Please see the definition of Hardy spaces on the unit disc [here](https://math.stackexchange.com/questions/2586370/if-a-function-belongs-to-the-hardy-space). Let $0<p\leq\infty$. Let $f\in H^p$ with $\|f-1\_e\|\_p<1$ (Where $1\_e$ Is the constant function one). Then is $f$ an **outer function**?



Regarding outer functions
This is true for $p=\infty$, but is not true for $p=2$. Let $g$ be an inner function. Then, the optimal polynomial approximant $p\_{n}^\*$ when minimizing $\|pg-1\|\_{2}$ over degree $n$ polynomials, $p$, satisfies $\|p\_{n}^{\*}g-1\|\_{2}^2=1-|g(0)|^2$ (see <http://shell.cas.usf.edu/~dkhavins/files/GenInnerFinal.pdf>, pg. 6, for details). 


To answer your question it suffices then to look at $g$ inner (and not also outer) with $|g(0)|>0$, since $f=p\_{n}^{\*}g$ need not be outer.


