International Journal of Nonlinear Analysis and Applications
On a certain subclass of analytic functions involving modified q-Opoola derivative operator
Document Type : Research Paper
Authors
Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, 43600, Selangor, Malaysia
Abstract
This paper introduces a new subclass in the open unit disc of analytic functions. It is mainly defined by the modified q-Opoola derivative operator. A coefficient inequality is obtained, and other properties like distortion and closure theorems are derived. Moreover, extreme points of the differential operator are also given. Additionally, Hadamard products (or convolution) of functions respective to the class are also included.
Keywords
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[2] F.M. Al-Oboudi, On univalent functions defined by a generalized S˘al˘agean operator, Int. J. Math. Math. Sci. 2004 (2004), no. 27, 1429–1436.
[3] A. Alsoboh and M. Darus, On Fekete-Szeg¨o problem associated with q-derivative operator, J. Phys.: Conf. Ser. 1212 (2019), 012003.
[4] A. Alsoboh and M. Darus, On q-starlike functions with respect to k-symmetric points, Acta Univer. Apulensis 60 (2019), 61–73.
[5] T.O. Opoola, On a subclass of univalent functions defined by a generalized differential operator, Int. J. Math. Anal. 11 (2017), no. 8, 869–876.
[6] S.H. Hadi, M. Darus and A. Alb Lupa¸s, A class of Janowski-type (p, q)-convex harmonic functions involving a generalized q-Mittag-Leffler function, Axioms 12 (2023), no. 2, 190.
[7] H.M. Srivastava, S.H. Hadi and M. Darus, Some subclasses of p-valent γ-uniformly type q-starlike and q-convex functions defined by using a certain generalized q-Bernardi integral operator, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117 (2023), 50.
[8] L.M. Fatunsin and T.O. Opoola, New results on subclasses of analytic functions defined by Opoola differential operator, J. Math. Syst. Sci. 7 (2017), 289–295.
[9] B.A. Frasin, Family of analytic functions of complex order, Acta Math. Acad. Paedagogicae Ny 22 (2006), no. 2, 179–191.
[10] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge university press, Cambridge, 2004.
[11] M.E. Ismail, E. Merkes, and S. David, A generalisation of starlike functions, Transyl. Complex Variables, Theory Appl. 61 (1990), 77–84.
[12] F.H. Jackson, On q-functions and a certain difference operator, Trans. Royal Soc. Edinb. 46 (1990), no. 2, 253–281.
[13] F.H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193–203.
[14] G. S˜al˜agean, Subclasses of univalent functions, Lecture Notes in Math, Springer-Verlag, Heidelberg (1983), 362– 372.
[15] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), no. 1, 109–116.
[2] F.M. Al-Oboudi, On univalent functions defined by a generalized S˘al˘agean operator, Int. J. Math. Math. Sci. 2004 (2004), no. 27, 1429–1436.
[3] A. Alsoboh and M. Darus, On Fekete-Szeg¨o problem associated with q-derivative operator, J. Phys.: Conf. Ser. 1212 (2019), 012003.
[4] A. Alsoboh and M. Darus, On q-starlike functions with respect to k-symmetric points, Acta Univer. Apulensis 60 (2019), 61–73.
[5] T.O. Opoola, On a subclass of univalent functions defined by a generalized differential operator, Int. J. Math. Anal. 11 (2017), no. 8, 869–876.
[6] S.H. Hadi, M. Darus and A. Alb Lupa¸s, A class of Janowski-type (p, q)-convex harmonic functions involving a generalized q-Mittag-Leffler function, Axioms 12 (2023), no. 2, 190.
[7] H.M. Srivastava, S.H. Hadi and M. Darus, Some subclasses of p-valent γ-uniformly type q-starlike and q-convex functions defined by using a certain generalized q-Bernardi integral operator, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117 (2023), 50.
[8] L.M. Fatunsin and T.O. Opoola, New results on subclasses of analytic functions defined by Opoola differential operator, J. Math. Syst. Sci. 7 (2017), 289–295.
[9] B.A. Frasin, Family of analytic functions of complex order, Acta Math. Acad. Paedagogicae Ny 22 (2006), no. 2, 179–191.
[10] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge university press, Cambridge, 2004.
[11] M.E. Ismail, E. Merkes, and S. David, A generalisation of starlike functions, Transyl. Complex Variables, Theory Appl. 61 (1990), 77–84.
[12] F.H. Jackson, On q-functions and a certain difference operator, Trans. Royal Soc. Edinb. 46 (1990), no. 2, 253–281.
[13] F.H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193–203.
[14] G. S˜al˜agean, Subclasses of univalent functions, Lecture Notes in Math, Springer-Verlag, Heidelberg (1983), 362– 372.
[15] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), no. 1, 109–116.
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Volume 14, Issue 5
May 2023Pages 9-16
- Receive Date: 28 November 2022
- Revise Date: 16 February 2023
- Accept Date: 24 February 2023