Let $h: S^1 \to S^1$ be a fixed oriention-preserving homeomorphism and let $p(z,t) = \frac{1}{2\pi}\frac{1-|z|^2}{|z-t|^2} $denote the Poisson kernel, $D$ denote the open unit disk in $\mathbb{C} $. Let us consider the function $ G: D\times D \to \bar{D} $ given by :
$ G (z,w) = \int_{S^1} \frac{h(t)-w}{1-\bar{w}h(t)} p(z,t) |dt| $.
Consider the quantity $ |G_w|^2 - |G_\bar{w}|^2$, where, a direct computation shows that :
$G_w(z,w) = \int_{S^1} \frac{-1}{1- \bar{w}h(t)} p(z,t)|dt| $
and $G_\bar{w}(z,w)= \int_{S^1}\frac{h(t)-w}{1-\bar{w}h(t)}.\frac{-h(t)}{1-\bar{w}h(t)} p(z,t)|dt|$
I was wondering whther there is a condition on $h: S^1 \to S^1$ [ for exzample, $h$ is real-analytic, smooth, bi-Lipchitz, Lipchitz, Holder ]m such that
$ |G_w(z,w)|^2 - |G_\bar{w}(z,w)|^2 \geq L(h) > 0 \forall (z,w) $, where $L(h)$ is a positive constant depending on $h$ only.
Any books/refernces/papers would be greatly appreciated ! Thank you !
