Logarithmic Sobolev Inequalities for Information Measures

For alpha ges 1, the new Vajda-type information measure J _alpha (X) is a quantity generalizing Fisher´s information (FI), to which it is reduced for alpha = 2 . In this paper, a corresponding generalized entropy power N _alpha (X) is introduced, and the inequality N _alpha (X) J _alpha(X) ges n is proved, which is reduced to the well-known inequality of Stam for alpha = 2. The cases of equality are also determined. Furthermore, the Blachman-Stam inequality for the FI of convolutions is generalized for the Vajda information J _alpha (X) and both families of results in the context of measure of information are discussed. That is, logarithmic Sobolev inequalities (LSIs) are written in terms of new more general entropy-type information measure, and therefore, new information inequalities are arisen. This generalization for special cases yields to the well known information measures and relative bounds.