In Terry Tao's notes https://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/#lind, the interpolation $B_0^{1-\theta} B_1^\theta$ appearing in the Hadamard 3-lines theorem comes from the fact that $h(s):= B_0^{1-s} B_1^s$ is an entire nowhere-vanishing function s.t. $|h|=B_0$ on the imaginary line $i\mathbb R$ and $|h|=B_1$ on the line $1+i\mathbb R$. Is this the only such function (up to constant multiple)?
If I had another such function $f$ satisfying just the modulus conditions, Entire function such that $|f(z)| = 1$ on the real line I think tells me that $f$ is automatically nowhere-vanishing. Since $h$ is nowhere-vanishing, $\frac fh$ is entire, and has modulus $1$ on $i\mathbb R$ and $1+i\mathbb R$, and nowhere-vanishing $\implies \frac fh = e^g$ for some entire $g$. Then on $i\mathbb R$ and $1+i\mathbb R$, $$1=|e^g| = e^{\Re(g)} \implies \Re(g) \in 2\pi i\mathbb Z \cap \mathbb R\implies \Re(g)=0.$$ There seem to be some results about situations like this, e.g. The real part of a holomorphic function and its link to the function being constant, but I don't really know where to go from here.
EDIT: I suppose from the very same theorem there is the function $h(s):= \exp(-i\exp(i\pi s))$, where for $s$ with $\Re s= 0,1$, we get $$\begin{aligned} |h(s)| &= \exp(\Re(-i \exp(i\pi s)) = \exp(\Im (\exp(i\pi s)) = \exp(\Im(\exp(i\pi \Re s) \exp(-\pi \Im s))\\ &= \exp(\Im(\pm \exp(-\pi \Im s)) = \exp(0) = 1 \end{aligned}$$ so $h(s) B_0^{1-s} B_1^s$ is also an entire nowhere-vanishing function s.t. $|h|=B_0$ on $i\mathbb R$ and $=B_1$ on $1+i\mathbb R$. Are these the unique functions with these properties? It seems like the facts I already found above via other MSE posts should severely restrict the class of such functions.
