Canonical Functions of Admissible Measures in the Half-Plane

For a $\gamma$-admissible measure $\lambda$ in the upper half-plane, we introduce the notion of canonical function, generalizing the canonical Nevanlinna product for analytic functions of finite order in the half-plane. It is shown that, for any growth function $\gamma$ defined by the Boutroux proximate order, the given definition and the canonical Nevanlinna product coincide.