Faber Polynomial Expansion for a New Subclass of Bi-univalent Functions Endowed with $(p,q)$ Calculus Operators

Abstract

In this paper, we use the Faber polynomial expansion techniques to get the general Taylor-Maclaurin coefficient estimates for $|a_n|,\ (n\geq 4)$ of a generalized class of bi-univalent functions by means of $(p,q)-$calculus, which was introduced by Chakrabarti and Jagannathan. For functions in such a class, we get the initial coefficient estimates for $|a_2|$ and $|a_3|.$ In particular, the results in this paper generalize or improve (in certain cases) the corresponding results obtained by recent researchers.

Keywords

• Faber polynomial expansion
• Bi-univalent functions
• (p,q)-calculus

References

1. [1] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer, New York, 1983.
2. [2] G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge University Press, 2004.
3. [3] V. Kac, P. Cheung, Quantum Calculus, Springer-Verlag, New York, 2002.
4. [4] R. Chakrabarti, R. Jagannathan, A (p;q)-oscillator realization of two parameter quantum algebras, J. Phys. A, 24 (1991), 711-718.
5. [5] F. H. Jackson, On q-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46 (1909), 253-281.
6. [6] F. H. Jackson, q-difference equations, Amer. J. Math., 32(4) (1910), 305-314.
7. [7] G. Faber, Uber polynomische Entwickelungen, Math. Annalen, 57 (1903), 389-408.
8. [8] H. Airault, A. Bouali, Differential calculus on the Faber polynomials, Bull. Sci. Math., 130 (2006), 179-222.

9. [9] S. Bulut, Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions, Filomat, 30(6) (2016), 1567-1575.
10. [10] B. A. Frasin, M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24 (2011), 1569-1573.
11. [11] H. M. Srivastava, A. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23 (2010), 1188-1192.
12. [12] H. M. Srivastava, S. S. Eker, R. M. Ali, Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29 (2015), 1839-1845.
13. [13] Ş. Altınkaya, S. Yalçın, Faber polynomial coefficient estimates for certain classes of bi-univalent functions defined by using the Jackson (p;q)-derivative operator, J. Nonlinear Sci. Appl., 10 (2017), 3067-3074.
14. [14] H. Airault, Remarks on Faber polynomials, Int. Math. Forum, 3(9) (2008), 449-456.
15. [15] S. G. Hamidi, J. M. Jahangiri, Faber polynomial coefficients of bi-subordinate functions, C. R. Acad. Sci. Paris, Ser I, 354 (2016), 365-370.
16. [16] J. M. Jahangiri, S. G. Hamidi, Coefficient estimates for certain classes of bi-univalent functions, Int. J. Math. Math. Sci., 2013 (2013), 1-4. Article ID 190560.