Contents Online

# Communications in Information and Systems

## Volume 14 (2014)

### Number 2

### Generalization of Mrs. Gerber’s lemma

Pages: 79 – 86

DOI: http://dx.doi.org/10.4310/CIS.2014.v14.n2.a1

#### Author

#### Abstract

Mrs. Gerber’s Lemma (MGL) hinges on the convexity of $H(p*H^{-1}(u))$, where $H(u)$ is the binary entropy function. In this work, we prove that $H(p*f(u))$ is convex in $u$ for every $p \in [0, 1]$ provided $H(f(u))$ is convex in $u$, where $f(u) \colon (a, b) \to [0, \frac{1}{2}]$. Moreover, our result subsumes MGL and simplifies the original proof. We show that the generalized MGL can be applied in binary broadcast channel to simplify some discussion.

#### Keywords

Mrs. Gerber’s lemma, binary channel