Extremality properties for Gallager's random coding exponent

We describe certain extremality properties for Gallager's reliability function E
_o
for binary input symmetric DMCs. In particular, we show that amongst such DMC's whose E
₀
(ρ
₁
) has a given value for a given ρ
₁
, the BEC and BSC have the largest and smallest value of the derivative of Eo(ρ
₂
) for any ρ
₂
≥ ρ
₁
. As the random coding exponent is obtained by tracing the map ρ → (E
₀
'(ρ), E
₀
(ρ) - pE'
₀
(ρ)) this conclusion includes as a special case the results of [1]. Furthermore, we show that amongst channels W with a given value of E
₀
(ρ) for a given ρ the BEC and BSC are the most and least polarizing under Arıkan's polar transformations in the sense that their polar transforms W
⁺
and W
^-
has the largest and smallest difference in their E
_o
values.