A spread relation for entire functions with negative zeros

Abstract:

Let $g$ be a canonical product having only real negative zeros and nonintegral order $\lambda$, and let $\phi$ be the set function defined by $2\pi \phi (E) = {\smallint _E}\pi \lambda \csc \pi \lambda \cos \lambda \theta d\theta$. It is shown that if $E(r)$ is the set of values of $\theta \in ( - \pi ,\pi ]$ where $|g(r{e^{i\theta }})| \geq 1,{r_n}$ is a sequence of Polya peaks of $g$ and $\delta$ is the deficiency of the value zero of $g$ then $\phi (E({r_n})) \geq 2{(1 - \delta )^{ - 1}}$. This inequality leads to a sharp spread relation for $g$.