On a property of the Nevanlinna characteristic

Abstract

UDC 517.53
We prove the existience of entire functions $f$ of an arbitrary lower order $\lambda\ge 0$ and the order $\rho=\lambda+1$ such that
$$\begin{equation*} \underset{r \to +\infty}{\overline{\lim}}T(r + 1, f)/T(r, f) > 1. \end{equation*}$$
Obtained results show that the condition $\rho - \lambda < 1$ of Valiron's theorem can not be improved.