/pdf/univalent-functions-and-non-convex-domains-1hjmgi2292.pdf

Some applications of the Weierstrass mean value theorem

Univalent functions and non-convex domains

A univalent function in a domain D is characterized by the property that it takes no value more than once in the domain and that, consequently, it maps the domain D onto a Schlicht domain, that is, one which is not self-overlapping and contains no branch points. In investigating the properties of analytic functions univalent in a domain D which is simply connected, one usually confines oneself to the case of the unit disc, for, by the Riemann mapping theorem, any simply connected domain with at least two boundary points can be mapped onto the unit disc, and any univalent function in D is associated with a corresponding univalent function in the unit disc. If f(z) is regular and univalent in the unit disc, so also is f(z) − f(0)}/f’(0), and this enables us to use the normalization f(0) = 0, f’(0) = 1 and study the class S of normalized regular functions having the Taylor expansion 

$$f(z) = z + {a_2}{z^2} + \; \cdot \cdot \cdot \;\;\;\;\;\left| z \right| < 1$$

We may observe that f’(0) does not vanish. If it did, f(z) could not be univalent. There also exist functions which are univalent but not regular in the unit disc. For example, if f(z) ∈ S, af(z) + b/ cf(z) + d, ad − bc ≠ 0 is obviously univalent in |z|<| but has a simple pole if −d/c is one of the values assumed by f(z) in |z| < 1.

Developments in the Theory of Univalent Functions

/pdf/some-applications-of-the-weierstrass-mean-value-theorem-3gq5pkrj51.pdf

/pdf/on-the-theory-of-univalent-functions-2qbm9p3sya.pdf

result

On the theory of univalent functions

exists for unrestricted approach in |z| gl, z^Zi, and in addition the conditions under which/'(z) is continuous at each point of an arc of \\z\\ =1. In the present paper we consider the corresponding questions for the higher derivatives and obtain results of similar nature, of which the following are the principal ones. Let 0(s) be an angle from the direction of the positive axis of reals to the tangent line, where 5 denotes arc length.' Let

Some applications of the Weierstrass mean value theorem

Citations

Univalent functions and non-convex domains

Univalent functions and non-convex domains

Developments in the Theory of Univalent Functions

References

On the theory of univalent functions

On the higher derivatives at the boundary in conformal mapping

On the theory of schlicht functions

Théorie des fonctions

On Umezawa's criteria for univalence II.

Related Papers (5)

A Mean Value Theorem

2742. A mean value theorem

On A Mean Value Theorem

On the mean value theorem

A generalization of Weierstrass' and Fekete's mean-value theorems