My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c\_1>c\_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic function in the right half of the complex plane with
$$ |f(k)|\leq c\_k \quad \forall\, k\in \mathbb N,$$
there holds $ f(z)=0$ on the right half plane.



On a variant of Carlson’s theorem
Condition
$$\lim\_{n\to\infty}\frac{\log|c\_n|}{n}=-\infty$$
is sufficient for $f=0$.


Since $f(z)=e^{-cz}$ and $c\_n=e^{-cn}$ satisfy all
conditions, we see that this is best possible in certain sense.


This follows for example from a (much more general) theorem of N. Levinson, Gap and density theorems, AMS, 1940, page 121.
Levinson's theorem allows some growth of $F$, and much more general class of
sequences instead of integers.


Remark. In fact Levinson generalizes a theorem of Vladimir Bernstein 1932
(Theorem 32 in Levinson's book), which also implies the result that I stated.


