Local and global average degree in bipartite graphs

Let $F_{\mathrm{b}\mathrm{i}\mathrm{p}}(n)$ denote the maximum, over all $n$-vertex bipartite graphs without isolated vertices, of the ratio of the minimum local average degree to the global average degree. We prove that $F_{\mathrm{b}\mathrm{i}\mathrm{p}}(n)=\frac{1}{4}\sqrt{n}+\frac{3}{8}+o(1)$. This answers a problem posed by Tuza.