HANKEL DETERMINANT FOR CERTAIN SUBCLASS OF UNIVALENT FUNCTIONS DEFINED BY q-DIFFERENCE OPERATOR
Keywords:
Analytic, q-difference operator, univalent functions, Hankel deter-minant, inverse Hankel determinant
Abstract
The most powerful tool that cannot be completely eroded in the history of Geometric Functions Theory (GFT) is determinant of any order. There is no gaining-saying that determinants have series of applications in Sciences, Engineering, Data analysis, Computing, and generally in other sectors of man’s endeavor. In particular, the Hankel determinant has attracted attention of numerous researchers possibly because of its distinct geometric structural sequence, and despite gaining so much attention there still exist some perceived gaps in knowledge that are yet to be explored. It is on this positive direction that this present study derived its interest so that a new development in knowledge can be reached. The method used the q- Difference Operator with the second Hankel determinant as well as its inverse functions of order two along with the concept of subordination principle. With this approach in focus,this study examined some new subclasses of analytic functions. The sharp initial coefficient bounds obtained were used to derive some new subclasses of the Second Hankel along with its inverse functions.
References
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[27] Hamzat, J.O., Oladipo, A.T. & Oros G. I. (2022). ”Application of a Multiplier transfor-mation to Libera integral operator associated with generalized distribution.” Symmetry , 14, 1934, 1-14.
[28] Jackson, F.H., On q-functions and certain q-difference operators, Transactions of the Royal Society of Edinburgh, 46 (1908), 253–281.
[29] Jackson, F.H., On definite integrals, The Quarterly Journal of Pure and Applied Mathe-matics, 28 (1910), 193–203.
[30] Khatter, K., Ravichandran, V. and Kumar, S.S., Starlike functions associated with ex-ponential function and lemniscate of Bernoulli, Revista de la Real Academia de Ciencias Exactas, Fsicas y Naturales. Serie A. Matemticas, 113 (2019), 233–253.
[31] Krishna, D.V. and Ramreddy, T., Hankel determinant for starlike and convex functions of order alpha, Thai Journal of Mathematics, 5 (2012), 65–76.
[32] Layman, J.W., The Hankel transform and some of its properties, Journal of Integer Se-quences, 4 (2001), Article 01.1.5.
[33] Moses O.B., James A.A., Oluwayemi O.M:Hankel determinant for certain subclass of mul-tivalent functions defined by Salagean differential operator Journal of Science, Technology, Mathematics and Education (JOSTMED), 16(3) (2020).
[34] Murugusundaramoorthy, G., Olatunji S.O.,Fadipe-Joseph O.A: Feketa-szego problems for analytic functions in the space of logistic sigmoid functions based on quasi-surbodination International Journal of non- linear analysis, 9(1) (2018), 55-68.
[35] Noor, J.W. and Thomas, D.K., On the second Hankel determinant of a real p-valent function, Transactions of the American Mathematical Society, 223 (1976), 337–346.
[36] Noor, K.I., Hankel determinant problem for the class of functions with bounded boundary rotation, Revue Roumaine de Mathmatiques Pures et Appliques, 28(8) (1983), 731–739.
[37] Nehad A, S., Nabeel, B.T., Sang-Ro, L., Seonhui, K. and Jae, D.C.I., On the Hankel determinant and inverse Hankel determinants of order two for some subclasses of analytic functions, Symmetry, 15 (2023), Article 1674.
[38] Olatunji, S.O., Oladapo, A.T. and Pangarali, T., On quasi-subordination for bi-univalent functions involving generalized distribution series associated with renormalized S-sigmoid function, International Journal of Nonlinear Analysis and Applications, 14(1) (2023), 2213–2222.
[39] Obradovi, M. and Tuneski, N., Hankel determinant of second order for some classes of analytic functions, Mathematica Pannonica, 32(1) (2021), 85–94.
[40] Obradovi, M. and Tuneski, N., Hankel determinant of second order for inverse functions of certain classes of univalent functions, Advances in Mathematics: Scientific Journal, 12(4) (2023), 519–528.
