# Boundedness of functions in complex interpolation method

In the method of complex interpolation one evaluates traces of suitable holomorphic functions on the strip. I have looked in the book of Lunardi and the one of Bergh/Löfström and in both this "suitable" meant for a function $f: S\to X+Y$ that $f$ is continuous, bounded, holomorphic on the interior of the strip and $f(it)\in \mathrm{C}_b(\mathbb{R},X)$, $f(1+it)\in \mathrm{C}_b(\mathbb{R},Y)$.

My question is, whether it is redundant to require that $f$ is bounded. Using the maximum principle for holomorphic functions, the continuous embeddings $X,Y \hookrightarrow X+Y$ and the boundedness of the restriction of $f$ to the lines bordering the strip this already is clear?

A typical counterexample: let $f(z) = \exp(\exp(\pi i (z-1/2)))$. When $\operatorname{Re} z \in \{0,1\}$, the imaginary part of $w = \pi i (z-1/2)$ is $\pm \pi /2$, hence $\exp(w)$ is purely imaginary, hence $\exp(\exp(w))$ has modulus $1$. Yet, at $z=1/2 - in$ we get $f(z) = \exp(\exp(\pi n))$ which grows to infinity rather rapidly.