BOUNDS FOR INITIAL MACLAURIN COEFFICIENTS FOR A NEW SUBCLASSES OF ANALYTIC AND M-FOLD SYMMETRIC BI-UNIVALENT FUNCTIONS

Abstract

In the present paper, we introduce and study two new subclasses of the function class Σm consisting of analytic and m-fold symmetric bi-univalent functions in the open unit disk U. We establish upper bounds for the initial coefficients |am+1| and |a2m+1| for functions in these subclasses. Certain special cases are also indicated.

Keywords

• Analytic functions,univalent functions,bi-univalent functions,m-fold symmetric bi-univalent functions,coefficient bounds

References

1. Altinkaya, S. and Yal¸cin, S., (2015), Coefficient bounds for certain subclasses of m-fold symmetric bi-univalent functions, Journal of Mathematics, Art. ID 241683, pp. 1-5.
2. Altinkaya, S. and Yal¸cin, S., (2018), On some subclasses of m-fold symmetric bi-univalent functions
3. Commun. Fac. Sci. Univ. Ank. Series A1, 67 (1), pp. 29-36. Caglar, M., Deniz, E. and Srivastava, H. M., (2017) Second Hankel determinant for certain subclasses of bi-univalent functions, Turkish J. Math., 41, pp. 694-706.
4. Duren, P. L., (1983), Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Band , Springer Verlag, New York, Berlin, Heidelberg and Tokyo.
5. Eker, S. S., (2016), Coefficient bounds for subclasses of m-fold symmetric bi-univalent functions, Turk. J. Math., 40, pp. 641-646.
6. Frasin, B. A. and Aouf, M. K., (2011), New subclasses of bi-univalent functions, Appl. Math. Lett., , pp. 1569–1573.
7. Goyal, S. P. and Goswami, P., (2012), Estimate for initial Maclaurin coefficients of bi-univalent func- tions for a class defined by fractional derivatives, J. Egyptian Math. Soc., 20, pp. 179-182.
8. Koepf, W., (1989), Coefficients of symmetric functions of bounded boundary rotations, Proc. Amer. Math. Soc., 105, pp. 324-329.

9. Li, X. F. and Wang, A. P., (2012), Tow new subclasses of bi-univalent functions, Int. Math. Forum, , pp. 1495-1504.
10. Murugusundaramoorthy, G., Selvaraj, C. and Babu, O. S., (2015), Coefficient estimates for pascu-type subclasses of bi-univalent functions based on subordination, International Journal of Nonliner Science, (1), pp. 47-52.
11. Srivastava, H. M. and Bansal, D., (2015), Coefficient estimates for a subclass of analytic and bi- univalent functions, J. Egyptian Math. Soc., 23, pp. 242–246.
12. Srivastava, H. M., Bulut, S., Caglar, M. and Yagmur, N., (2013), Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat, 27 (5), pp. 831–842.
13. Srivastava, H. M., Eker, S. S. and Ali, R. M., (2015), Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29, pp. 1839–1845.
14. Srivastava, H. M., Eker, S. S., Hamidi, S. G., Jahangiri, J. M., (2018), Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator, Bull.
15. Iranian Math. Soc., 44 (1), pp. 149-157. Srivastava, H. M., Gaboury, S. and Ghanim, F., (2016), Initial coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions, Acta Math. Sci. Ser. B Engl. Ed., 36, pp. 863-871.
16. Srivastava, H. M., Gaboury, S. and Ghanim, F., (2017), Coefficient estimates for some general sub- classes of analytic and bi-univalent functions, Africa Math., 28, pp. 693-706.
17. Srivastava, H. M., Mishra, A. K. and Gochhayat, P., (2010), Certain subclasses of analytic and bi- univalent functions, Appl. Math. Lett., 23, pp. 1188–1192.
18. Srivastava, H. M., Sivasubramanian, S. and Sivakumar, R., (2014), Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions, Tbilisi Math. J., 7 (2), pp. 1-10.
19. Tang, H., Srivastava, H. M., Sivasubramanian, S. and Gurusamy, P., (2016), The Fekete-Szeg¨o func- tional problems for some subclasses of m-fold symmetric bi-univalent functions, J. Math. Inequal., 10, pp. 1063-1092.
20. Wanas, A. K. and Majeed, A. H., (2018), Certain new subclasses of analytic and m-fold symmetric bi-univalent functions, Applied Mathematics E-Notes, 18, pp. 178-188.