Initial coefficient estimates for a new subclasses of analytic and m-fold symmetric bi-univalent functions

Abbas Kareem Wanas; Sibel Yalçın

doi:10.26637/MJM0703/0018

Abstract

In the present investigation, we define two new subclasses of the function class $\Sigma_m$ of analytic and $m$-fold symmetric bi-univalent functions defined in the open unit disk $U$. Furthermore, for functions in each of the subclasses introduced here, we etermine the estimates on the initial coefficients $\left|a_{m+1}\right|$ and $\left|a_{2 m+1}\right|$. Also, we indicate certain special cases for our results.

Keywords:

Analytic functions, univalent functions, bi-univalent functions, m-fold symmetric bi-univalent functions, coefficient estimates

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Abbas Kareem Wanas Department of Mathematics, College of Science, University of Al-Qadisiyah, Diwaniyah, Iraq. https://orcid.org/0000-0001-5838-7365
Sibel Yalçın Department of Mathematics, Faculty of Arts and Science, Bursa Uludag University, Bursa, Turkey. https://orcid.org/0000-0002-0243-8263

Pages: 472-476
Date Published: 01-07-2019
Vol. 7 No. 03 (2019): Malaya Journal of Matematik (MJM)

Ş. Altınkaya and S. Yalçın, Coefficient bounds for certain subclasses of $mathrm{m}$-fold symmetric bi-univalent functions, Journal of Mathematics, Art. ID 241683 (2015), 1-5.

Ş. Altınkaya and S. Yalçın, On some subclasses of mfold symmetric bi-univalent functions, Commun. Fac. Sci. Univ. Ank. Series Al, 67(1)(2018), 29-36.

D. A. Brannan and T. S. Taha, On Some classes of biunivalent functions, Studia Univ. Babes-Bolyai Math., 31(2)(1986), 70-77.

P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Band 259, Springer Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.

S. S. Eker, Coefficient bounds for subclasses of mfold symmetric bi-univalent functions, Turk. J. Math., 40(2016), 641-646.

B. A. Frasin and M. K. Aouf, New subclasses of biunivalent functions, Appl. Math. Lett., 24(2011), 15691573.

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.

W. Koepf, Coefficients of symmetric functions of bounded boundary rotations, Proc. Amer. Math. Soc., 105(1989), 324-329.

X. F. Li and A. P. Wang, Two new subclasses of biunivalent functions, Int. Math. Forum, 7(2012), 14951504.

H. M. Srivastava and D. Bansal, Coefficient estimates for a subclass of analytic and bi-univalent functions, $J$. Egyptian Math. Soc., 23(2015), 242-246.

H. M. Srivastava, S. Bulut, M. Caglar and N. Yagmur, Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat, 27(5)(2013), 831842.

H. M. Srivastava, S. S. Eker and R. M. Ali, Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29(2015), 1839-1845.

H. M. Srivastava, S. Gaboury and F. Ghanim, Initial coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions, Acta Mathematica Scientia, $36 mathrm{~B}(3)(2016), 863-871$.

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, S. Sivasubramanian and R. Sivakumar, Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions, Tbilisi Math. J., 7(2)(2014), $1-10$.

H. Tang, H. M. Srivastava, S. Sivasubramanian and P. Gurusamy, The Fekete-Szegö functional problems for some subclasses of m-fold symmetric bi-univalent functions, $J$. Math. Inequal., 10(2016), 1063-1092.

A. K. Wanas and A. H. Majeed, Certain new subclasses of analytic and m-fold symmetric bi-univalent functions, Applied Mathematics E-Notes, 18(2018), 178-188.