Document Type : Research Paper
Author
- Arzu Akgul ^{}
Department of Mathematics, Faculty of Arts and Science, Kocaeli University, Kocaeli, Turkey.
Abstract
In this work, the subclass of the function class S of analytic and bi-univalent functions is defined and studied in the open unit disc. Estimates for initial coefficients of Taylor- Maclaurin series of bi-univalent functions belonging these class are obtained. By choosing the special values for parameters and functions it is shown that the class reduces to several earlier known classes of analytic and biunivalent functions studied in the literature. Coclusions are given for all special parameters and the functions. And finally, some relevant classes which are well known before are recognized and connections to previus results are made.
Keywords
- Analytic functions
- Biunivalent functions
- Coefficient bounds and coefficient estimates
- Taylor-Maclaurin coefficients
Main Subjects
[2] A. Akgul, Finding initial coefficients for a class of bi-univalent functions given by q-derivative, AIP Conference Proceedings, Vol. 1926. No. 1. AIP Publishing, 2018.
[3] A. Akgul, New subclasses of analytic and bi-univalent functions involving a new integral operator defined by polylogarithm function, Theory Appl. Math. Comput. Sci., 7 (2017), pp. 31-40.
[4] A. Akgul, Coefficient estimates for certain subclass of bi-univalent functions obtained with polylogarithms, Mathematical Sciences And Applications E-Notes, 6 (2018), pp. 70-76.
[5] S. Altinkaya, S. Yalcin, Coefficient bounds for a general subclass of bi-univalent functions, Le Matematiche, LXXI, Fasc., I (2016), pp. 89-97
[6] D.A. Brannan, T.S. Taha, On some classes of bi-univalent functions, in: S.M. Mazhar, A. Hamoui, N.S. Faour (Eds.), Math. Anal. and Appl., Kuwait; February 18--21, 1985, in: KFAS Proceedings Series, vol.3, Pergamon Press, Elsevier Science Limited, Oxford, 1988, pp. 53-60. see also Studia Univ. Babes-Bolyai Math., 31 (1986), pp. 70-77.
[7] D. A. Brannan and J. G. Clunie, Aspects of comtemporary complex analysis, (Proceedings of the NATO Advanced Study Instute Held at University of Durham:July 1-20, 1979). New York: Academic Press, (1980).
[8] S. Bulut, Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions, Filomat, 30 (2016), pp. 1567-1575.
[9] M. Caglar, Halit Orhan, Nihat Yagmur, Coefficient bounds for new subclasses of bi-univalent functions, Filomat, 27 (2013), pp. 1165-1171.
[10] E. Deniz, M. and H. Orhan, The Fekete-Szego problem for a class of analytic functions defined by Dziok-Srivastava operator, Kodai Math. J., 35 (2012), pp. 439-462.
[11] B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24 (2011), pp. 1569-1573.
[12] M. Lewin, On a coefficient problem for bi-univalent functions, Proceeding of the American Mathematical Society, 18 (1967), pp. 63-68.
[13] E. Netanyahu, The minimal distance of the image boundary from the orijin and the second coefficient of a univalent function in |z|<1, Arch. Ration. Mech. Anal., 32 (1969), pp. 100-112.
[14] Ch. Pommerenke, Univalent functions, Vandenhoeck and Rupercht, Gottingen, 1975.
[15] H. M. Srivastava, A. K. Mishra, and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Applied Mathematics Letters, 23 (2010), pp. 1188-1192.
[16] H. M. Srivastava, S. Sumer Eker and R. M. Ali, Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29 (2015), pp. 1839-1845.
[17] Q.-H. Xu, Y.-C. Gui and H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and biunivalent functions, Appl. Math. Lett., 25 (2012), pp. 990-994.
[18] Q.-H. Xu, H.-G. Xiao and H. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput., 218 (2012), pp. 461- 465.