×
Mahmood, Shahid; Srivastava, Hari M.; Khan, Nazar; Ahmad, Qazi Zahoor; Khan, Bilal; Ali, Irfan

Upper bound of the third Hankel determinant for a subclass of \(q\)-starlike functions. (English) Zbl 1423.05030

Symmetry 11, No. 3, Paper No. 347, 13 p. (2019).
Summary: The main purpose of this article is to find the upper bound of the third Hankel determinant for a family of \(q\)-starlike functions which are associated with the Ruscheweyh-type \(q\)-derivative operator. The work is motivated by several special cases and consequences of our main results, which are pointed out herein.

Cited in 29 Documents

MSC:

05A30 \(q\)-calculus and related topics
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
11B65 Binomial coefficients; factorials; \(q\)-identities
47B38 Linear operators on function spaces (general)

Keywords:

analytic functions; Hadamard product; starlike functions; \(q\)-derivative operator; \(q\)-difference operator; Hankel determinant; \(q\)-starlike functions
Cite Review PDF
Full Text: DOI
OA License

References:

[1] Miller, S.S.; Mocanu, P.T.; Differential subordination and univalent functions; Mich. Math. J.: 1981; Volume 28 ,157-171. · Zbl 0439.30015
[2] Miller, S.S.; Mocanu, P.T.; Mocanu; Differential Subordination: Theory and Applications: New York, NY, USA 2000; . · Zbl 0954.34003
[3] Jackson, F.H.; On q-definite integrals; Quart. J. Pure Appl. Math.: 1910; Volume 41 ,193-203. · JFM 41.0317.04
[4] Jackson, F.H.; q-difference equations; Am. J. Math.: 1910; Volume 32 ,305-314. · JFM 41.0502.01
[5] Ismail, M.E.H.; Merkes, E.; Styer, D.; A generalization of starlike functions; Complex Var. Theory Appl.: 1990; Volume 14 ,77-84. · Zbl 0708.30014
[6] Srivastava, H.M.; Univalent functions, fractional calculus, and associated generalized hypergeometric functions; Univalent Functions, Fractional Calculus, and Their Applications: New York, NY, USA 1989; ,329-354. · Zbl 0693.30013
[7] Srivastava, H.M.; Ahmad, Q.Z.; Khan, N.; Khan, N.; Khan, B.; Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain; Mathematics: 2019; Volume 7 .
[8] Srivastava, H.M.; Tahir, M.; Khan, B.; Ahmad, Q.Z.; Khan, N.; Some general classes of q-starlike functions associated with the Janowski functions; Symmetry: 2019; Volume 11 . · Zbl 1416.30007
[9] Mahmood, S.; Jabeen, M.; Malik, S.N.; Srivastava, H.M.; Manzoor, R.; Riaz, S.M.J.; Some coefficient inequalities of q-starlike functions associated with conic domain defined by q-derivative; J. Funct. Spaces: 2018; Volume 2018 ,8492072. · Zbl 1408.30015
[10] Srivastava, H.M.; Bansal, D.; Close-to-convexity of a certain family of q-Mittag-Leffler functions; J. Nonlinear Var. Anal.: 2017; Volume 1 ,61-69. · Zbl 1400.30035
[11] Aldweby, H.; Darus, H.; Some subordination results on q-analogue of Ruscheweyh differential operator; Abstr. Appl. Anal.: 2014; Volume 2014 ,958563. · Zbl 1474.30044
[12] Ezeafulukwe, U.A.; Darus, M.; A note on q-calculus; Fasc. Math.: 2015; Volume 55 ,53-63. · Zbl 1338.30009
[13] Ezeafulukwe, U.A.; Darus, M.; Certain properties of q-hypergeometric functions; Int. J. Math. Math. Sci.: 2015; Volume 2015 ,489218. · Zbl 1476.30044
[14] Sahoo, S.K.; Sharma, N.L.; On a generalization of close-to-convex functions; Ann. Pol. Math.: 2015; Volume 113 ,93-108. · Zbl 1308.30020
[15] Uçar, H.E.Ö.; Coefficient inequality for q-starlike functions; Appl. Math. Comput.: 2016; Volume 276 ,122-126. · Zbl 1410.30012
[16] Kanas, S.; Răducanu, D.; Some class of analytic functions related to conic domains; Math. Slovaca: 2014; Volume 64 ,1183-1196. · Zbl 1349.30054
[17] Ruscheweyh, S.; New criteria for univalent functions; Proc. Am. Math. Soc.: 1975; Volume 49 ,109-115. · Zbl 0303.30006
[18] Janteng, J.; Abdulhalim, S.; Darus, M.; Hankel determinant for starlike and convex functions; Int. J. Math. Anal.: 2007; Volume 1 ,619-625. · Zbl 1137.30308
[19] Mishra, A.K.; Gochhayat, P.; Second Hankel determinant for a class of analytic functions defined by fractional derivative; Int. J. Math. Math. Sci.: 2008; Volume 2008 ,1-10. · Zbl 1158.30308
[20] Singh, G.; Singh, G.; On the second Hankel determinant for a new subclass of analytic functions; J. Math. Sci. Appl.: 2014; Volume 2 ,1-3.
[21] Srivastava, H.M.; Altinkaya, Ş.; Yalçın, S.; Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric q-derivative operator; Filomat: 2018; Volume 32 ,503-516. · Zbl 1488.30130
[22] Babalola, K.O.; On H31 Hankel determinant for some classes of univalent functions; Inequal. Theory Appl.: 2007; Volume 6 ,1-7.
[23] Ma, W.C.; Minda, D.A.; unified treatment of some special classes of univalent functions; Proceedings of the Conference on Complex Analysis (Tianjin, 1992): Cambridge, UK 1994; ,157-169. · Zbl 0823.30007
[24] Libera, R.J.; Zlotkiewicz, E.J.; Early coefficient of the inverse of a regular convex function; Proc. Am. Math. Soc.: 1982; Volume 85 ,225-230. · Zbl 0464.30019
[25] Libera, R.J.; Zlotkiewicz, E.J.; Coefficient bounds for the inverse of a function with derivative in 𝒫; Proc. Am. Math. Soc.: 1983; Volume 87 ,251-257. · Zbl 0488.30010
[26] Duren, P.L.; ; Univalent Functions: New York, NY, USA 1983; . · Zbl 0514.30001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.