Using majorization theory via "Robin Hood" elementary operations, optimal lower and upper bounds are derived on Rényi and guessing entropies with respect to either error probability (yielding reverse-Fano and Fano inequalities) or total variation distance to the uniform (yielding reverse-Pinsker and Pinsker inequalities). This gives a general picture of how the notion of randomness can be measured in many areas of computer science.
Keywords: Fano inequality; Pinsker inequality; Rényi entropy; Schur concavity; data processing inequality; entropy; error probability; guessing entropy; guessing moments; majorization; total variation distance.