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Theorem 2.1 in the book ‘Theory of Hp spaces by Peter. L Duren states that : Any function $f$ analytic on the unit disc belongs to the Nevanlinna class iff it is of the form $\frac{g}{h}$ where $g$ and $h$ are bounded analytic functions.

I have read the proof and know the construction of the functions $g$ and $h$. My question is, suppose $f$ is an outer function, then is the constructed $g$, the contant function 1? If yes, then can you tell me how?

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No, g and h are the exp of the Poisson integral of the parts of log|f| that are less than 1 and bigger than 1 (last one taken with a minus and put in the denominator) respectively

For example, f(z)=(1-z)/(1+z) is outer and is the ratio of the bounded functions 1-z and 1+z as obviously 1/f is not bounded either!

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