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

Second main theorem and uniqueness problem of meromorphic functions with finite growth index sharing five small functions on a complex disc