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.
Variant of Carlson’s Theorem – Analysis
analytic-continuationanalytic-functionscv.complex-variablesfa.functional-analysisreal-analysis
Best Answer
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.