Recall that an inner function on the disk $D$ is a bounded analytic function on $D$ having radial limits of modulus one almost everywhere.
There has been many works on the growth of the inner functions. I wonder if the following question is known or not.
$\bf Question:$ Does there exists an inner function $\theta$ on $D$ such that $\theta(0) = 0$, and $$|\theta(w)|^2 \le 1 - (1-|w|^2)^{1/2}, \quad \forall w \in D \quad ?$$ (Or equivalently $$(1-|w|^2)^{1/2} \le 1-|\theta(w)|^2 \quad \forall w \in D.$$)
$\bf Remark:$ I know that quite a few works are of the following form: Estimate $$\Delta(r, \theta) : = 1-\int|\theta(re^{it})|^2 dm(e^{it}).$$ And it is known that there is inner function $\theta$ such that $$\Delta(r, \theta) \ge c(1-r^2)^{1/2}.$$
If you have related reference, thank you for telling me ^_^
