Higher-order Szegő theorems with two singular points

We consider probability measures, dμ = w(θ)^(dθ)_(2π) + dμ_s, on the unit circle, ∂D, with Verblunsky coefficients, {αj}_(j=0)^∞. We prove for θ_1 ≠ θ_2 in [0,2π) that ∫[1-cos(θ-θ_1)][1-cos(θ-θ_2)]log w(θ)^(dθ)_(2π > -∞if and only if ∑_(j=0)^∞ │{(δ-e^(-iθ2))(δ-e^(-iθ1))α}_j^2 +|α_j|^4 < ∞,where δ is the left shift operator (δβ)_j = β_(j+1). We also prove that ∫(1-cosθ)^2 log w (θ)^(dθ)_(2π) > - ∞ if and only if ∑_(j=0)^∞|α_(j+2) - 2α_(j+1) + α_j|^2 + |αj|^ 6 <∞.