Faber polynomials coefficients estimates for a certain subclass of Bazilevic functions

For a certain subclass of Bazilevic functions, Faber polynomials expansions are used to obtain bi-univalent properties. Estimates on the $n$th Taylor-Maclaurin coefficients of functions in this class are found. Moreover, some special cases are also indicated.

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  • [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] S. Altınkaya, Bounds for a new subclass of bi-univalent functions subordinate to the Fibonacci numbers, Turkish J. Math. 44 (2), 553–560, 2020.
  • [4] S. Altınkaya and S. Yalcın, Faber polynomial coefficient bounds for a subclass of bi- univalent functions, C. R. Math. 353 (12), 1075–108, 2015.
  • [5] A. Bouali, Faber polynomials, Cayley-Hamilton equation and Newton symmetric func- tions, Bull. Sci. Math. 130, 49–70, 2006.
  • [6] M. Caglar, H. Orhan and N. Yagmur, Coefficient bounds for new subclasses of bi- univalent functions, Filomat, 27 (7), 1165–1171, 2013.
  • [7] E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate condi- tions, J. Class. Anal. 2, 49–60, 2013.
  • [8] E. Deniz, J.M. Jahangiri, S.G. Hamidi and S.K. Kina, Faber polynomial coefficients for generalized bi-subordinate functions of complex order, J. Math. Inequal. 12 (3), 645–653, 2018.
  • [9] P.L. Duren, Univalent Functions, Grundlehren Math. Wissenschaften, Band 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.
  • [10] G. Faber, Über polynomische Entwickelungen, Math. Ann. doi: 10.1007/BF01444293, 1903.
  • [11] P. Goel and S.S. Kumar, Certain class of starlike functions associated with modified sigmoid function, Bull. Malays. Math. Sci. Soc. 43, 957–991, 2020.
  • [12] S.P. Goyal and R. Kumar, Coefficient estimates and quasi-subordination properties associated with certain subclasses of analytic and bi-univalent functions, Math. Slo- vaca, 65 (3), 533–544, 2015.
  • [13] T. Hayami and S. Owa, Coefficient bounds for bi-univalent functions, Pan Amer. Math. J. 22 (4), 15–26, 2012.
  • [14] S. Kanas, V.S. Masih and A. Ebadian, Relations of a planar domain bounded by hyperbola with family of holomorphic functions, J. Inequl. Appl. 246, 1–14, 2019.
  • [15] A.Y. Lashin, On certain subclasses of analytic and bi-univalent functions, J. Egyptian Math. Soc. 24 (2), 220–225, 2016.
  • [16] A.Y. Lashin, Coefficient estimates for two subclasses of analytic and bi-univalent functions, Ukr. Math. J. 70 (9), 1484–1492, 2019.
  • [17] A.Y. Lashin and F.Z. EL-Emam, Faber polynomial coefficients for certain subclasses of analytic and biunivalent functions, Turkish J. Math. 44, 1345–1361, 2020.
  • [18] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18, 63–68, 1967.
  • [19] N. Magesh, T. Rosy and S. Varma, Coefficient estimate problem for a new subclass of bi-univalent functions, J. Complex Anal. 2013, Art. ID 474231, 3pp., 2013.
  • [20] N. Magesh and J. Yamini, Coefficient bounds for certain subclasses of bi-univalent functions, Int. Math. Forum, 8, 1337–1344, 2013.
  • [21] R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38 (1), 365–386, 2015.
  • [22] G. Murugusundaramoorthy, N. Magesh and V. Prameela, Coefficient bounds for cer- tain subclasses of bi-univalent function, Abstr. Appl. Anal. Art. ID 573017, 3 pp., 2013.
  • [23] Z.-G. Peng and Q.-Q. Han, On the coefficients of several classes of bi-univalent func- tions, Acta Math. Sci. Ser. B Engl. Ed. 34, 228–240, 2014.
  • [24] S. Porwal and M. Darus, On a new subclass of bi-univalent functions, J. Egypt. Math. Soc. 21 (3), 190–193, 2013.
  • [25] R.K. Raina and J. Sokół, Some properties related to a certain class of starlike func- tions, C. R. Acad. Sci. Paris Ser. I 353 (11), 973–978, 2015.
  • [26] J. Sokół and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19, 101–105, 1996.
  • [27] H.M. Srivastava, S. Bulut, M. Caglar and N. Yagmur, Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat, 27 (5), 831–842, 2013.
  • [28] A. Zireh, E.A. Adegani and M. Bidkham, Faber polynomial coefficient estimates for subclass of bi-univalent functions defined by quasi-subordinate, Math. Slovaca, 68 (2), 369–378, 2018.
  • [29] Z-G.Wang and Y-P. Jiang, Notes on certain subclass of p-valently Bazilevic functions, J. Inequl. Pure Appl. Math. 9 (3), Art. 70, 7pp., 2008.