Consider the Bregman divergence. $$ D_F(p, q) = F(p)-F(q)-\langle \nabla F(q), p-q\rangle. $$ And its dual norm: $D_{F*}(p, q) $ where $ F^*(y) = \arg\min_x \left\{ \langle x, y\rangle - F(x) \right\} $ based on the definition of convex conjugacy.
The Bregman divergence can be seen as a distance measure between its arguments. Can we say something like Hölder's inequality or perhaps using the Cauchy-Schwarz with Bregman Divergence? Something like this? $$ f(x) g(x) \leq D_{F^*}(f(x)||0). D_{F}(g(x)||0) $$
Here is a special case: If $F(x) = \frac{1}{2}x^\top x$, $D_R(x,y) = \frac{1}{2} \| x- y \|^2$. Similarly dual norm is also $\ell 2$, and Cauchy-Schwarz follows.