  Let $\hom(H,G)$ denote the number of homomorphisms from a graph $H$ to a
graph $G$. Sidorenko's conjecture asserts that for any bipartite graph $H$, and
a graph $G$ we have $$\hom(H,G)\geq
v(G)^{v(H)}\left(\frac{\hom(K_2,G)}{v(G)^2}\right)^{e(H)},$$ where $v(H),v(G)$
and $e(H),e(G)$ denote the number of vertices and edges of the graph $H$ and
$G$, respectively. In this paper we prove Sidorenko's conjecture for certain
special graphs $G$: for the complete graph $K_q$ on $q$ vertices, for a $K_2$
with a loop added at one of the end vertices, and for a path on $3$ vertices
with a loop added at each vertex. These cases correspond to counting colorings,
independent sets and Widom-Rowlinson colorings of a graph $H$. For instance,
for a bipartite graph $H$ the number of $q$-colorings $\textrm{ch}(H,q)$
satisfies $$\textrm{ch}(H,q)\geq q^{v(H)}\left(\frac{q-1}{q}\right)^{e(H)}.$$
In fact, we will prove that in the last two cases (independent sets and
Widom-Rowlinson colorings) the graph $H$ does not need to be bipartite. In all
cases, we first prove a certain correlation inequality which implies
Sidorenko's conjecture in a stronger form.
