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) This is regarding outer functions on a Hardy space. I know that outer functions can have no zeroes in the open unit disc since it is the exponential of something. I know that if $f\in H^p$ with $Re(f)>0$, then $f$ is outer. Can we say that any function in $H^p$ that has no zeroes in the open unit is outer?
Is there any other characterisation of outer functions in Hardy spaces?



Regarding characterisation of outer functions in a Hardy space

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> Can we say that any function in Hp that has no zeroes in the open unit is outer? 
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> 
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No. $\exp\left(\frac{z-1}{z+1}\right)$ is inner and belongs to $H^\infty$ and thus to all $H^p$.


