Extremality for Gallager's Reliability Function $E_0$

We describe certain extremalities for Gallager’s E0 function evaluated under the uniform input distribution for binary input discrete memoryless channels. The results characterize the extremality of the E0(ρ) curves of the binary erasure channel and the binary symmetric channel among all the E0(ρ) curves that can be generated by the class of binary discrete memoryless channels whose E0(ρ) curves pass through a given point (ρ0, e0), for some ρ0 > −1.