Is there a definition of relative Renyi entropy? Is there a Renyi entropy analogue of ``$H(X \vert Y)$" ? 
If yes then is there any known meaning to that? 

Googling around I found a few different notions,


*

*equation 18 here, http://arxiv.org/abs/1505.06980

*equation 2.5 here, http://rgmia.org/papers/v14/v14a17.pdf

*The notion of ``Renyi divergence" as given here, http://en.wikipedia.org/wiki/R%C3%A9nyi_entropy#R.C3.A9nyi_divergence 
I wonder if for continuous probability distributions $f$ and $g$ if one can write, $D_\alpha (p \vert \vert g) = \frac{1}{ \alpha - 1} log \left [ \int \frac{f^\alpha}{g^{\alpha -1}} dx \right ]  $ and if this has any meaning. 
Though I don't know as to how it can be ensure that $g > 0$...
(The question could have arisen with Kullback-Leibler divergence too?) 
 A: Normally $H(X|Y)$ means conditional entropy. In this case I don't think there is any generally accepted definition of Renyi counterpart. There is, however, a recent Master thesis which lists some possibilities including a new propositions by the author, which seems to be quite reasonable:
http://web.math.leidenuniv.nl/scripties/MasterBerens.pdf 
If you really mean "relative Renyi entropy" then the divergence you found is the correct answer. 
Relation between relative entropy and conditional entropy:
Note that Kullback-Leibler divergence $D_{KL}(P||Q)=D_1(P||Q)$ (a.k.a. relative entropy) is a measure of how two probability distributions are similar. Roughly speaking, it's equal to zero when two probability distributions are the same ($P=Q$). It does not include any information about how two random variables are related. If we talk about discrete probability measure we can think of $P$ and $Q$ as a vectors. In that case they should both have the same size if one wants to compute K-L divergence.
In contrast, conditional entropy $H(X|Y)$ is a measure of how two random variables are related. One has to define joint probability over these two random variables in order to be able to calculate conditional entropy. It is equal to zero when knowledge of $Y$ determines the value of $X$ completely. Also, these two random variables may have different number of possible values.
