  We consider the Courtade-Kumar most informative Boolean function conjecture
for balanced functions, as well as a conjecture by Li and M\'edard that
dictatorship functions also maximize the $L^\alpha$ norm of $T_pf$ for
$1\leq\alpha\leq2$ 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 $L^\alpha$-norm related quantity can cross zero
as a function of $\alpha$, and show that these two conjectures are essentially
equivalent.
