Coefficient estimates for general subclasses of mm-fold symmetric analytic bi-univalent functions

In this work, we introduce and investigate two new subclasses of the bi-univalent functions in which both f and f −1 are m-fold symmetric analytic functions. For functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for |am+1| and |a2m+1| .

PDF

___

[1] Aouf MK, El-Ashwah RM, Abd-Eltawab AM. New subclasses of biunivalent functions involving Dziok-Srivastava operator. ISRN Math Anal 2013; 2013: 387178.
[2] Brannan DA, Clunie JG, editors. Aspects of Contemporary Complex Analysis (Proceedings of the NATO Advanced Study Institute held at the University of Durham). Academic Press: New York, NY, USA, 1980.
[3] Brannan DA, Taha TS. On some classes of bi-univalent functions. Studia Univ Babe¸s-Bolyai Math 1986; 31: 70-77.
[4] Bulut S. Coefficient estimates for initial Taylor-Maclaurin coefficients for a subclass of analytic and bi-univalent functions defined by Al-Oboudi differential operator. Scientific World Journal 2013; 2013: 171039.
[5] Bulut S. Coefficient estimates for new subclasses of analytic and bi-univalent functions defined by Al-Oboudi differential operator. J Funct Spaces Appl 2013; 2013: 181932.
[6] Bulut S. Coefficient estimates for a class of analytic and bi-univalent functions. Novi Sad J Math 2013; 43: 59-65.
[7] Bulut S. Coefficient estimates for a new subclass of analytic and bi-univalent functions. An S¸tiint¸ Univ Al I Cuza Ia¸si Mat (N.S.) (in press).
[8] C¸ a˘glar M, Orhan H, Ya˘gmur N. Coefficient bounds for new subclasses of bi-univalent functions. Filomat 2013; 27: 1165-1171.
[9] Deniz E. Certain subclasses of bi-univalent functions satisfying subordinate conditions. J Class Anal 2013; 2: 49-60.
[10] Deniz E, C¸ a˘glar M, Orhan H. Second Hankel determinant for bi-starlike and bi-convex functions of order β . Appl Math Comput 2015; 271: 301-307.
[11] Duren PL. Univalent Functions. Grundlehren der Mathematischen Wissenschaften, Vol. 259. Springer: New York, NY, USA, 1983.
[12] Frasin BA, Aouf MK. New subclasses of bi-univalent functions. Appl Math Lett 2011; 24: 1569-1573.
[13] Goyal SP, Goswami P. Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives. J Egyptian Math Soc 2012; 20: 179-182.
[14] Hamidi SG, Jahangiri JM. Unpredictability of the coefficients of m-fold symmetric bi-starlike functions. Internat J Math 2014; 25: 1450064 .
[15] Hayami T, Owa S. Coefficient bounds for bi-univalent functions. Pan Amer Math J 2012; 22: 15-26.
[16] Lewin M. On a coefficient problem for bi-univalent functions. P Am Math Soc 1967; 18: 63-68.
[17] Murugusundaramoorthy G, Magesh N, Prameela V. Coefficient bounds for certain subclasses of bi-univalent function. Abstr Appl Anal 2013; 2013: 573017.
[18] Netanyahu E. The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| _____ 1. Arch Rational Mech Anal 1969; 32: 100-112.
[19] Porwal S, Darus M. On a new subclass of bi-univalent functions. J Egyptian Math Soc 2013; 21: 190-193.
[20] Srivastava HM, Bulut S, C¸ a˘glar M, Ya˘gmur N. Coefficient estimates for a general subclass of analytic and biunivalent functions. Filomat 2013; 27: 831-842.
[21] Srivastava HM, Mishra AK, Gochhayat P. Certain subclasses of analytic and bi-univalent functions. Appl Math Lett 2010; 23: 1188-1192.
[22] Srivastava HM, Sivasubramanian S, Sivakumar R. Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions. Tbilisi Math J 2014; 7: 1-10.
[23] S¨umer Eker S. Coefficient bounds for subclasses of m-fold symmetric bi-univalent functions. Turkish J Math 2016; 40: 641-646.
[24] Taha TS. Topics in univalent function theory. PhD, University of London, London, UK, 1981.
[25] Xu QH, Gui YC, Srivastava HM. Coefficient estimates for a certain subclass of analytic and bi-univalent functions. Appl Math Lett 2012; 25: 990-994.
[26] Xu QH, Xiao HG, Srivastava HM. A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems. Appl Math Comput 2012; 218: 11461-11465.