  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 \[
\sum_{j=0}^\infty \abs{\alpha_{j+2}-2\alpha_{j+1} +\alpha_j}^2 +
\abs{\alpha_j}^6 <\infty \]
