Revisiting Gallager´s error exponent analysis technique

We demonstrate a tight analysis of an expectation of a sum of exponents raised to some power, prevalent in, but not confined to, Gallager´s bounding techniques. We show that the traditional analysis that uses Jensen´s inequality, although tight in Gallager´s random coding error exponent, might not be tight in general. Using the binary symmetric channel as an example, we show that R_c - the lowest rate at which Gallager´s bound agrees with the sphere packing bound is the lowest rate for which Jensen´s inequality is tight for a range of possible parameters.