Proof of a combinatorial conjecture posed in "The Blimpy Shape of Heady-s and Taily-s Bit Strings"

We demonstrate three properties conjectured to hold for a certain function by Levin (2025) in a study of the blimpy graphical shape of the number of bit strings with a given score under an interesting scoring system. The properties include discrete convexity, a simple formula for the greatest argument at which the function is negative, and a positive expectation under a certain probability function. A new set of inequalities which imply the latter is presented and proved under some monotonicity assumptions.