A generalization of the $\textrm {cos}\ \pi \rho$ theorem

Abstract:

Let f be an entire function, and let $\beta$ and $\lambda$ be positive numbers with $\beta \leq \pi$ and $\beta \lambda < \pi$. Let $E(r) = \{ \theta :\log |f(r{e^{i\theta }})| > \cos \beta \lambda \log M(r)\}$. It is proved that either there exist arbitrarily large values of r for which $E(r)$ contains an interval of length at least $2\beta$, or else ${\lim _{r \to \infty }}{r^{ - \lambda }}\log M(r,f)$ exists and is positive or infinite. For $\beta = \pi$ this is Kjellberg’s refinement of the cos $\pi \rho$ theorem.