient inequality for starlike and
convex functions. Int. J. Ineq. Math. Anal. 2007; 1; 619 625:
Janowski, W. Extremal problems for a family of functions with positive real
part and for some related families. Ann. Pol. Math. 1971; 23; 159 177:
Keough, F.; Merkes, E. A coe¢ cient inequality for certain subclasses of analytic
functions. Proc. Am. Math. Soc. 1969; 20; 8 12:
Kowalczyk, B.; Lecko, A.; Sim, Y.J.The sharp bound of the hankel determinent
of the third kind for convex functions. Bull. Aust. Math. Soc. 2018; 97; 435445:
Krishna, D. V.; Venkateswarlu, B.; RamReddy, T. Third Hankel determinant
for bounded turning functions of order alpha. J. Niger. Math. Soc. 2015; 34;
127:
Krishna, D.V.; RamReddy, T. Hankel determinant for starlike and convex functions
of order alpha. Tbil. Math. J. 2012; 5; 65 76:
Krishna, D.V.; RamReddy, T. Second Hankel determinant for the class of
Bazilevic functions. Stud. Univ. Babe ‚s-Bolyai Math. 2015; 60; 413 420:
Kumar, S.S., Kumar, V., Ravichandran, V., Cho, N.E.: Su¢ cient conditions
for starlike functions associated with the lemniscate of Bernoulli. J. Inequal.
Appl. 2013; 176(2013):
Lecko, A.; Sim, Y. J.;
Smiarowska, B. The sharp bound of the Hankel determi-
nant of the third kind for starlike functions of order 1/2. Complex anal. Oper.
theory. 2018; 1 8:
Lee, S. K.; Ravichandran, V.; Supramaniam, S. Bounds for the second Hankel
determinant of certain univalent functions. J. Inequal. Appl. 2013; 2013; 281:
Liu, M. S.; Xu, J. F.; Yang, M. Upper bound of second Hankel determinant for
certain subclasses of analytic functions. Abstr. Appl. Anal. 2014; 2014; 603180.
Livingston, A.E. The coe¢ cients of multivalent close-to-convex functions.
Proc.Am. Math. Soc. 1969; 21(3); 545 552(1969):
Ma, W.; Minda, D. A unied treatment of some special classes of univalent
functions.In Proceedings of the Conference on Complex Analysis; Li, Z., Ren, F.,
Yang, L., Zhang, S. Eds.; Int. Press: Cambridge, MA, USA, 1992; pp:157169:
Mendiratta, R.; Nagpal, S.; Ravichandran, V. On a subclass of strongly starlike
functions associated with exponential function. Bull. Malays. Math. Sci. Soc.
; 38; 365 386:
Noonan, J.W.; Thomas, D.K. On the second Hankel determinant of areally
mean p-valent functions. Trans. Am. Math. Soc. 1976; 233; 337 346:
Noor, K. I.; Bukhari, S. Z. H.; Arif, M.; Nazir, M. Some properties of p-valent
analytic functions involving Cho-Kwon-Srivastava integral operator. Journal of
classical analysis. 2013, 3(1); 35 43.
Orhan, H.; Magesh, N.; Yamini, J. Bounds for the second Hankel determinant
of certain bi-univalent functions. Turk. J. Math. 2016; 40; 679 687:
Pommerenke, C. On the coe¢ cients and Hankel determinants of univalent functions.
J. Lond. Math. Soc. 1966; 41; 111 122:
Pommerenke, C. On the Hankel determinants of univalent functions. Mathematika.
; 14108 112:
Pommerenke, C. Univalent Functions. Vandenhoeck and Ruprecht: Gottingen,
Germany, 1975:
Raza, M.; Arif, M.; Darus, M. Fekete-Szego inequality for a subclass of p-valent
analytic functions, Journal of Applied Mathematics, Volume 2013, Article ID
, 7 pages
Raza, M.; Malik, S.N. Upper bound of third Hankel determinant for a class
of analytic functions related with lemniscate of Bernoulli. J. Inequal. Appl.
; 2013; 412:
DOI: http://dx.doi.org/10.21015/vtm.v8i1.273
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