  We study a type of calculus for proving inequalities between subgraph
densities which is based on Jensen's inequality for the logarithmic function.
As a demonstration of the method we verify the conjecture of Erd\"os-Simonovits
and Sidorenko for new families of graphs. In particular we give a short
analytic proof for a result by Conlon, Fox and Sudakov. Using this, we prove
the forcing conjecture for bipartite graphs in which one vertex is complete to
the other side.
