Univalent Function
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Univalent Function

In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective.[1]

Examples

show that the function

 f(z)=2z+z^2, |z|<1

Is univalent in the domain. Solve..

 Let z1 and z2 be any two points in |z|<1. then

f(z1)=f(z2)->(z1-z2)(z1+z2+2)=0 ->z1-z2=0 and z1+z2+2?0 in |z|<1 This is clear from the fact Re(z1+z2+2)=Rez1+Rez2+2>-1-1+2=0 Thus f is one-one in |z|<1 and hence univalent in the domain.

Basic properties

One can prove that if and are two open connected sets in the complex plane, and

is a univalent function such that (that is, is surjective), then the derivative of is never zero, is invertible, and its inverse is also holomorphic. More, one has by the chain rule

for all in

Comparison with real functions

For real analytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function

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).

See also

References

  1. ^ John B. Conway (1996) Functions of One Complex Variable II, chapter 14: Conformal equivalence for simply connected regions, page 32, Springer-Verlag, New York, ISBN 0-387-94460-5. Definition 1.12: "A function on an open set is univalent if it is analytic and one-to-one."

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