[41] Pommerenke, C., On the coefficients and Hankel determinants of univalent functions. J. Lond. Math. Soc. 38 (1963), 1,111122.
[42] Ponnusamy, S. and Rajasekaran, V., New sufficient conditions for the class of univalent functions, Soochow Journal of Mathematics, 21(2) (1995), 193–201.
[43] Ramachandran, C., Dhanalakshmi, K. and Vanitha, L., Hankel determinant for a subclass of analytic functions associated with error functions bounded by conical regions, Interna-tional Journal of Mathematical Analysis, 11(12) (2017), 571–581.
[44] Sakaguchi, K., On a certain univalent mapping, Journal of the Mathematical Society of Japan, 11 (1959), 72–75.
[45] Salagean, G.S. Subclass of univalent functions, Lecture notes in math, 2(1) (2020), 24–36.
[46] Saritha, G.P., New subclasses of bi-univalent functions with respect to q-geometric points defined by Bernoulli polynomials, Journal of Applied Mathematics and Informatics, 40(6) (2022), 1362–1377.
[47] Shi, L.; Arif, M.; Raza, M.; Abbas, M. , Hankel determinant containing logarithmic coeffi-cients for bounded turning functions connected to a three-leaf-shaped domain, Mathemat-ics, 10 (2022), 2924 https://doi.org/10.3390/math10162924
[48] Sim,Y.J, Thomas, D.K. and Zaprasca, P., The second Hankel determinant for starlike and convex functions of order alpha, Complex Variables and Elliptic Equations, 67 (2022), 2423–2443.
[49] Thomas, D.K., Tuneski, N. and Vasudevarao, A., Univalent Functions, in: A Primer, De Gruyter Studies in Mathematics, Volume 69, De Gruyter, Berlin, Germany, 2015 Volume 69.
[2] Atinkaya, S. and Yalan, S., On the (p,q)-Lucas polynomial coefficient bounds of the bi-univalent function class, Boletn de la Sociedad Matemtica Mexicana, 25 (2019), 567–575.
[3] Babalola, K.O., On a peculiar subclass of analytic functions, International Journal of Mathematical Analysis, 3(20) (2013), 137–147.
[4] Cantor, D.G., Power series with integral coefficients Bull Am. Math. Soc., 38 (1963), 69, 362–366.
[5] Catas . A., Oros G.I, Oros G, : Differential surbordinations associated with mutiplier transformation Abstract Application, ID845724 (2018), 1–11.
[6] Denis, L., The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable, Dover Publications, New York, NY, USA, 1957.
[7] Dzoik g, A second solution of the FeketeSzeg problem Bound value problem, 98 (2013), .
[8] Edrei, A., Sur les dterminants rcurrents et les singularits dune fonction donne par son dveloppement de Taylor, Compositio Mathematica, 7 (1940), 20–88.
[9] Ehrenborg, R., The Hankel determinant of exponential polynomials, American Mathemat-ical Monthly, 107 (2000), 557–560.
[10] Fekete-Szeg, .M Eine Bemurking ber ungerade schlichte Funktionen, Journal of the London Mathematical Society, 6 (1935), 85–89.
[11] Hamzat, J.O., & Fagbemiro, O. (2024). ”Brief Study on Some Properties of Symmetric Cardioid-Bazilevic Functions.”Unilag J. Math. Appl., 4(2), 1-16.
[12] Hamzat, J. O. & Adeyemo A. A. (2019). ”New Subclasses of analytic functions with respect to symmetric and conjugate points defined by extended Salagean derivative derivative operator.” Int. J. Math. Anal. Opt.: Theory and Appl., vol.2019, no.2, 631-643.
[13] Hamzat, J. O. & Adeleke O. J. (2016). ”A new subclass of p-valent functions defined by Aouf et al. derivative operator.” Int. J. Pure and Appl. Math., vol.110, no.2, 257-264.
