## For the given map $G$ arising in the context of Hyperbolic Geometry, is $|G_w|^2 - |G_\bar{w} |^2$ bounded away from zero ?

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 !

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