In a recent paper of Salazar et al. (2017), The Diversification Delta: A different perspective, in the Journal of Portfolio Management (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2518705), the authors make the following claim:
If we define $$H(X)= -\int_x f(x) \ln(f(x)) \,\mathrm{d}x$$ to be the differential Shannon entropy, then we have that for random variables $X$ and $Y$: $$ \exp(H(X+Y)) \leqslant \exp(H(X)) + \exp(H(X)) $$ (pp.10). After thorough research I haven't been able to find any proof of this sub-additive property which makes me wonder: is this property really correct? And would you have any proof of it? Moreover is seems that this sub-additive property is at odds with the entropy power inequality that says that for independent $X$ and $Y$: \begin{align} \exp(2H(X+Y)) \geqslant \exp(2H(X)) + \exp(2H(X)). \end{align}
Also, I am interested in knowing how this property would generalize to Rényi entropy defined as $$H_\alpha(X)=\frac{1}{1-\alpha}\ln \int_x (f_X(x))^\alpha \,\mathrm{d}x.$$ Is $\exp(H_\alpha(X))$ also sub-additive for some values of $\alpha$?
I would really appreciate any insights you might have on this problem.