Given a pair of random variables $(X,Y)\sim P_{XY}$ and two convex functions
$f_1$ and $f_2$, we introduce two bottleneck functionals as the lower and upper
boundaries of the two-dimensional convex set that consists of the pairs
$\left(I_{f_1}(W; X), I_{f_2}(W; Y)\right)$, where $I_f$ denotes
$f$-information and $W$ varies over the set of all discrete random variables
satisfying the Markov condition $W \to X \to Y$. Applying Witsenhausen and
Wyner's approach, we provide an algorithm for computing boundaries of this set
for $f_1$, $f_2$, and discrete $P_{XY}$. In the binary symmetric case, we fully
characterize the set when (i) $f_1(t)=f_2(t)=t\log t$, (ii)
$f_1(t)=f_2(t)=t^2-1$, and (iii) $f_1$ and $f_2$ are both $\ell^\beta$ norm
function for $\beta \geq 2$. We then argue that upper and lower boundaries in
(i) correspond to Mrs. Gerber's Lemma and its inverse (which we call Mr.
Gerber's Lemma), in (ii) correspond to estimation-theoretic variants of
Information Bottleneck and Privacy Funnel, and in (iii) correspond to Arimoto
Information Bottleneck and Privacy Funnel.