# 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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(see ) Let q be a convex univalent function in U and let [delta], [gamma] [member of] C with
The univalent function q(z) is called a dominant of the solution of the differential subordination (1.5) or more simply, a dominant if p(z) [??] q(z) for all p(z) satisfying (1.5).
then the function [F.sub.n,[alpha]] (z) defined by (8) is in the univalent function class S.
Motivated by the above-mentioned work, in this paper we have introduced [lambda]-bi-pseudo-starlike functions subordinate to a starlike univalent function whose range is symmetric with respect to the real axis and estimated second Hankel determinants.
Let [phi] be analytic in a domain containing f(E), [phi](0) = 0, [phi]'(0) = 1 and [phi](w) [not equal to] 0 for w [member of] f (E) \ {0}, then the function f [member of] A is called [phi]-like with respect to a univalent function q,q(0) = 1, if
MacGregor  constructed a univalent function f such that f [not member of] [F.sub.2].
A univalent function q(z) is called a dominant of the solutions of the differential subordination or, more simply, a dominant if p(z) < q(z) for all p(z) satisfying (22).
Suppose also that p ia convex univalent function in U satisfying the inequality that
Owa, Note on a class of starlike functions, Proceeding Of the International Short Joint Work on Study on Calculus Operators in Univalent Function Theory Kyoto (2006), 1-10.
The Koebe distortion theorem gives a series of bounds for a univalent function and its derivative.
The univalent function q(z) is called a dominant of the solutions of the differential subordination (1.3) if p(z) < q(z) for all p(z) satisfying (1.3).
Let q (z) be a convex univalent function in U and let [sigma] [member of] C, [eta] [member of] [C.sup.*] = C\ {0} with

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