Any mapping of the open unit disc to itself, : where is univalent.
One can prove that if and are two open connected sets in the complex plane, and
for all in
Comparison with real functions
given by ƒ(x) = x3. This function is clearly injective, but its derivative is 0 at x = 0, and its inverse is not analytic, or even differentiable, on the whole interval (−1, 1). Consequently, if we enlarge the domain to an open subset G of the complex plane, it must fail to be injective; and this is the case, since (for example) f(εω) = f(ε) (where ω is a primitive cube root of unity and ε is a positive real number smaller than the radius of G as a neighbourhood of 0).
- John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, 1978. ISBN 0-387-90328-3.
- John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, 1996. ISBN 0-387-94460-5.