Coefficient Bounds for a New Class of Bi-Univalent Functions Associated with Subordination

Document Type : Original Article

Author
Nafya Hameed Mohammed
Department of Mathematics, College of Basic Education, University of Raparin, Kurdistan Region-Iraq.
10.22034/maco.2.2.8
Abstract
The main purpose of this article is to introduce and investigate the subcategory $mathcal{H}_{Sigma}(n,beta;phi)$ of bi-univalent functions in the open unit disk $mathbb{U}$ related to subordination. Moreover,  estimates on coefficient $|a_n|$ for functions belong to this subcategory are given applying different technique. In addition,  smaller upper bound and more accurate estimation than the previous outcomes are obtained. 

Keywords

[1] H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bull. Sci. Math., 130:179–222, 2006.
[2] H. Airault and J. Ren, An algebra of differential operators and generating functions on the set of univalent functions, Bull. Sci. Math., 126:343–367, 2002.
[3] R.M. Ali, S.K. Lee, V. Ravichandran and S. Subramaniam, Coefficient estimates for bi-univalent MaMinda starlike and convex functions, Appl. Math. Lett., 25:344-351, 2012.
[4] H.S. Al-Amiri, On Ruscheweh derivatives. Ann. Poln. Math., 38:87–94, 1980.
[5] O. Alrefal and M. Ali, General coefficient estimates for bi-univalent functions: a new approach, Turk. J. Math., 44:240–251, 2020.
[6] S. Altınkaya and S. Yalçın, On the some subclasses of bi-univalent functions related to the Faber polynomial expansions and the Fibonacci numbers, Rend. Mat. Appl., 41:105–116, 2020.
[7] E.A. Adegani, S. Bulut and A. Zireh, Coefficient estimates for a subclass of analytic bi-univalent functions, Bull. Korean Math. Soc., 55:405–413, 2018.
[8] E.A. Adegani, N.E. Cho, D. Alimohammadi and A. Motamednezhad, Coefficient bounds for certain two subclasses of bi-univalent functions, AIMS Math., 6(9):9126—9137, 2021
[9] E.A. Adegani, A. Motamednezhad and S. Bulut, Coefficient estimates for a subclass of meromorphic bi-univalent functions defined by subordination, Stud. Univ. Babeş-Bolyai Math., 65:57–66, 2020.
[10] M.K. Aouf, S.M. Madian and A. O. Mostafa, Bi-univalent properties for certain class of Bazilevič functions defined by convolution and with bounded boundary rotation, J. Egyptian Math. Soc., 27(11):1–9, 2019.
[11] S. Bulut, Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions, Filomat, 30:1567–1575, 2016.
[12] S. Bulut, S. Salehian and A. Motamednezhad, Comprehensive subclass of m-fold symmetric bi-univalent functions defined by subordination, Afr. Mat., 32(3): 531–541, 2021.
[13] P.L. Duren, Univalent Functions. Grundlehren der mathematischen Wissenschaften, Springer-Verlag New York, Berlin, Heidelberg, Tokyo, 259, 1983.
[14] G. Faber, Über polynomische Entwickelungen, Math. Ann., 57:389–408, 1903.
[15] B.A. Frasin, Coefficient bounds for certain classes of bi-univalent functions, Hacet. J. Math. Stat., 43:383–389, 2014.
[16] B.A. Frasin and M.K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24:1569–1573, 2011.
[17] S.G. Hamidi and J.M. Jahangiri, Faber polynomial coefficient estimates for bi-univalent functions defined by subordinations, Bull. Iran. Math. Soc., 41(5):1103–1119, 2015.
[18] S.G. Hamidi and J.M. Jahangiri, Faber polynomial coefficients of bi-subordinate functions, C. R. Math. Acad. Sci. Paris, 354:365–370, 2016.
[19] S. Hussain, S. Khan, M. A. Zaighum, M. Darus and Z. Shareef, Coefficients bounds for certain subclass of biunivalent functions associated with Ruscheweyh-Differential operator, J. Complex Anal., 2017, Article ID 2826514, 2017.
[20] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18:63–68, 1967.
[21] W.C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992); Internat. Press, Cambridge, MA, USA, 157–169,1992.
[22] R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc., 38:365–386, 2015.
[23] S. Salehian and A. Zireh, Coefficient estimates for certain subclass of meromorphic and bi-univalent functions, Commun. Korean Math. Soc., 32:389–397, 2017.
[24] G. Murugusundaramoorthy and T. Bulboacă, Estimate for initial MacLaurin coefficients of certain subclasses of bi-univalent functions of complex order associated with the Hohlov operator, Ann. Univ. Paedagog. Crac. Stud. Math., 17(1):27–36, 2018.
[25] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49:109–115, 1975.
[26] G. Saravanan and K. Muthunagai, Coefficient estimates and Fekete-Szegö inequality for a subclass of Bi-univalent functions defined by symmetric Q-derivative operator by using Faber polynomial techniques, Period. Eng. Nat. Sci., 6(1):241–250, 2018.
[27] H.M. Srivastava, Ş. Altınkaya and S. Yalçin, Certain subclasses of bi-univalent functions associated with the Horadam polynomials, Iran. J. Sci. Technol. Trans. A Sci., 43:1873–1879, 2019.
[28] H.M. Srivastava, A.K. Mishra and P. Gochhayat, Certain subclasses of analytic and biunivalent functions, Appl. Math. Lett., 23:1188–1192, 2010.
[29] H.M. Srivastava, S. Sümer Eker and R.M. Ali, Coeffcient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29:1839–1845, 2015.
[30] H.M. Srivastava, A. Zireh and S. Hajiparvaneh, Coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions, Filomat, 32(9): 3143–3153, 2018.
[31] P.G. Todorov, On the Faber polynomials of the univalent functions of class Σ, J. Math. Anal. Appl., 162:268–276, 1991.
[32] A. Zireh, E. A. Adegani, M. Bidkham, Faber polynomial coefficient estimates for subclass of bi-univalent functions defined by quasi-subordinate, Math. Slovaca, 68:369–378, 2018.
Mathematical Analysis and Convex Optimization
Volume 2, Issue 2
Autumn 2021
Pages 73-82