The Courtade-Kumar Most Informative Boolean Function Conjecture and a Symmetrized Li-Médard Conjecture are Equivalent

We consider the Courtade-Kumar most informative Boolean function conjecture for balanced functions, as well as a conjecture by Li and Médard that dictatorship functions also maximize the $L^α$ norm of $T_pf$ for $1\leqα\leq2$ where $T_p$ is the noise operator and $f$ is a balanced Boolean function. By using a result due to Laguerre from the 1880's, we are able to bound how many times an $L^α$-norm related quantity can cross zero as a function of $α$, and show that these two conjectures are essentially equivalent.

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