On a Theorem of Sunouchi

This result is best possible in the sense that δ cannot be dropped.

The special case of this theorem in which α is an integer has been generalised as follows by introducing a general sequence of factors:

Theorem S'. [7], [11]. Let λ_n > 0 and non-decreasing and let α be a non-negative integer and let

Σ |σ^α_n-σ^α_n-1| = O(λ_m).