Harmonic and Superharmonic Majorants on the Disk

We prove that a positive function on the unit disk admits a harmonic majorant if and only if a certain explicit upper envelope of it admits a superharmonic majorant. We provide examples to show that mere superharmonicity of the data does not help with the problem of existence of a harmonic majorant. 1. Definitions and statements Let D stand for the open unit disk in the complex plane, and H(D) for the cone of positive harmonic functions on D. We would like to describe the functions φ : D 7−→ R+ which admit a harmonic majorant, that is, h ∈ H (D) such that h ≥ φ. This question arises in problems about the decrease of bounded holomorphic functions in the unit disk, as well as in the description of free interpolating sequences for the Nevanlinna class (see [HMNT]). In that paper, an answer is given in terms of duality with the measures that act on positive harmonic functions. The aim of this note is to reduce this problem first to the finiteness of a certain best Lipschitz majorant function, and then to the existence of a merely superharmonic (nontrivial) majorant. Let the hyperbolic (or Poincaré) distance ρ on the disk be defined by dρ(z) := (1− |z|)|dz|. This is invariant under biholomorphic maps from the disk to itself. Explicitly, if we first define the pseudohyperbolic (or Gleason) distance by