Orthogonal polynomials for weights close to indeterminacy

We obtain estimates for Christoffel functions and orthogonal polynomials for even weights W : R → [0, ∞) that are 'close' to indeterminate weights. Our main example is exp fenced(- || fenced(x) fenced(log || fenced(x))^β), with β real, possibly modified near 0, but our results also apply to exp fenced(- || fenced(x)^α fenced(log || fenced(x))^β), α < 1. These types of weights exhibit interesting properties largely because they are either indeterminate or are close to the border between determinacy and indeterminacy in the classical moment problem.