COEFFICIENT PROBLEMS ON THE CLASS U(lambda)

For 0 < lambda <= 1, let U(lambda) denote the family of functions f (z) = z + Sigma(infinity)(n=2) alpha(n)z(n) analytic in the unit disk D satisfying the condition vertical bar(z/f(z))(2) f'(z) - 1 vertical bar <lambda in D. Although functions in this family are known to be univalent in D, the coe ffi cient conjecture about alpha(n) for n >= 5 remains an open problem. In this article, we shall fi rst present a non-sharp bound for vertical bar alpha(n)vertical bar. Some members of the family U(lambda) are given by z/f(z) = 1 - (1 + lambda)phi(z) + lambda(phi(z))(2) with phi(z) = e(i theta) z, that solve many extremal problems in U(lambda). Secondly, we shall consider the following question: Do there exist functions phi analytic in D with vertical bar phi(z)vertical bar < 1 that are not of the form phi(z) = e(i theta) z for which the corresponding functions f of the above form are members of the family U(lambda)? Finally, we shall solve the second coe ffi cient (alpha(2)) problem in an explicit form for f is an element of U(lambda) of the form f (z) = z/1 - alpha(2)z + lambda z integral(z)(0) omega(t) dt, where omega is analytic in D such that vertical bar omega(z)vertical bar <= 1 and omega(0) = alpha, where alpha is an element of (D) over bar.