Does the Lesche condition for stability validate generalized entropies

We reconsider the Lesche condition for a stability of entropies, which has been used for supporting validity of a generalized entropy over the others in recent papers. The fabric of an argument for applying the stability criterion with specific distributions to generalized entropies is shown to be interrogative. We show that a local stability analysis for the generalized entropies does not lead to the same conclusion as the one derived from the Lesche condition. It is also pointed out that the bound on the Shannon entropy by Lesche with an auxiliary function, and that on its generalized version are weak compared to those on the variational distance of their entropies against small changes in distribution. We conclude that there is no good reason for putting much emphasis on the criterion in the sense of Lesche when we advocate the validity of a generalized entropy.

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