By Bregman Divergence, if $F$ is a convex function, then we can define :
$$D(x,y) = F(y) - F(x) - (y - x) F^{'}(x),$$
whereby, $D(x,y) \ge 0.$
Here, $F{'}(x)$ is the derivative of the function $F$ at $x$.
My question is, do $D(x,y)$ and $D(x^A,y^A)$ have a one-to-one correspondence, where $A \ge 1, A \in \mathrm{R} \;\;\;$ ($\mathrm{R}$ being the real number line) ?
Here, $x$ and $y$ are scalar variables and $A$ is a scalar constant.