# On a variant of Carlson’s theorem

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.

• The answer is certainly "yes" (by the standard compactness argument) but I suspect that much more is known to the experts, so I'll leave it to them to answer your question properly. Apr 8, 2022 at 22:43
• For example $c_k=\exp(-k^{1+\epsilon})$ will do. Apr 8, 2022 at 23:29

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.
• See @fedja's comment to the question. Quntitave estimates are quite different for the condition that $f(n)$ is small, vs the condition $f(n)=0$. Apr 9, 2022 at 18:58