Univalent Function


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univalent function

[¦yü·nə¦vā·lənt ′fəŋk·shən]
(aerospace engineering)

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 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.

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In fact, the Koebe one-quarter theorem [10] ensures that the image of U under every univalent function f [member of] S contains a disk of radius 1/4.
The univalent function q(z) is called a dominant of the solution of the differential subordination (1.
Thus every such univalent function has an inverse [f.
Univalent function is a function which does not take the same value twice, f([z.
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The univalent function q(z) is said to be a dominant of the differential subordination (1.
s] denote the subclass of [summation] consisting of univalent function in E.
Netanyahu, "The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in [absolute value of (z)] < 1," Archive for Rational Mechanics and Analysis, vol 32, pp.
Keywords-: Analytic function, Univalent function, hyper geometric distribution.
In fact, the Koebe one-quarter theorem [1] ensures that the image of U under every univalent function f [member of] S contains a disk of radius 1/4.
where q(z) is a given univalent function in U with q(z) [not equal to] 0.