A meromorphic function g on a complex disc \(\Delta (R)\) is said to have a finite growth index \(c_g\) if \(c_g\) is the infimum of all numbers c such that \(\int _{0}^R\exp (cT(r,g))=+\infty \) . In this paper, we investigate the relation between the characteristic functions of two meromorphic functions weighted sharing three values. We show that if two non-constant meromorphic functions f and g with finite growth index on \(\Delta (R)\) weakly share three distinct values \(a_1,a_2,a_3\) with weights \(n_1,n_2,n_3\) respectively, then \(\begin{aligned} \Vert \ (1-3\varepsilon )T(r,f)\le (2+3\varepsilon +\delta _\varepsilon c_g)T(r,g)+o(T(r,g)), \end{aligned}\) for every \(\varepsilon \in (0,\frac{1}{2})\) , where \(\delta _\varepsilon \) is explicitly estimated depending only on \(n_1,n_2,n_3\) and \(\varepsilon \) . Our result is the extension of the previous results of Li and Yang (J Math Anal Appl 220:132–145, 1998) and others. Our proof also provides a new proof for the case of meromorphic functions on \({\mathbb {C}}\) , which is simpler and clearer than that of the previous ones.
