INITIAL TAYLOR-MACLAURIN COEFFICIENT BOUNDS AND THE FEKETE-SZEG ÖPROBLEM FOR SUBCLASSES OF m-FOLD SYMMETRIC ANALYTIC BI-UNIVALENT FUNCTIONS

In the present paper, we introduce two new subclasses of the m-fold symmetric, analytic and bi-univalent function class sigma m defined in the open unit disk D1 := {z : z is an element of C and |z| < 1}. These two subclasses are denoted by S-Sigma m (alpha) and S*(Sigma m )(beta). For the functions f belong to both of these subclasses, we obtain estimates on the first two Taylor-Maclaurin coefficients |a(m+1)| and |a(2m+1)|. Also, we obtain estimate on the Fekete-Szego functional |a(2m+1) - ka(m+1)(2)|, k E R. It is interesting to see that the geometrical similarities in these two subclasses also reflects in their coefficient estimates. Further, we pointed out interconnection of these results with some of the earlier known results.