MacLane Class Research Articles

The MacLane class and the Eremenko–Lyubich class

In 1970 G. R. MacLane asked if it is possible for a locally univalent function in the class $\mathcal{A}$ to have an arc tract. This question remains open, but several results about it have been given. We significantly strengthen these results, in particular replacing the condition of local univalence by the more general condition that the set of critical values is bounded. Also, we adapt a recent powerful technique of C. J. Bishop in order to show that there is a function in the Eremenko-Lyubich class for the disc that is not in the class $\mathcal{A}$.

The MacLane class and complex differential equations in the unit disk

The interaction between the coefficients and solutions for linear differential equations and for Riccati differential equations is investigated in the unit disk in terms of their asymptotic values as described by the MacLane class. This interaction is further explored within the context of a meromorphic version of the MacLane class.

Nonvanishing derivatives and the MacLane class

Nonvanishing derivatives and the MacLane class

Nonvanishing derivatives and the MacLane class $\mathcal{A}$

Let k ≥ 2 and let f be meromorphic in the unit disc Δ, such that f(z)f(k)(z) �= 0 for all z ∈ Δ and the poles of f in Δ have bounded multiplicities. Then f has asymptotic values on a dense subset of ∂Δ.

On a Problem of MacLane Concerning Arc Tracts

G. R. MacLane posed the question of whether a locally univalent function in the MacLane class A can have an arc tract. We show that the behaviour of any such example must be extremely irregular. We also indicate one possible approach to constructing an example of such a function, which relates MacLane’s question to the ‘type problem’ for certain Riemann surfaces.

Positive cases to the branch point problem

Fatou's Theorem states that a bounded analytic function in the unit disk admits radial limits almost everywhere on the unit circle C If we relax the condition of boundedness of f in the hypothesis of Fatou's theorem to unboundedness of the maximum modulus function M f (r) such that M f r < q(r),where q (r) has a slow growth to ∞ we can only hope to obtain asymptotic values instead of radial limits at points of C. The MacLane class A of analytic functions is the class of non constant analytic functions in the unit disk D that have asymptotic values at a dense subset of the unit circle C. The main purpose of this paper is to consider a question posed by MacLane [M2] in 1970 on whether functions in A with an arc tract have branch points. We answer this question positively in the special cases, where the finite asymptotic values are bounded, or where infinity is linearly accessible at least at two points of the end of the tract

Towards a higher-dimensional MacLane class

Let D be a bounded region in ℝm, m ≥ 2. We say that a function u defined in D has asymptotic value α if there is a boundary path Γ:x(t), 0≤t&lt;1, in D (that is, dist (x(t), ∂D)→0 as t→1), such that u(x(t))→α as t→1. If in addition, x(t)→ξ as t→1, then u has asymptotic value α at ξ.

Angular limits, maximum principles, and level domains in the MacLane class

Let A denote the MacLane Class, that is, the class of (nonconstant) analytic functions on the unit disk D having asymptotic limits in a dense subset of the unit circle T. Of central interest here are three desirable properties that a function f∊A may possess: f has angular limits in a metrically dense subset of T f obeys a certain maximum principle, and the components of {|f|>λ} are Jordan regions for each λ∊(inf|f|,sup|f|). It is known, for example, that if f ∊A and f has no infinite asymptotic values, then all of these properties hold. In this note we consider various subclasses of A and give less restrictive conditions which guarantee that the properties are valid. In particular, we show that for the class N of normal functions, relatively weak radial conditions are sufficient.

Some Problems Related to Maclane's Class

Some Problems Related to Maclane's Class

On a Growth Condition Related to the Maclane Class

On a Growth Condition Related to the Maclane Class

Two sufficient conditions for the MacLane class A

Two sufficient conditions for the MacLane class A

Boundary Behavior and Quasi-Normality of Finitely Valent Holomorphic Functions

A function denned in a domain D is n-valent in D if f(z) — w0 has at most n zeros in D for each complex number w0. Let denote the class of nonconstant, holomorphic functions f in the unit disc that are n-valent in each component of the set . MacLane's class is the class of nonconstant, holomorphic functions in the unit disc that have asymptotic values at a dense subset of |z| = 1.

Addendum to our paper: Exponentiation of functions in Mac Lane's Class A

Article Addendum to our paper: Exponentiation of functions in Mac Lane's Class A was published on January 1, 1972 in the journal Journal für die reine und angewandte Mathematik (volume 1972, issue 252).

Integrals and derivatives of functions in MacLane's class and of normal functions

If f is meromorphic in D = {,|z| &lt; 1}, we say that f has the asymptotic value a at ζ, ζ ∈ C(={|z| = 1}), if there exists an arc Γ such that Γ ends at ζ, (Γ − {ζ}) ⊂ D, and f has the limit a as |z| → 1 on Γ. The class (M) originally defined by G. R. MacLane ((8), p. 8), consists of those functions that are non-constant and holomorphic (meromorphic) in D and that have asymptotic values at a dense subset of C. In (8), p. 36, MacLane has shown the following three conditions are sufficient for a non-constant holomorphic function f to be in :

A Growth Condition for the Maclane Class A

A Growth Condition for the Maclane Class <i>A</i>

Sums and products of functions in the MacLane class A

MathematikaVolume 16, Issue 1 p. 86-87 Research Article Sums and products of functions in the MacLane class A D. A. Brannan, D. A. Brannan University of Maryland, Imperial College, LondonSearch for more papers by this authorR. Hornblower, R. Hornblower University of Maryland, Imperial College, LondonSearch for more papers by this author D. A. Brannan, D. A. Brannan University of Maryland, Imperial College, LondonSearch for more papers by this authorR. Hornblower, R. Hornblower University of Maryland, Imperial College, LondonSearch for more papers by this author First published: 01 June 1969 https://doi.org/10.1112/S0025579300004630Citations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL No abstract is available for this article.Citing Literature Volume16, Issue1June 1969Pages 86-87 RelatedInformation

Exponentiation of functions in MacLane's class A.

Exponentiation of functions in MacLane's class A.