Differential calculus for p-norms of complex-valued vector functions with applications

For complex-valued n-dimensional vector functions t↦s(t), supposed to be sufficiently smooth, the differentiability properties of the mapping t↦∥s(t)∥p at every point t=t0∈R0+ ≔ {t∈R | t⩾0} are investigated, where ∥·∥p is the usual vector norm in Cn resp. Rn, for p∈[1,∞]. Moreover, formulae for the first three right derivatives D+k∥s(t)∥p, k=1,2,3 are determined. These formulae are applied to vibration problems by computing the best upper bounds on ∥s(t)∥p in certain classes of bounds. These results cannot be obtained by the methods used so far. The systematic use of the differential calculus for vector norms, as done here for the first time, could lead to major advances also in other branches of mathematics and other sciences.