Higher-order Szegő theorems with two singular points

We consider probability measures, dμ=w(θ)dθ2π+dμs, on the unit circle, ∂D, with Verblunsky coefficients, {αj}j=0∞. We prove for θ1≠θ2 in [0,2π) that∫[1-cos(θ-θ1)][1-cos(θ-θ2)]logw(θ)dθ2π>-∞if and only if∑j=0∞(δ-e-iθ2)(δ-e-iθ1)αj2+|αj|4<∞,where δ is the left shift operator (δβ)j=βj+1. We also prove that∫(1-cosθ)2logw(θ)dθ2π>-∞if and only if∑j=0∞|αj+2-2αj+1+αj|2+|αj|6<∞.