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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 ^_^

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  • $\begingroup$ An inner function can "grow" (approach $1$) as slowly as you wish. The construction of A.B. Aleksandrov is one of the easiest ways to get such examples. $\endgroup$
    – fedja
    May 25, 2013 at 16:02

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