A mathematical characterization of the Hirsch-index by means of minimal increments

The minimum configuration to have a h-index equal to h is h papers each having h citations, hence h2 citations in total. To increase the h-index to h + 1 we minimally need (h + 1)2 citations, an increment of I1(h) = 2h + 1. The latter number increases with 2 per unit increase of h. This increment of the second order is denoted I2(h) = 2. define I1 and I2 for a general Hirsch configuration (say n papers each having f(n) citations) we calculate I1(f) and I2(f) similarly as for the h-index. We characterize all functions f for which I2(f) = 2 and show that this can be obtained for functions f(n) different from the h-index. We show that f(n) = n (i.e. the h-index) if and only if I2(f) = 2, f(1) = 1 and f(2) = 2. e a similar characterization for the threshold index (where n papers have a constant number C of citations). Here we deal with second order increments I2(f) = 0.