Sharp estimates of generalized zalcman functional of early coeffcients for Ma-Minda type functions

Abstract

Let ? be an analytic function in the unit disk D := {z ? C : |z| < 1} which has the form ?(z) = 1 + p1z + p2z2 + p3z3 + ... with p1 > 0, p2, p3 ? R. For given such ?, let S*(?), K(?) and R(?) denote the classes of standardly normalized analytic functions f in D which satisfy zf'(z)/f(z) < ?(z), 1 + zf''(z)/f'(z) < ?(z) f'(z)< ?(z), z ? D, respectively, where < means the usual subordination. In this paper, we find the sharp bounds of |a2a3-a4|, where an := f(n)(0)/n!; n ? N, over classes S*(?), K(?) and R(?).