Inner-Outer and Riesz Factorization

Prove likewise that $f \in N$ if and only if $\log^{+}\left|f\right|$ has a harmonic majorant in $U$.

Suppose $f \in H^{p}, \varphi \in H(U)$, and $\varphi (U) \subset U$. Does it follow that $f \circ \varphi \in H^{p}$? Answer the same question with $N$ in place of $H^{p}$.

Suppose $\varphi$ is a nonconstant inner function, $\left|\alpha \right| < 1$, and $\alpha \notin \varphi (U)$. Prove that $\lim_{r \rightarrow 1} \varphi \left(r e^{i \theta }\right) = \alpha$ for at least one $e^{i \theta } \in T$.

Suppose $f \in H^{1}$ and $1 / f \in H^{1}$. Prove that $f$ is then an outer function.

Suppose $f \in H^{1}$ and $\mathrm{Re}\left(f(z)\right) > 0$ for all $z \in U$. Prove that $f$ is an outer function.

Prove that $f \in N$ if and only if $f = g / h$, where $g$ and $h \in H^{\infty }$ and $h$ has no zero in $U$.