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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Vaezpour, On the commutant of operators of multiplication by univalent functions, Proc.
Let S denote the class of all analytic univalent functions f defined in E and normalized by the conditions f (0) = f '(0) - 1 = 0.
An equivalent definition of CL by using Kaplan class and some related sets of univalent functions can be found in [6].
Khairnar and Meena More, Subclass of analytic and univalent functions in the unit disk, Scientia Magna, 3 (2007), No.
Complex-valued harmonic univalent functions have recently been studied from the perspective of geometric function theory.
Using the above Carlson-Shaffer operator, we introduce the following subclasses of analytic and univalent functions defined as follows:
Keywords Ruschewayh derivatives, analytic and univalent functions, quasi-subordinate.
In [2], Frasin and Jahangiri introduced the class B([mu], [alpha]) of analytic and univalent functions to give some properties for this class.
On three classes of univalent functions with real coefficients.
Srivastava, Distortion inequalities for analytic and univalent functions associated with certain fractional calculus and other linear operators Analytic and Geometric Inequalities and Applications, eds.
univalent functions f on the unit disk D with f(D) convex in the direction [e.