Local and global average degree in bipartite graphs

Let $F_{\mathrm{bip}}(n)$ 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 $F_{\mathrm{bip}}(n)=\frac14\sqrt n+\frac38+o(1)$.

This answers a problem posed by Tuza.