Upper and lower bounds for the Bregman divergence

In this paper we study upper and lower bounds on the Bregman divergence ${\mathrm{\Delta }}_{\mathcal{F}}^{\xi }(y,x):=\mathcal{F}(y)-\mathcal{F}(x)-⟨\xi ,y-x⟩$ for some convex functional $\mathcal{F}$ on a normed space $\mathcal{X}$, with subgradient $\xi \in \mathrm{\partial }\mathcal{F}(x)$. We give a considerably simpler new proof of the inequalities by Xu and Roach for the special case $\mathcal{F}(x)={\parallel x\parallel }^p,p>1$. The results can be transfered to more general functions as well.