# Univalent Function

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## univalent function

[¦yü·nə¦vā·lənt ′fəŋk·shən]## Univalent Function

an analytic function that effects one-to-one mapping of one region in the complex plane onto another region. The study of a function that is univalent in some simply connected region can be reduced to the study of two functions that are univalent within the circle ǀ*z*ǀ ≤ 1. A function that is univalent in the circle ǀ*z*ǀ < 1 is said to be normalized if *f(0*) = 0 and *f’(0*) = 1. The family *S* of normalized functions that are univalent in the circle ǀ*z*ǀ < 1 has been studied quite thoroughly. Estimates valid for any function of *S* can be given for certain quantities associated with univalent functions. If the function *f(z*) of the family *S* is expanded into a Taylor series

*f* (*z*) = *z* + *a*_{2}*z*^{2} + *a*_{3}*z*^{3} + …

then the inequalities ǀ*a*^{2}ǀ ≤ 2 and ǀ*a*^{3}ǀ ≤ 3 will be satisfied. The well-known coefficient problem from the theory of univalent functions consists in finding the necessary and sufficient conditions that must be imposed on the complex numbers *a*_{2}, *a*_{3}, *a*_{4}, … in order that the series *z* + *a*_{2}*z*^{2} + *a*_{3}*z*^{3} + … be the Taylor series of some univalent function. The coefficient problem has still not been solved.