Variations on the theme of the Carlson theorem (Q1324902)

The main result of the paper is the following theorem. Let \[ f(z) = \sum^ \infty_{k=0} {a_ k \over k!} z^ k\;(z \in \mathbb{C}) \] be an entire function of exponential type. Assume that there exists \(\delta \in (0,\pi)\) such that: the Borel-associated function of \(f\) is holomorphic on \(\{z \in \mathbb{C} : | \text{Im} z | \geq \pi - \delta\}\) and \(\text{Im} f(n_ k) = \text{Re} f(-m_ k)=0\), \(k \in \mathbb{N}\), where \(n_ k\), \(m_ k \in \mathbb{N}\) are such that \[ \lim_{k \to + \infty} {k \over n_ k} \geq 1 - {\delta \over \pi},\quad \lim_{k \to + \infty} {k \over m_ k} \geq 1-{\delta \over \pi}. \] Then \(f \equiv 0\).