  This paper has twofold. The first is to establish a second main theorem for
meromorphic functions on the complex disc $\Delta (R_0)\subset\mathbb C$ with
finite growth index and small functions, where the counting functions are
truncated to level $1$ and the small term is more detailed estimated. The
second is to prove a generalization and improvement of the five values theorem
of Nevanlinna for the case of five small functions on the complex disc $\Delta
(R_0)$.
