We prove inequalities between the densities of various bipartite subgraphs in
signed graphs and graphons. One of the main inequalities is that the density of
any bipartite graph with girth r cannot exceed the density of the r-cycle. This
study is motivated by Sidorenko's conjecture, which states that the density of
a bipartite graph F with m edges in any graph G is at least the m-th power of
the edge density of G. Another way of stating this is that the graph G with
given edge density minimizing the number of copies of F is, asymptotically, a
random graph. We prove that this is true locally, i.e., for graphs G that are
"close" to a random graph.Comment: 20 page