On the Growth of Entire Functions of Exponential Type near a Straight Line

Let $\Lambda =\{\lambda _n\}$ be a sequence of complex numbers such that $\lambda _n\to\infty$, as $n\to+\infty$. For $\Lambda$ close to the imaginary axis, upper bounds of the indicator of a nonzero entire function of exponential type with minimal growth vanishing on $\Lambda$ is obtained. These bounds give sufficient conditions for the system of exponents to be incomplete in an unbounded domain in $\{\exp(\lambda _nz)\}$.