An inequality is a statement of the form or . (What's up with vertical alignment of formulas in WP?) An inequation is . I can’t think of any adequate Russian word for “inequation”, but that’s besides the point. The literal analog “неравенство” is already used for “inequality”.
Suppose we want to prove that a map is Lipschitz, that is, such that for all . All we know is that the map is injective for all with a large modulus, i.e., for . In the following we consider only such values of .
Fix distinct and record the inequation: . Rearrange as . Note that we can multiply by any unimodular complex number, since the inequation holds whenever . Thus, we have a stronger inequation: . Yes, putting absolute values in an inequation makes it stronger, opposite to what happens with equations.
Still keeping and fixed, increase the modulus of until the inequality holds. Continuity with respect to tells us that was indeed all the way. So we bring back down to , and happily record the inequality , the desired Lipschitz continuity.
Self-promotion: a paper in which the above trick was used.