Given a compact convex planar domain Omega with non-empty interior, the classical Neumann's configuration constant c(R)(Omega) is the norm of the Neumann-Poincar & eacute; operator K Omega acting on the space of continuous real-valued functions on the boundary partial derivative Omega, modulo constants. We investigate the related operator norm cC(Omega) of K Omega on the corresponding space of complex-valued functions, and the norm a(Omega) on the subspace of analytic functions. This change requires introduction of techniques much different from the ones used in the classical setting. We prove the equality c(R)(Omega)=cC(Omega), the analytic Neumann-type inequality a(Omega)<1, and provide various estimates for these quantities expressed in terms of the geometry of Omega. We apply our results to estimates for the holomorphic functional calculus of operators on Hilbert space of the type parallel to p(T)parallel to <= Ksup(z is an element of Omega)|p(z)|, where p is a polynomial and Omega is a domain containing the numerical range of the operator T. Among other results, we show that the well-known Crouzeix-Palencia bound K <= 1+root 2- can be improved to K <= 1+root 1+a(Omega). In the case that Omega is an ellipse, this leads to an estimate of K in terms of the eccentricity of the ellipse.