  Building on the inequalities for homogeneous tetrahedral polynomials in
independent Gaussian variables due to R. Lata{\l}a we provide a concentration
inequality for non-necessarily Lipschitz functions $f\colon \R^n \to \R$ with
bounded derivatives of higher orders, which hold when the underlying measure
satisfies a family of Sobolev type inequalities $\|g- \E g\|_p \le C(p)\|\nabla
g\|_p.$
  Such Sobolev type inequalities hold, e.g., if the underlying measure
satisfies the log-Sobolev inequality (in which case $C(p) \le C\sqrt{p}$) or
the Poincar\'e inequality (then $C(p) \le Cp$). Our concentration estimates are
expressed in terms of tensor-product norms of the derivatives of $f$.
  When the underlying measure is Gaussian and $f$ is a polynomial
(non-necessarily tetrahedral or homogeneous), our estimates can be reversed (up
to a constant depending only on the degree of the polynomial). We also show
that for polynomial functions, analogous estimates hold for arbitrary random
vectors with independent sub-Gaussian coordinates.
  We apply our inequalities to general additive functionals of random vectors
(in particular linear eigenvalue statistics of random matrices) and the problem
of counting cycles of fixed length in Erd\H{o}s-R{\'e}nyi random graphs,
obtaining new estimates, optimal in a certain range of parameters.
