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 E₀(ρ₁) has a given value for a given ρ₁, the BEC and BSC have the largest and smallest value of the derivative of Eo(ρ₂) for any ρ₂ ≥ ρ₁. As the random coding exponent is obtained by tracing the map ρ → (E₀´(ρ), E₀(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 E₀(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.