  We prove the Courtade-Kumar conjecture, for several classes of n-dimensional
Boolean functions, for all $n \geq 2$ and for all values of the error
probability of the binary symmetric channel, $0 \leq p \leq 1/2$. This
conjecture states that the mutual information between any Boolean function of
an n-dimensional vector of independent and identically distributed inputs to a
memoryless binary symmetric channel and the corresponding vector of outputs is
upper-bounded by $1-\operatorname{H}(p)$, where $\operatorname{H}(p)$
represents the binary entropy function. That is, let $\mathbf{X}=[X_1 \ldots
X_n]$ be a vector of independent and identically distributed Bernoulli(1/2)
random variables, which are the input to a memoryless binary symmetric channel,
with the error probability in the interval $0 \leq p \leq 1/2$ and
$\mathbf{Y}=[Y_1 \ldots Y_n]$ the corresponding output. Let $f:\{0,1\}^n
\rightarrow \{0,1\}$ be an n-dimensional Boolean function. Then,
$\operatorname{MI}(f(X),Y) \leq 1-\operatorname{H}(p)$. Our proof employs
Karamata's theorem, concepts from probability theory, transformations of random
variables and vectors and algebraic manipulations.
