SUCCESSIVE COEFFICIENTS AND TOEPLITZ DETERMINANT FOR CONCAVE UNIVALENT FUNCTIONS

Let Co(p) be the class of all functions f defined in the unit disc D having a simple pole at z = p where 0 < p < 1 and analytic in D \ { p } with f(0) = 0 = f'(0) - 1 such that f maps D onto a domain whose complement with respect to the extended complex plane is a bounded convex set. These functions are called concave univalent functions. Each f is an element of Co(p) has the following Taylor expansion: f(z ) = z + Sigma(infinity)(n = 2)a(n)z(n) , | z | < p . In this article, we first determine the regions of variability of the difference of successive coefficients ( a(n+1) - a(n) ) for n >= 3. We also find sharp upper bounds of the Toeplitz determinants, the entries of which are the Taylor coefficients of functions in Co(p) .