Entropy Inequalities for the generalized Gaussian

The target of this paper is to discuss the existent Poincaré and Logarithm Sobolev Inequalities (PI and LSI resp.) for the Gaussian (normal) distribution which is essential in theoretical Statistics and plays an important role in Information Theory and Statistics. The adopted Mathematical backround is usually simplified in practical applications. The entropy, energy and variance are related through some order due to PI and LSI. The extended multivariate normal, being a generalized Gaussian, also obeys to LSI.