Extremality for Gallager’s Reliability Function $E_{0}$

We describe certain extremalities for Gallager's E
₀
function evaluated under the uniform input distribution over the class of binary input discrete memoryless channels; The results characterize the extremality of the E
₀
(ρ) curves of the binary erasure channel and the binary symmetric channel among all the E
₀
(ρ) curves that can be generated by the class of binary discrete memoryless channels whose E
₀
(ρ) curves pass through a given point (ρ
₀
, ε
₀
), for some ρ
₀
> -1.