Infinitesimal generators of semigroups with prescribed boundary fixed points

We study infinitesimal generators of one-parameter semigroups in the unit disk $\mathbb{D}$ having prescribed boundary regular fixed points. Using an explicit representation of such infinitesimal generators in combination with Krein-Milman Theory we obtain new sharp inequalities relating spectral values at the fixed points with other important quantities having dynamical meaning.vWe also give a new proof of the classical Cowen-Pommerenke inequalities for univalent self-maps of $\mathbb{D}$.