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We introduce a variant of the Rényi entropy definition that aligns it with the well-known Hölder mean: in the new formulation, the r -th order Rényi Entropy is the logarithm of the inverse of the r -th order Hölder mean. This brings about new insights into the relationship of the Rényi entropy to quantities close to it, like the information potential and the partition function of statistical mechanics. We also provide expressions that allow us to calculate the Rényi entropies from the Shannon cross-entropy and the escort probabilities. Finally, we discuss why shifting the Rényi entropy is fruitful in some applications.