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An Italian dominating function (IDF) on a graph G = (V,E) is a function f: V → {0,1,2} satisfying the property that for every vertex \(v \in V\), with f(v) = 0, \(\sum_{u \in N_{(v)}}f(u)\geq2\). The weight of an IDF f is the value \(w(f) = f(V) = \sum_{u \in V}f(u)\). The minimum weight of an IDF on a graph G is called the Italian domination number of G, denoted by \(\gamma_I(G)\). For a graph G = (V,E), a double Roman dominating function (or just DRDF) is a function f: V → {0, 1, 2,3} having the property that if \(f(v) = O\) for a vertex \(u\), then \(u\) has at least two neighbors assigned 2 under for one neighbor assigned 3 under f, and if \(f(v) = 1\), then u has at least one neighbor with f(w) ≥ 2. The weight of a DRDF f is the sum \(f(V) = \sum_{u \in V}f(v)\), and the minimum weight of a DRDF on G is the double Roman domination number oi G, denoted by \(\gamma_{dR}(G)\). In this paper we show that \(\gamma_dR(G)/2 ≤ \gamma_I(G) ≤ 2\gamma_dR(G)/3\), and characterize all trees T with \(\gamma_I(T) = 2\gamma_dR (T)/3\).