English reference for Ostrowski's theorem for Julia exceptional function

Question

In complex analysis, almost all of the book on Nevanlinna theory will mention that Ostrowski first constructed an example of mermorphic function without Julia direction.

Furthermore, recently, from Segal's book "Nine introduction in complex analysis", I know that not only did Ostrowski construct the example, but also he characterized very explicity all meromorphic funcitons which have no Julia direction. This is very interesting at least from my point of view, I really want to know his method for this argument.

The book of Segal did not contain Ostrowski's theorem (explicitly) on characterization of meromorphic function without Julia direction. The original paper of Ostrowski was the following

Alexander Ostrowski, Über Folgen analytischer Funktionen und einige Verschärfungen des Picardschen Satzes, Math. Z. 24 (1926), no. 1, 215–258; MR 1544761

which I can only find a small part (10 pages) of this paper from the internet. From the small part, I only know that the class of functions is the ratio of Weirstrass products with 0 order satisfying certain auxillary conditions on poles and zeros. However, I can not find the accurate statment for Ostroski's result on Julia exceptional function.

I really want to know whether Ostrowski's result has also contained somewhere, especially, in some English reference.

Any comment and reference will be appreciated.

Answer 1

Ostrowski's knew form the theory of normal families that if $f(z)$ is a Julia exceptional function, it has to be a meromorphic function of order zero. Thus it can be expressed as

$$f(z)=z^m \frac{\prod (1-\frac{z}{a_\alpha})}{\prod (1-\frac{z}{b_\beta})}$$

His result was that such a $f(z)$ is Julia exceptional if and only if the following conditions (denoted a), b), c), d) in his 1926 paper) hold

• there exists a constant $c_1$ such that $|n(r,\infty)-n(r,0)|<c_1$.
• there exists constants $c_2$, $c_3$ independent of $r$ such that $n(2r,\infty)-n(r,0)<c_2$ and $n(2r,0)-n(r,0)<c_3$.

• there exists constants $c_4$, $c_5$ such that $$|a_p|^m\prod_{|a_\alpha|<|a_p|}|\frac{a_p}{a_\alpha}|\bigg/{\prod_{|b_\beta|<|a_p|}|\frac{a_p}{b_\beta}|}<c_4$$ $$|b_q|^{-m}\prod_{|b_\beta|<|b_q|}|\frac{b_q}{a_\beta}|\bigg/{\prod_{|a_\alpha|<|b_q|}|\frac{b_q}{a_\alpha}|}<c_4$$ for any $p$ and $q$.

• there exists a positive $\epsilon$ such that for any $\lambda$, $\mu$,

$$|\frac{a_\lambda}{b_\mu}-1|\geq \epsilon > 0$$

This is explained without much more detail in Chung-Chun Yang and Chuang-Gan Hu's book "Vector-Valued Functions and their Applications".

Ostrowski's paper is avaible in full here, but for some reason the pdf version only has 10 pages. This result seem to be on pages 245-249.

Answer 2

Another exposition of Ostrowski's result in English is here: arXiv:0710.1281 and here arXiv:1208.0779.