Given an inner function $\Theta \in H^\infty(\mathbb D)$ and $[g]$ in the quotient algebra $H^\infty/ \Theta H^\infty$,

its quotient norm is 

$\|[g]\|:= \inf \left\{ \|g+\Theta h\|_\infty, h \in H^\infty \right\}$. We show that 

when $g$ is normalized so that $\|[g]\|=1$, the quotient norm of its inverse can be made 

arbitrarily close to $1$ by imposing $|g(z)|\ge 1- \delta$ when $\Theta(z)=0$, with \delta>0 small enough, 

(call this property SIP)

 if and only if the function $\Theta$ satisfies the following growth property:

$</p><p>\lim_{t\to 1} \inf\left\{ |\Theta(z)|: z \in \mathbb D, \rho(z, \Theta^{-1} \{0\} ) \ge t \right\} =1,</p><p>$

where $\rho$ is the usual pseudohyperbolic distance in the disc, $\rho(z,w):= \left| \frac{z-w}{1-z\bar w}\right|$.

We prove that an inner function is SIP if and only if for any \eps&gt;0, the set \{ z: 0&lt; |\Theta (z) | &lt; 1-\eps\}

cannot contain hyperbolic disks of arbitrarily large radius.

Thin Blaschke products provide an example of such functions. Some SIP Blaschke products fail to be interpolating

(and thus aren't thin), while there exist Blaschke products which are interpolating and fail to be SIP. 

We also study the functions which can be divisors of SIP inner functions.