Extremality for Gallager's Reliability Function $E_0$

We describe certain extremalities for Gallager's $E_0$ function evaluated under the uniform input distribution for binary input discrete memoryless channels. The results characterize the extremality of the $E_0(\rho)$ curves of the binary erasure channel and the binary symmetric channel among all the $E_0(\rho)$ curves that can be generated by the class of binary discrete memoryless channels whose $E_0(\rho)$ curves pass through a given point $(\rho_0, e_0)$, for some $\rho_0 > -1$.