On One Property of the Nevanlinna Characteristic

We prove the existence of entire functions f of an arbitrary lower order ⋋ ≥ 0 and the order 𝜌 = ⋋ + 1 such that

$$ \underset{r\to +\infty }{\lim \kern0.5em \operatorname{inf}T}\left(r+1,f\right)/T\left(r,f\right)>1. $$

The obtained results show that the condition 𝜌 − ⋋ < 1 in Valiron’s theorem cannot be improved.