On Angular Limits of Normal Meromorphic Functions: A Geometric Aspect

We will prove the assertions which give necessary and sufficient conditions for a normal meromorphic function on the open unit disk to have an angular limit. The results obtained show that the conditions from the classical Lindelöf theorem, as well as the theorems of Lehto and Virtanen and Bagemihl and Seidel, concerning angular limit values of meromorphic functions, can be weakened.

Texture and Magnetic Levitation Force of Melt-Textured YBaCuO Cylinders

To investigate the correlation between the levitation force of YBaCuO cylinders when approached by a permanent magnet and the texture of these YBaCuO cylinders some of these specimens were selected which are free of microcracks, have the same oxygen load, show no isotropic fraction or other undesired parasitic c-axis orientations. To gain a maximum levitation force the c-axis orientation of the crystalline domains has to have a small divergence between the c-axis and the cylinder axis. By measurements of the (002) pole figures by neutron diffraction with a mesh width no more than 6° undesired c-axis orientations have been analyzed. The distribution of the c-axis near the cylinder axis has been measured at the (005) reflection of 7 times higher intensity than the (002) reflection. The angular limit within which lie 94% of all c-axis orientations is a practical value for comparison of the c-axis orientations and the levitation forces of the different YBaCuO cylinders.

On Metric Properties of Sets of Angular Limits of Meromorphic Functions

Let f be a nonconstant function meromorphic in the unit disc , with circumference C, and let Ez be a subset of C with positive (linear) measure. Suppose that at each ζ ∈ Ezf has an angular limit aζ and let It is known that Ew contains a closed set with positive harmonic measure (see Priwalow [6, p. 210] or Tsuji [7, p. 339]).

Some Remarks on Angular Derivatives and Julia's Lemma

Let w = f(z) be holomorphic on the unit disk D = { z: | z | < 1}, with the additional restrictions that | f ( z ) | < l and , where denotes the (outer) angular limit of f (z) at z = 1. Let us now define and then focus our attention on the behaviour of g(z) in an arbitrary angular neighbourhood of z = 1. Whenever exists, this limit is commonly referred to as the angular derivative of f(z) at z = 1.

On the Relation between a Cluster set Introduced by Constantinescu and Cornea, and the Fine Cluster set of Cartan, Brelot, and Naïm

The field of boundary limit theorems in analytic function theory is usually considered to have begun about 1906, with the publication of Fatou's thesis [8]. In this remarkable memoir a theorem is proved, that now bears the author's name, which implies that any bounded holomorphic function defined on the unit disk possesses an angular limit almost everywhere (Lebesgue measure) on the frontier. Outstanding classical contributions to this field can be attributed to F. and M. Riesz, R. Nevanlinna, Lusin, Privaloff, Frostman, Plessner, and others.

Radial and Angular Limits of Meromorphic Functions

Let us say that a function denned in the open unit disk D has the Montel property if the set of those points eiθ on the unit circle C where the radial limit exists coincides with the set where the angular limit exists. By a classical theorem of Montel (4), every bounded holomorphic function has this property. Meromorphic functions omitting at least three values and, more generally, the normal functions recently introduced by Lehto and Virtanen (3) also enjoy the Montel property (also see 1).