> There are two equivalent statements of the same problem . The first is to > prove that for complex z, |z - 1| > 2 implies |z^3 - 1| > 1 , the second is > made by substituting z with |i*y - x| , and proving that one > x^2 + 2x + y^2 > 3 (which is equivalent to |z-1| > 2| > implies x^3 - 3xy^2 + 1)^2 + (3x^2 y - y^3)^2 > 1. (which is equivalent to > |z^3 - 1| > 1 ) .
Let z = -x + yi. Then |z - 1| = |-1 - x + yi| = sqr((-1 - x)^2 + y^2) . . = sqr(x^2 + 2x + y^2) |z^3 - 1| = |-x^3 + 3x^2 yi - 3xy^2 - y^3 i| . . = sqr((x^3 + 3xy^2)^2 + (3x^2 y - y^3)^2)