## Tbilisi Mathematical Journal

### Initial coefficient bounds for a subclass of $m$-fold symmetric bi-univalent functions

#### Abstract

Let $\Sigma$ denote the class of functions $$f(z)=z+\sum_{n=2}^{\infty}a_nz^n$$ belonging to the normalized analytic function class $\mathcal{A}$ in the open unit disk $\mathbb{U}$, which are bi-univalent in $\mathbb{U}$, that is, both the function $f$ and its inverse $f^{-1}$ are univalent in $\mathbb{U}$. The usual method for computation of the coefficients of the inverse function $f^{-1}(z)$ by means of the relation $f^{-1}\big(f(z)\big)=z$ is too difficult to apply in the case of $m$-fold symmetric analytic functions in $\mathbb{U}$. Here, in our present investigation, we aim at overcoming this difficulty by using a general formula to compute the coefficients of $f^{-1}(z)$ in conjunction with the residue calculus. As an application, we introduce two new subclasses of the bi-univalent function class $\Sigma$ in which both $f(z)$ and $f^{-1}(z)$ are $m$-fold symmetric analytic functions with their derivatives in the class $\mathcal{P}$ of analytic functions with positive real part in $\mathbb{U}$. For functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for $|a_{m+1}|$ and $|a_{2m+1}|$.

#### Article information

Source
Tbilisi Math. J., Volume 7, Issue 2 (2014), 1-10.

Dates
Received: 9 June 2014
Accepted: 13 October 2014
First available in Project Euclid: 12 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1528768970

Digital Object Identifier
doi:10.2478/tmj-2014-0011

Mathematical Reviews number (MathSciNet)
MR3313050

Zentralblatt MATH identifier
1304.30026

#### Citation

Srivastava, H. M.; Sivasubramanian, S.; Sivakumar, R. Initial coefficient bounds for a subclass of $m$-fold symmetric bi-univalent functions. Tbilisi Math. J. 7 (2014), no. 2, 1--10. doi:10.2478/tmj-2014-0011. https://projecteuclid.org/euclid.tbilisi/1528768970

#### References

• D. A. Brannan and J. G. Clunie (Editors), Aspects of contemporary complex analysis (Proceedings of the NATO Advanced Study Institute held at the University of Durham, Durham; July 1–20, 1979), Academic Press, New York and London, 1980.
• P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften, Band 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.
• B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24 (2011), 1569–1573.
• A. W. Goodman, Univalent functions, Vols. 1 and 2, Mariner Publishing Company, Tampa, Florida, 1983.
• 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 (2012), 179–182.
• T. Hayami and S. Owa, Coefficient bounds for bi-univalent functions, Pan Amer. Math. J. 22 (4) (2012), 15–26.
• J. G. Krzyż, R. J. Libera and E. J. Złotkiewicz, Coefficients of inverses of regular starlike functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 33 (1979), 103–110.
• M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63–68.
• W. Koepf, Coefficients of symmetric functions of bounded boundary rotations, Proc. Amer. Math. Soc. 105 (1989), 324–329.
• C. Pommerenke, On the coefficients of close-to-convex functions, Michigan Math. J. 9 (1962), 259–269.
• H. M. Srivastava, Some inequalities and other results associated with certain subclasses of univalent and bi-univalent analytic functions, in Nonlinear Analysis$:$ Stability$,$ Approximation$,$ and Inequalities (Panos M. Pardalos, Pando G. Georgiev and Hari M. Srivastava, Editors.), Springer Series on Optimization and Its Applications, Vol. 68, Springer-Verlag, Berlin, Heidelberg and New York, 2012, pp. 607–630.
• H. M. Srivastava, S. Bulut, M. Çağlar and N. Yağmur, Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat 27 (2013), 831–842.
• H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010), 1188–1192.
• H. M. Srivastava, G. Murugusundaramoorthy and N. Magesh, Certain subclasses of bi-univalent functions associated with the Hohlov operator, Global J. Math. Anal. 1 (2) (2013), 67–73.
• Q.-H. Xu, H. M. Srivastava and Z. Li, A certain subclass of analytic and close-to-convex functions, Appl. Math. Lett. 24 (2011), 396–401.
• Q.-H. Xu, Y.-C. Gui and H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett. 25 (2012), 990–994.
• Q.-H. Xu, H.-G. Xiao, and H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput. 218 (2012), 11461–11465.
• Q.-H. Xu, C.-B. Lv and H. M. Srivastava, Coefficient estimates for the inverses of a certain general class of spirallike functions, Appl. Math. Comput. 219 (2013), 7000–7011.