Inequalities for sums of Green potentials and Blaschke products

We study inequalities for the infima of Green potentials on a compact subset of an arbitrary domain in the complex plane. The results are based on a new representation of the pseudohyperbolic farthest-point distance function via a Green potential. We also give applications to sharp inequalities for the supremum norms of Blaschke products. 1. Green potentials Let G ⊂ C be a domain possessing the Green function gG(z, ζ) with pole at ζ ∈ G. For the positive Borel measures νk, k = 1, . . . ,m, with compact supports in G, define their Green potentials [1, p. 96] by Uk G (z) := ∫ gG(z, ζ) dνk(ζ), z ∈ G. Note that Green potentials are superharmonic and nonnegative in G. Suppose that ν := ∑m j=1 νj is a unit measure. We study inequalities of the following type m ∑ j=1 inf E Uk G ≥ A+B inf E m ∑