Let $T$ be the unit circle. Let $\phi\in C(T)$ and let $\psi$ be a function in $L^2(T)$ such that $\phi+i\psi\in H^2$. Assume both $\psi$ and $\phi$ are real-valued. Show $e^{\phi+i\psi}\in H^\infty$.
I saw this in a paper by Sarason but it is not proved there. I tried proving it by writing $e^{\phi+i\psi}=\sum_{n=0}^\infty \frac{(\phi+i\psi)^n }{n!}$ but this I do not think will help since $H^2$ is not an algebra. Can someone give me a hint on how to prove this? I mostly need help in showing $\frac{1}{2\pi}\int_0^{2\pi} \left(e^{\phi(e^{i\theta})+i\psi(e^{i\theta})}\right)\chi_n(e^{i\theta})d\theta=0$ for $n>0$.
Note that $|e^z| = e^{\operatorname{Re} z}$, so $|e^{\phi+i\psi}| = e^\phi$ is bounded (since $\phi$ and $\psi$ are real-valued and $\phi \in C(T)$). Hence $e^{\phi+i\psi} \in H^\infty$.