INVARIANCE OF THE COEFFICIENTS OF STRONGLY CONVEX FUNCTIONS

Let the function f be analytic in D = {z : vertical bar z vertical bar < 1} and given by f (z) = z + Sigma(infinity)(n=2) a(n)Z(n). For 0 < beta <= 1, denote by C(beta) the class of strongly convex functions. We give sharp bounds for the initial coefficients of the inverse function of f is an element of C(beta), showing that these estimates are the same as those for functions in C(beta), thus extending a classical result for convex functions. We also give invariance results for the second Hankel determinant H-2 = vertical bar a(2)a(4) - a(3)(2)vertical bar, the first three coefficients of log(f(z)/z) and Fekete-Szego theorems.