Coefficient estimates for $m$-fold symmetric bi-subordinate functions

Year 2019, , 365 - 371, 01.04.2019

Ebrahim A. Adegani , Samaneh G. Hamidi , Jay M. Jahangiri , Ahmad Zireh

https://doi.org/10.15672/hujms.545237

Abstract

A function is said to be bi-univalent in the open unit disk $\mathbb{U}$ if both the function and its inverse map are univalent in $\mathbb{U}$. By the same token, a function is said to be bi-subordinate in $\mathbb{U}$ if both the function and its inverse map are subordinate to a given function in $\mathbb{U}$. In this paper, we consider the m-fold symmtric transform of such functions and use their Faber polynomial expansions to find upper bounds for their n-th ($n\geq 3$) coefficients subject to a given gap series condition. We also determine bounds for the first two coefficients of such functions with no restrictions imposed.

Keywords

Faber polynomials, $m$-fold symmetric, bi-univalent functions

References

• H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bull. Sci. Math. 130, 179-222, 2006.
• 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.
• P.L. Duren, Univalent Functions, in: Grundlehren der Mathematischen Wissenschaften, Band 259, Springer-Verlag, New York, NY, 1983.
• B.A. Frasin and M.K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24, 1569-1573, 2011.
• S.G. Hamidi and J.M. Jahangiri Unpredictability of the coefficients of m-fold symmetric bi-starlike functions, Internat. J. Math. 25, 8 pp, 2014.
• S.G. Hamidi and J.M. Jahangiri, Faber polynomial coefficients of bi-subordinate functions, C. R. Acad. Sci. Paris. 354, 365-370, 2016.
• J.M. Jahangiri and S.G. Hamidi, Coefficient estimates for certain classes of biunivalent functions, Int. J. Math. Math. Sci. Article ID: 190560, 1-4, 2013.
• W.C. Ma and D.A. Minda, Unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis, Int Press, 157- 169, 1994.
• H.M. Srivastava, A.K. Mishra and P. Gochhayat, Certain subclasses of analytic and biunivalent functions, Appl. Math. Lett. 23, 1188-1192, 2010.
• H.M. Srivastava, S. Sivasubramanian and R. Sivakumar, Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions, Tbilisi. Math. J. 7, 1-10, 2014.
• S. Sümer Eker, Coefficient bounds for subclasses of m-fold symmetric bi-univalent functions, Turk. J. Math. 1, 1-6, 2015.
• P.G. Todorov, On the Faber polynomials of the univalent functions of class Σ, J. Math. Anal. Appl. 162, 268-276, 1991.