Extended inequalities for weighted Renyi entropy involving generalized Gaussian densities

In this paper the author analyzes the weighted Renyi entropy in order to derive several inequalities in weighted case. Furthermore, using the proposed notions α-th generalized deviation and (α, p)-th weighted Fisher information, extended versions of the moment-entropy, Fisher information and Cramér-Rao inequalities in terms of generalized Gaussian densities are given. 1 The weighted p-Renyi entropy In 1960, Renyi was looking for the most general definition of information measures that would preserve the additivity for independent events and was compatible with the axioms of probability. He started with Cauchy’s functional equation and he ended up with the general theory of means. The consequence of this investigation derived the definition of Renyi entropy, see [16]. In addition, Stam [19] showed that a continuous random variable with given Fisher information and minimal Shannon entropy must be Gaussian. However the moment-entropy inequality established the result saying: a continuous random variable with given second moment and maximal Shannon entropy must be Gaussian as well. Cf. [10, 25, 14, 6]. Furthermore, The Cramér-Rao inequality shows that the second moment of a continuous random variable is bounded by the reciprocal of its Fisher information. Cf. [7]. Currently in [13], the notions of relative Renyi entropy, (α,p)-th Fisher information, as a general form of Fisher information associated with Renyi entropy including α-th moment and deviation has been introduced. More results and certain applications emerged in [5, 11, 2, 3]. Later, an interesting generalization of Stam’s inequality involving the Renyi entropy was given, check again [13], wherein it has been asserts that the generalized Gaussian densities maximizes the Renyi entropy with given generalized Fisher information. The initial concept of the weighted entropy as another generalization of entropy was proposed in [1, 8, 4]. Certain applications of the weighted entropy has been presented in information theory and computer science (see [17, 18, 21, 24, 22]). Let us now give the definition of the weighted entropy. For given function x ∈ R 7→ φ(x) ≥ 0, and an RV X : Ω → R, with a PM/DF f , the weighted entropy (WE) of X (or f) with weight function (WF) φ is defined by hφ (X) = h w φ(f) = − ∫ R φ(x)f(x) log f(x)dx = −EX(φ log f) (1.1) 2010 Mathematics Subject Classification: 60A10, 60B05, 60C05.