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

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

S. D. JADHAV, A. B. PATIL, I. A. WANI

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Abstract. In the present paper, we introduce two new subclasses of the m-fold symmetric, analytic and bi-univalent function class ∑_m defined in the open unit disk D₁ := {z : z ∈ C and |z| < 1}. These two subclasses are denoted by S_∑m(α) and  S*_∑m(β). For the functions f belong to both of these subclasses, we obtain estimates on the first two Taylor-Maclaurin coeffcients |a_m+1| and |a_2m+1|. Also, we obtain estimate on the Fekete-Szegö functional |a_2m+1 - ka² _m+1|, k ∈ R. It is interesting to see that the geometrical similarities in these two subclasses also re ects in their coefficient estimates. Further, we pointed out interconnection of these results with some of the earlier known results.

Keywords:Analytic function, univalent function, bi-univalent function, coefficient bound, m-fold symmetric function, Fekete-Szegö functional.

AMS Subject Classification:30C45, 30C50.