Generalized Grunsky coefficients and inequalities (Q1093766)

Let \(\alpha =\{\alpha_ n\}^{\infty}_{n=1}\) be a sequence of complex numbers and \[ \alpha (w)^{\ell}=(\sum^{\infty}_{n=1}\alpha_ nw^ n)^{\ell}=\sum^{\infty}_{k=1}A_{k,\ell}(\alpha)w^ k, \] \(A_{k,\ell}(\alpha)\) are the Bell polynomials. In this article the author investigates the connection between the Bell polynomials and some invariants introduced by the reviewer. These invariants are defined by the generating function \[ f'(z)/(f(z+w)- f(w))=(1/w)+\sum^{\infty}_{n=0}\phi_ n(w)z^ n, \] where \(\{\phi_ j\}^{\infty}_{j=1}\) are known to be invariant under Möbius transformation and \(\phi_ 1\) is the Schwarzian derivative multiplied by a constant. Further the author uses these investigations to improve some of the reviewer's results concerning the univalence of a meromorphic function as well as conditions for quasiconformal extension.