If $f$ is analytic on $\mathbb{C}_+$ and continuous on $\overline{\mathbb{C}_+}$ such that $|f|\leq 1$ on $i\mathbb{R}$ and $|f|$ has 'slow' growth on $\mathbb{C}_+$, for a basic example, polynomial growth, we can use Phragmen-Lindelof to say that $|f|\leq 1$ on all $\overline{\mathbb{C}_+}$.
Is there an application or version of Phragmen-Lindelof (or any other method, perhaps harmonic measure) that allows us to say something about $|f|$ on $\mathbb{C}_+$ if we don't have boundedness on the imaginary axis, but instead have some controlled growth? For example, if $|f(it)|\leq a|t|^n + b$ for $t\in \mathbb{R}$ and we also have polynomial growth on $\mathbb{C}_+$, can we say something about $|f(\lambda)|$ in relation to $a|\Im(\lambda)|^n+b$ for $\lambda \in \mathbb{C}_+$?
A related question which would provide an answer, is if $|f|\leq 1$ on the interval $\{it : |t|\leq c\}$ for some $c>0$ and has polynomial growth on the half-strip $\mathbb{C}_+$. Can we say $|f| \leq 1$ on $\{s+it : s >0, |t|\leq c\}$?
I'm aware that if I also had $|f| \leq 1$ on the boundary of the half-strip (i.e. the parallel lines $\{s \pm ic : s >0\}$) I could also apply Phragmen-Lindelof, but the point here is that I don't have this condition.
I'm trying to adjust an argument in a paper I'm reading for a slightly more general case. This isn't homework or an assignment. Thanks!