On Bloch's theorem

Mathematische Zeitschrift -     

Identity theorem for bounded p-adic meromorphic functions

Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences (an)_n∈N in a disk d(0,R⁻) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f(an)=0∀n∈N implies f=0), identity sequences (when limn→+∞f(an)=0 implies f=0) and analytic boundaries (when lim supn→∞|f(an)|=‖f‖). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions.     

An extension of Picard's theorem for meromorphic functions of small hyper-order

A version of the second main theorem of Nevanlinna theory is proved, where the ramification term is replaced by a term depending on a certain composition operator of a meromorphic function of small hyper-order. As a corollary of this result it is shown that if n∈N and three distinct values of a meromorphic function f of hyper-order less than 1/n² have forward invariant pre-images with respect to a fixed branch of the algebraic function τ(z)=z+αn−1z1−1/n+?+α₁z1/n+α₀ with constant coefficients, then f○τ≡f. This is a generalization of Picard's theorem for meromorphic functions of small hyper-order, since the (empty) pre-images of the usual Picard exceptional values are special cases of forward invariant pre-images.     

Generalizations of the area theorem for meromorphic univalent functions with nonzero pole

In this article, we consider meromorphic univalent functions f in the unit disc of the complex plane having a simple pole at z = α ∈ (0, 1) with nonzero residue b at z = α. In 1969, P.N. Chichra proved an area theorem for such functions. In this note, we generalize this theorem and prove an interesting consequence of this result.     

Forms for meromorphic functions

An infinite circular arithmetic is defined and used to define power sum and centered forms for meromorphic functions. The properties of these forms are investigated and it is shown that both the power sum and the centered forms form including chains with respect to their orders and that they possess a kind of quadratic convergence with respect to the domain circle. An example of the use of the forms is given.     

Difference analogues of the second main theorem for meromorphic functions in several complex variables

In this paper, we obtain difference analogues of the second main theorem for meromorphic functions in several complex variables from which difference analogues of Picard‐type theorems are also obtained. Our results are improvements or extensions of some recent results of papers [Proc. Royal Soc. Edinburgh Section A Math. 137 , 457–474 (2007); Comput. Meth. Funct. Theor. 12 , No. 1, 343–361 (2012)]. The method we used is very different from theirs.     

亚纯函数的唯一性定理

应用奈望里纳理论研究亚纯函数的唯一性问题,得到若干结果,推广了某些已知定理．     

Majorization principles for meromorphic functions

Supplements to the Lindelöf principle on the behavior of Green’s function and the Nevanlinna principle on the behavior of the harmonic measure under meromorphic maps are proposed; these supplements go back to Mityuk’s work on the change of the inner radius of a domain under the action of regular functions.     

On the decomposition theorem for meromorphic Fredholm resolvents

In this note results of B. Gramsch and W. Kaballo [8] on the decomposition of meromorphic (semi-) Fredholm resolvents are sharpened. A condition on an Orlicz function is given, under which the singular part in this decomposition can be chosen meromorphic inN , the ideal of -nuclear operators. Then the necessity of this condition is studied. Moreover, it is shown that for the rather steep Orlicz functions relevant to this question,N equalsS , the ideal of -approximable operators.Dedicated to Professor Albert Schneider on the occasion of his 60 ^th birthdayresearch supported by a grant from DAAD