Range of validity of alpha and beta for a generalized diversity index H (alpha, beta) due to Good.

Good (I. J. Good, Biometrika 40 (1953) 237; J. Am. Statist. Assoc. 77 (1982) 561) proposed a diversity index H (alpha, beta), defined for non-negative integer alpha and beta, which generalized the well-known indices of Shannon and Simpson. In this paper we further generalize Good's index by allowing the parameters alpha and beta to take values in the real plane. For two simple properties which a 'good' diversity index should possess, we determine the range of valid alpha and beta values for this generalized index.