From Diff(S1) to Univalent Functions. Cases of Degeneracy

We explain in detail how to obtain the Kirillov vector fields (L-k)(k is an element of Z) on the space of univalent functions inside the unit disk. Following Kirillov, they can be produced from perturbations by vectors (e(ik theta))(k is an element of Z) of diffeomorphisms of the circle. We give a second approach to the construction of the vector fields. In our approach, the Lagrange series for the inverse function plays an important part. We relate the polynomial coefficients in these series to the polynomial coefficients in Kirillov vector fields. By investigation of degenerate cases, we look for the functions f(z) = z + Sigma(n >= 1) a(n)z(n+1) such that L-k f = L(-k)f for k >= 1. We find that f (z) must satisfy the differential equation: [z(2)w/1-zw+zw/w-z] f'(Z) - f(z) - w(2)f'(w)(2)/f(w)(2) x f(z)(2)/f(w) - f(z) = 0 (*) We prove that the only solutions of (*) are Koebe functions. On the other hand, we show that the vector fields (T-k)(k is an element of Z) image of the (L-k)(k is an element of Z) through the map g(z) = 1/f(i/z) can be obtained directly as the (L-k) from perturbations of diffeomorphisms of the circle.