Generalized Zalcman conjecture for starlike and typically real functions

Zalcman conjectured that \a(n)(2) - a(2n-1)\ less than or equal to (n - 1)(2), n = 2, 3,... for f(z) = z + a(2)z(2) + a(3)z(3) + ... is an element of S, the class of normalized holomorphic and univalent functions f(z) in the unit disk D. We propose a generalized conjecture as follows. For f(z) = z + a(2)z(2) + a(3)z(3) + ... is an element of S, \a(n)a(m) - a(n+m-1)\less than or equal to(n - 1)(m - 1), n, m = 2, 3,.... We prove that this conjecture holds for starlike functions and univalent functions with real coefficients. We also obtain sharp bounds on \a(n)a(m) - a(n+m-1)\ for typically real functions. (C) 1999 Academic Press.