Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The Hardy spaces on $\mathbb{D}$ are defined as: $$H^{p}:= \left\{ f\in Hol\left( \mathbb{D}\right) :\sup _{r < 1}\int ^{2\pi }_{0}\left| f\left( re^{i\theta}\right) \right| ^{p}d\theta < \infty \right\} \;\;\;\;(0<p<\infty), $$ $$H^{\infty }:= \left\{ f\in Hol\left( \mathbb{D}\right) :\sup_{z\in D}\left| f\left( z\right) \right| < \infty \right\}.$$ A function $g\in H^p(\mathbb{D})$ is outer if there exists a function $G:\mathbb{T}\longrightarrow [0,\infty)$ with $G\in L^1(\mathbb{T})$ such that $$g\left( z\right) =\alpha \text{exp}\left( \int ^{2\pi }_{0}\dfrac {e^{i\theta }+z}{e^{i\theta }-z}G\left( e^{i\theta }\right) \dfrac {d\theta }{2\pi }\right) \qquad(z\in \mathbb{D})$$ and $|\alpha|=1$. Let $\mathscr{P}[h]$ denote the closed subspace generated by the functions $z^n h(z),\; n=0,1,2,....$, ie $\mathscr{P}[h]$ consists of all $H^p$ functions that can be approximated by polynomial multiples of $h$. Note that $\mathscr{P}[1]=H^p$, since polynomials are dense in $H^p$.

I wanted to ask:

I know that if $h$ is not outer, then $\mathscr{P}[h]\neq H^p $. Will this also imply that $1\notin \mathscr{P}[h]$ with dist$(1,\mathscr{P}[h])>0.$? Will this distance be bounded below by some +real number(irrespective of given outer function h)?