univalent function

[¦yü·nə¦vā·lənt ′fəŋk·shən]

(aerospace engineering)

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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.

Mentioned in

injection

References in periodicals archive

(see [11]) Let q be a convex univalent function in U and let [delta], [gamma] [member of] C with

A class of multivalent non-Bazilevic functions involving the Cho-Kwon-Srivastava operator

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).

Some Subordination Theorems of Univalent Functions Defined by Linear Operators

then the function [F.sub.n,[alpha]] (z) defined by (8) is in the univalent function class S.

Univalence of a general integral operator associated with the generalized hypergeometric function

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.

Second Hankel Determinants for Some Subclasses of Biunivalent Functions Associated with Pseudo-Starlike Functions

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

Some sufficient conditions for [phi]-like functions

MacGregor [3] constructed a univalent function f such that f [not member of] [F.sub.2].

Weighted Composition Operators between the Fractional Cauchy Spaces and the Bloch-Type Spaces

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).

Third-order differential subordination and superordination results for meromorphically multivalent functions associated with the Liu-Srivastava operator

Suppose also that p ia convex univalent function in U satisfying the inequality that

On a Hurwitz-Lerch zeta type function and its applications

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.

Note on new general integral operators of p-valent functions

The Koebe distortion theorem gives a series of bounds for a univalent function and its derivative.

Distortion of quasiregular mappings and equivalent norms on Lipschitz-type spaces

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).

Some sandwich type theorems for analytic functions involving the Dziok-Srivastava operator and other related linear operators

Let q (z) be a convex univalent function in U and let [sigma] [member of] C, [eta] [member of] [C.sup.*] = C\ {0} with

On a certain subclass of multivalent analytic functions defined by the Liu-Owa operator