On meromorphic functions without Julia directions

Abstract

It is proved that for any positive number $λ$, $1<λ<2$; there exists a meromorphic function $f$ with logarithmic order $λ$= $\displaystyle\limsup_{r\to+\infty}\frac{\log T(r,f)}{\log\log r}$ such that $f$ has no Julia directions, where $T(r,f)$ is the Nevanlinna characteristic function of $f$. (Note that A. Ostrowski has proved a {\it similar} result for $λ=2$ in 1926.)