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univalent function[¦yü·nə¦vā·lənt ′fəŋk·shən]
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 ǀzǀ ≤ 1. A function that is univalent in the circle ǀzǀ < 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 ǀzǀ < 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 + a2z2 + a3z3 + …
then the inequalities ǀa2ǀ ≤ 2 and ǀa3ǀ ≤ 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 a2, a3, a4, … in order that the series z + a2z2 + a3z3 + … be the Taylor series of some univalent function. The coefficient problem has still not been solved.