  For a class of density functions $q^n(x^n)$ on $\Bbb R^n$ we prove an
inequality between relative entropy and the sum of average conditional relative
entropies of the following form: For any density function $p^n(x^n)$ on $\Bbb
R^n$, $D(p^n||q^n)\leq Const. \sum_{i=1}^n \Bbb E D(p_i(\cdot|Y_1,...,
Y_{i-1},Y_{i+1},..., Y_n) || Q_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n)),$
where $p_i(\cdot|y_1,..., y_{i-1},y_{i+1},..., y_n)$ and $Q_i(\cdot|x_1,...,
x_{i-1},x_{i+1},..., x_n)$ denote the local specifications for $p^n$ resp.
$q^n$, i.e., the conditional density functions of the $i$'th coordinate, given
the other coordinates. The constant depends on the properties of the local
specifications of $q^n$. The above inequality implies a logarithmic Sobolev
inequality for $q^n$. We get an explicit lower bound for the logarithmic
Sobolev constant of $q^n$ under the assumptions that: (i) the local
specifications of $q^n$ satisfy logarithmic Sobolev inequalities with constants
$\rho_i$, and (ii) they also satisfy some condition expressing that the mixed
partial derivatives of the Hamiltonian of $q^n$ are not too large relative to
the logarithmic Sobolev constants $\rho_i$. Condition (ii) may be weaker than
that used in Otto and Reznikoff's recent paper on the estimation of logarithmic
Sobolev constants of spin systems.
