Definition (Renyi Entropy)

Given parameter $\alpha>0, \ \alpha\not=1$, a RV $X\sim p_X$, then the Renyi entropy with parameter $\alpha$ is $$\begin{align*} H_\alpha(X)&=\frac{1}{1-\alpha}\log_2\left(\sum_{a\in\mathscr{X}}p_X(a)^\alpha\right)\\ &=\frac{\alpha}{1-\alpha}\log_2\left(\|p_X\|_\alpha\right) \end{align*}$$ where $\|p_X\|_\alpha$ is the “$\alpha$-norm of $p_X$” or $$\|p_X\|_\alpha=\left[\sum_{a\in\mathscr{X}}p_X(a)^\alpha\right]^{\frac{1}{\alpha}}$$