Assume $f$ is entire. I want to know if
$$|f(z) |<Ce^{\tau |z|} \: \& \: |f(iy)|<Ce^{c|y|}, y\in \mathbb{R}$$ for some constants $C,\tau$ and $c<\pi$, is equivalent to
$$ |f(z) |<Ce^{P |Re(z)|+Q|Im(z)|}$$ for some constants $C,P$ and $Q<\pi$.
The second condition implies the first, this is clear. The other direction is not (to me). This question is of interest since I am writing a thesis on Ramanujan's master theorem and the latter is precisely the condition of this result. There is a uniqueness theorem that follows directly from the master theorem and I would like to know if it is equivalent to Carlson's theorem (https://en.wikipedia.org/wiki/Carlson%27s_theorem).