[14] Hamzat, J. O. & Sangoniyi S. O. (2021). ”Fekete Szego inequalities associated with certain generalized class of non-Bazilevic functions.” Int. J. Math. Anal. Opt.: Theory and Appl., vol.2021, no.2, 874-885.
[15] Hamzat, J. O. (2021). ”Some properties of a new subclass of m-fold symmetric bi-Bazilevic functions associated with modified sigmoid function.” Tbilisi Mathematical J., 14(1), 107-118.
[16] Hamzat, J. O. & Makinde D. O. (2018). ”Coefficient bounds for Bazilevic functions involv-ing Logistic Sigmoid function associated with conic Domains.” Int. J. Math. Anal. Opt.: Theory and Appl., vol.2018, no.2, 392-400.
[17] Hamzat, J.O., Oladipo A. T. & Fagbemiro, O. (2018). ”Coefficient bounds for certain new subclass of m-fold symmetric bi-univalent functions associated with conic domains.”FUW Trends Sci. Tech. J., 43, 1-8.
[18] Hamzat, J.O. (2019). ”Coefficient inequalities for bounded turning functions associated with conic domain.” Electronic J. Math. Anal. Appl., 7(2), 73-78.
[19] Hamzat, J.O. & Fagbemiro, O. (2017). ”Fekete Szego inequalityy for lambda-pseudo-Bazilevic functions.”J. Nig. Math. Phy., 43, 1-8.
[20] Hamzat, J. O. (2017). ”Coefficient Bounds for Bazilevic Functions Associated with Modi-fied Sigmoid Function.” Asian Research J. Math., 5(3), 1-10.
[21] Hamzat, J. O. & Olayiwola M. A. (2017). ”Initial Coefficients Estimates for Certain Gener-alized Class of Analytic Functions Involving Sigmoid Function.” Asian Research J. Math., 5(2), 1-11.
[22] Hamzat, J. O. (2019). ”Hankel Determinant for Lambda-pseudo-spiralike functions.” Transaction Nig. Assoc. Math. Phy., 8, 1-4.
[23] Hamzat, J. O. (2023). ”Estimates of second and third Hankel determinants for Bazilevic functions of order gamma.” Unilag J. Math. Appl., vol.3, 102-112.
[24] Hamzat, J. O., Raji M. T. &Oni A. A. (2018). ”Hankel determinant associated with logis-tic sigmoid functions in the space of Lambda-pseudo-starlike functions.” Asian J. Math. Comput. Research, 25(2), 74-84.
[25] Hamzat, J. O. & Oni A. A.(2017). ”Hankel Determinant for Certain Subclasses of Analytic Function.” Asian Research J. Math., 5(2), 1-10.
[26] Hamzat, J.O, & Fagbemiro, O. (2017). ”Second Hankel determinant for certain generalized subclasses of Modified Bazilevic functions.”J. Nig. Math. Phy., 42, 27-36.
[27] Hamzat, J.O., Oladipo, A.T. & Oros G. I. (2022). ”Application of a Multiplier transfor-mation to Libera integral operator associated with generalized distribution.” Symmetry , 14, 1934, 1-14.
[28] Jackson, F.H., On q-functions and certain q-difference operators, Transactions of the Royal Society of Edinburgh, 46 (1908), 253–281.
[29] Jackson, F.H., On definite integrals, The Quarterly Journal of Pure and Applied Mathe-matics, 28 (1910), 193–203.
[30] Khatter, K., Ravichandran, V. and Kumar, S.S., Starlike functions associated with ex-ponential function and lemniscate of Bernoulli, Revista de la Real Academia de Ciencias Exactas, Fsicas y Naturales. Serie A. Matemticas, 113 (2019), 233–253.
[31] Krishna, D.V. and Ramreddy, T., Hankel determinant for starlike and convex functions of order alpha, Thai Journal of Mathematics, 5 (2012), 65–76.
[32] Layman, J.W., The Hankel transform and some of its properties, Journal of Integer Se-quences, 4 (2001), Article 01.1.5.
[33] Moses O.B., James A.A., Oluwayemi O.M:Hankel determinant for certain subclass of mul-tivalent functions defined by Salagean differential operator Journal of Science, Technology, Mathematics and Education (JOSTMED), 16(3) (2020).
