A Matrix Variance Inequality for k-Functions

In this paper a course of solving variational problem is considered. [2] obtained what appears to be specialized inequality for a variance, namely, that for a standard normal variable X , V ar[g (x)] ¸ E[g (x)]^2 . However both of the simplicity and usefulness of the inequality has generated a plethora of extensions, as well as alternative proofs. [5] had focused on a result of two random variables for the normal and gamma distribution. They obtained the result of normal distribution with k functions, without proving and the proof is presented here. This paper also extend the result obtained by [5] to the k functions for the gamma distribution.