If f is outer, 1/f is outer (more generally f raised to any real - say non-zero to avoid constants - power is outer) - any outer function has no zeros as those are factored out with Blaschke products, so we can talk about log(f) and any complex power of f in the disk.
1/f always belongs to the Nevanlinna class N btw and if for example Re(f)>0, than Re(1/f)>0 so 1/f belongs to all hardy classes of exponent less than 1
Applying the above factorization with bounded analytic functions for class N for the ratio of any two outer functions, we get your answer