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We introduce orderings between total functions f,g: N -> N which refine thepointwise "up to a constant" ordering <=cte and also insure that f(x) is oftenmuch less thang(x). With such orderings, we prove a strong hierarchy theoremfor Kolmogorov complexities obtained with jump oracles and/or Max or Min ofpartial recursive functions. We introduce a notion of second order conditionalKolmogorov complexity which yields a uniform bound for the "up to a constant"comparisons involved in the hierarchy theorem.

Refinment of the "up to a constant" ordering using contructive  co-immunity and alike. Application to the Min/Max hierarchy of Kolmogorov  complexities