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.