  This paper reports on constructive approximation methods for three classes of
holomorphic functions on the unit disk which are closely connected each other:
the class of starlike and spirallike functions, the class of semigroup
generators, and the class of functions with positive real part. It is
more-or-less known that starlike or spirallike functions can be defined as
solutions of singular differential equations which, in general, are not stable
under the motion of interior singular points to the boundary. At the same time,
one can establish a perturbation formula which continuously transforms a
starlike (or spirallike) function with respect to a boundary point to a
starlike (or spirallike) function with respect to an interior point. This
formula is based on an appropriate approximation method of holomorphic
generators which determine the above-mentioned differential equations. In turn,
the well-known Berkson--Porta parametric representation of holomorphic
generators leads us to study an approximation-interpolation problem for the
class of functions with positive real part. While this problem is of
independent interest, the solution we present here is again based on the
Berkson--Porta formula. Finally, we apply our results to solve a natural
perturbation problem for one-parameter semigroups of holomorphic self-mappings,
as well as the eigenvalue problem for the semigroup of composition operators.
