An extension of Carlson's theorem for analytic functions (Q1098969)

The aim of this paper is, to give an improvement of an identity theorem, of the second author, for entire functions of exponential type less than \(\pi\) in the following form: Given any entire function f, \(\tau (f)<\pi\), \(\sum_{n\in Z}| f(n)| <\infty\), such that Re f(n)\(=0\). If Re f(n\(+i)=0\) for all negative integers n and Im f(n\(+i)=0\) for all nonnegative integers n, then \(f=0\).