Coefficients estimates of a new class of analytic bi-univalent functions with bounded boundary rotation

Abstract

In this paper, we introduce a new subclass of analytic bi-univalent functions defined by using $q$-derivative operator. Further, we obtain both some initial and general coefficient bounds, and also Fekete-Szegö inequalities for bi-univalent functions that belong to this class. Our results extend and improve some known results as special cases.

Keywords

• univalent functions
• bi-univalent functions
• bounded boundary rotations
• coefficient estimates
• fekete-Szegö problem
• q-calculus

References

1. [1] H. Airault, Symmetric sums associated to the factorization of Grunsky coefficients, in Conference, Groups and Symmetries, Montreal, Canada, April 2007.
2. [2] H. Airault, Remarks on Faber polynomials, Int. Math. Forum 3(9-12), 449–456, 2008.
3. [3] H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bull. Sci. Math. 20(3), 179–222, 2006.
4. [4] R.M. Ali, S.K. Lee, V. Ravichandran and S. Supramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett. 25(3), 344–351, 2012.
5. [5] A. Aljouiee and P. Goswami, Coefficients estimates of the class of bi-univalent functions, J. Function Spaces 2016, Article ID 3454763, 4 pages, 2016.
6. [6] D.A. Brannan and T.S. Taha, On some classes of bi-univalent functions, Studia Univ. Babeş-Bolyai Math. 31(2), 70–77, 1986.
7. [7] P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer, New York, 1983.
8. [8] G. Faber, Über polynomische Entwickelungen, Math. Ann. 57 (3), 389–408, 1903.

9. [9] B.A. Frasin and M.K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24(9), 1569–1573, 2011.
10. [10] S.P. Goyal and P. Goswami, Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives, J. Egyptian Math. Soc. 20(3), 179–182, 2012.
11. [11] S.G. Hamidi, S.A. Halim and J.M. Jahangiri, Coefficient of bi-univalent functions with positive real parts, Bull. Malaysian Math. Sci. Soc. 37(3), 633–640, 2014.
12. [12] F.H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math. 41, 193–203, 1910.
13. [13] F.H. Jackson, q-difference equations, Amer. J. Math. 32, 305–314, 1910.
14. [14] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18, 63–68, 1967.
15. [15] K. Padmanabhan and R. Parvatham, Properties of a class of functions with bounded boundary rotation, Ann. Polon. Math. 31, 311–323, 1975.
16. [16] B. Pinchuk, Functions with bounded boundary rotation, Israel J. Math. 10, 7–16, 1971.
17. [17] M. Schiffer, Faber polynomials in the theory of univalent functions, Bull. Amer. Math. Soc. 54, 503–517, 1948.
18. [18] P.G. Todorov, On the Faber polynomials of the univalent functions of class $\Sigma$, J. Math. Anal. Appl. 162(1), 268-267, 1991.
19. [19] H.M. Srivastava, A.K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23(10), 1188–1192, 2010.
20. [20] P. Zaprawa, Estimates of initial coefficients for bi-univalent functions, Abstr. Appl. Anal. Art. ID 357480, 6 pp. 2014.
21. [21] P. Zaprawa, On the Fekete-Szegö problem for classes of bi-univalent functions, Bull. Belg. Math. Soc. Simon Stevin 21(1), 169–178, 2014.