Energy of a graph and Randić index of subgraphs

We give a new inequality between the energy of a graph and a weighted sum over the edges of the graph. Using this inequality we prove that $E\left(G\right)\ge 2R\left(H\right)$, where $E\left(G\right)$ is the energy of a graph $G$ and $R\left(H\right)$ is the Randić index of any subgraph of $G$ (not necessarily induced). In particular, this generalizes well-known inequalities $E\left(G\right)\ge 2R\left(G\right)$ and $E\left(G\right)\ge 2\mu \left(G\right)$ where $\mu \left(G\right)$ is the matching number. We give other inequalities as applications to this result.