Univalent function

Univalent in the area $A\subset \mathbb {C}$ ${\ displaystyle A \ subset \ mathbb {C}}$ $A \ subset {\ mathbb C}$ function in complex analysis - holomorphic function $f(z)$ ${\ displaystyle f (z)}$ $f (z)$ defined in $A$ ${\ displaystyle A}$ $A$ and setting the injection between the prototype $A$ ${\ displaystyle A}$ $A$ and manner $f(A)$ ${\ displaystyle f (A)}$ $f (A)$ .

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Local univalence

Analytic function $f$ ${\ displaystyle f}$ called locally univalent at the point $z_{0}\in \mathbb {C}$ ${\ displaystyle z_ {0} \ in \ mathbb {C}}$ if some neighborhood exists ${\mathcal {U}}_{z_{0}}$ ${\ displaystyle {\ mathcal {U}} _ {z_ {0}}}$ where $f$ ${\ displaystyle f}$ univalent.

The principle of univalence

Assume that the function $f$ ${\ displaystyle f}$ analytic in $G$ ${\ displaystyle G}$ , moreover, continues continuously to the Jordan curve $\partial G$ ${\ displaystyle \ partial G}$ . Then if $f$ ${\ displaystyle f}$ realizes one-to-one mapping $\partial G$ ${\ displaystyle \ partial G}$ on $f(\partial G)$ ${\ displaystyle f (\ partial G)}$ then $f$ ${\ displaystyle f}$ will be univalent in $G$ ${\ displaystyle G}$ .

Maximum area of univalence

Maximum univalence region for a function $f(z)$ ${\ displaystyle f (z)}$ Is an area $A\subset \mathbb {C}$ ${\ displaystyle A \ subset \ mathbb {C}}$ , wherein $f(z)$ ${\ displaystyle f (z)}$ univalent, but in any field $A'\supset A$ ${\ displaystyle A '\ supset A}$ the function is no longer univalent.

Univalent function

Content

Local univalence

The principle of univalence

Maximum area of univalence

See also

More articles:

Univalent function

Content

Local univalence

The principle of univalence

Maximum area of ​​univalence

See also

More articles:

Maximum area of univalence