For f ∈ S, the class of normalized functions, analytic and univalent in the unit diskD and given by f (z) = z + ∑∞ n=2 anzn for z ∈ D, we give an upper bound for the coefficient difference |a4| − |a3|when f ∈ S. This provides an improved bound in the case n = 3 of Grinspan’s 1976 general bound ||an+1| − |an|| ≤ 3.61 . . . . Other coefficients bounds, and bounds for the second and third Hankel determinants when f ∈ S are found when either a2 = 0, or a3 = 0