[34] Murugusundaramoorthy, G., Olatunji S.O.,Fadipe-Joseph O.A: Feketa-szego problems for analytic functions in the space of logistic sigmoid functions based on quasi-surbodination International Journal of non- linear analysis, 9(1) (2018), 55-68.
[35] Noor, J.W. and Thomas, D.K., On the second Hankel determinant of a real p-valent function, Transactions of the American Mathematical Society, 223 (1976), 337–346.
[36] Noor, K.I., Hankel determinant problem for the class of functions with bounded boundary rotation, Revue Roumaine de Mathmatiques Pures et Appliques, 28(8) (1983), 731–739.
[37] Nehad A, S., Nabeel, B.T., Sang-Ro, L., Seonhui, K. and Jae, D.C.I., On the Hankel determinant and inverse Hankel determinants of order two for some subclasses of analytic functions, Symmetry, 15 (2023), Article 1674.
[38] Olatunji, S.O., Oladapo, A.T. and Pangarali, T., On quasi-subordination for bi-univalent functions involving generalized distribution series associated with renormalized S-sigmoid function, International Journal of Nonlinear Analysis and Applications, 14(1) (2023), 2213–2222.
[39] Obradovi, M. and Tuneski, N., Hankel determinant of second order for some classes of analytic functions, Mathematica Pannonica, 32(1) (2021), 85–94.
[40] Obradovi, M. and Tuneski, N., Hankel determinant of second order for inverse functions of certain classes of univalent functions, Advances in Mathematics: Scientific Journal, 12(4) (2023), 519–528.
[41] Pommerenke, C., On the coefficients and Hankel determinants of univalent functions. J. Lond. Math. Soc. 38 (1963), 1,111122.
[42] Ponnusamy, S. and Rajasekaran, V., New sufficient conditions for the class of univalent functions, Soochow Journal of Mathematics, 21(2) (1995), 193–201.
[43] Ramachandran, C., Dhanalakshmi, K. and Vanitha, L., Hankel determinant for a subclass of analytic functions associated with error functions bounded by conical regions, Interna-tional Journal of Mathematical Analysis, 11(12) (2017), 571–581.
[44] Sakaguchi, K., On a certain univalent mapping, Journal of the Mathematical Society of Japan, 11 (1959), 72–75.
[45] Salagean, G.S. Subclass of univalent functions, Lecture notes in math, 2(1) (2020), 24–36.
[46] Saritha, G.P., New subclasses of bi-univalent functions with respect to q-geometric points defined by Bernoulli polynomials, Journal of Applied Mathematics and Informatics, 40(6) (2022), 1362–1377.
[47] Shi, L.; Arif, M.; Raza, M.; Abbas, M. , Hankel determinant containing logarithmic coeffi-cients for bounded turning functions connected to a three-leaf-shaped domain, Mathemat-ics, 10 (2022), 2924 https://doi.org/10.3390/math10162924
[48] Sim,Y.J, Thomas, D.K. and Zaprasca, P., The second Hankel determinant for starlike and convex functions of order alpha, Complex Variables and Elliptic Equations, 67 (2022), 2423–2443.
[49] Thomas, D.K., Tuneski, N. and Vasudevarao, A., Univalent Functions, in: A Primer, De Gruyter Studies in Mathematics, Volume 69, De Gruyter, Berlin, Germany, 2015 Volume 69.
Published
2026-04-16
How to Cite
OLATUNJI , S. O., & FAGBEMIRO , O. (2026). HANKEL DETERMINANT FOR CERTAIN SUBCLASS OF UNIVALENT FUNCTIONS DEFINED BY q-DIFFERENCE OPERATOR. Unilag Journal of Mathematics and Applications, 6(2), 48 - 59. Retrieved from https://lagjma.unilag.edu.ng/article/view/3037
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Copyright (c) 2026 SUNDAY OLUWAFEMI OLATUNJI , OLALEKAN FAGBEMIRO
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