Higher-Order Szego Theorems With Two Singular Points

We consider probability measures, $d\mu=w(\theta) \f{d\theta}{2\pi} +d\mu_\s$, on the unit circle, $\partial\bbD$, with Verblunsky coefficients, $\{\alpha_j\}_{j=0}^\infty$. We prove for $\theta_1\neq\theta_2$ in $[0,2\pi)$ and $(\delta\beta)_j=\beta_{j+1}$ that $$\int [1-\cos(\theta-\theta_1)][1-\cos(\theta-\theta_2)] \log w(\theta) \f{d\theta}{2\pi} >-\infty$$ if and only if $$\sum_{j=0}^\infty \bigl|\bigl\{(\delta -e^{-i\theta_2}) (\delta -e^{-i\theta_1}) \alpha\bigr\}_j\bigr|^2 +\abs{\alpha_j}^4 <\infty$$ We also prove that $$\int (1-\cos\theta)^2 \log w(\theta) \f{d\theta}{2\pi} >-\infty$$ if and only if $$\sum_{j=0}^\infty \abs{\alpha_{j+2}-2\alpha_{j+1} +\alpha_j}^2 + \abs{\alpha_j}^6 <\infty$$