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Recently I come cross a question about deficient values of entire functions.

I find that many examples in the book about functions $f$ whose deficient values are singularities of the inverse $f^{-1}$.

I want to know whether there exist an example with the following property (in some sense it aks whether the concept of deficient value has a geometric explanation):

Is there an entire function $f$ with deficient value $a$ such that $a$ is not in the closure of the singular value set?

Another question: For entire functions in the Eremenko-Lybuich class, is it true that the the set of finite difficiency value is also bounded?

Any comments will be appreciated.

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  • $\begingroup$ Perhaps you mean 'deficient values of entire functions'? $\endgroup$
    – Stopple
    Commented Nov 24, 2014 at 16:15
  • $\begingroup$ you are right, I mean the dificient values of entire function. $\endgroup$
    – yaoxiao
    Commented Nov 25, 2014 at 0:41

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The book of Goldberg and Ostrovskii MR2435270 contains several examples of functions whose deficient value is not asymptotic. And in fact there are such functions without asymptotic values at all. But deficient value must be in the closure of the singular set.

This follows from a theorem of E. Collingwood, which has many generalizations, by Tumura, H. Selberg, Kakutani and H. Cartan. You can find one generalization as Theorem 1.2 in Chapter 7 of Goldberg - Ostrovskii. In particular, for a function of class $S$, only singular values can be deficient. If you search carefully, you can find this book on the Internet.

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  • $\begingroup$ Thank you, professor. I find the book you metioned. It is really an interested thm, which help me to understand some geometric intuition of difficient value. $\endgroup$
    – yaoxiao
    Commented Nov 25, 2014 at 0:40
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    $\begingroup$ Another nice exposition of this matter is in H. Wittich, Neuere Untresuchungen uber eindeutige analytische Funktionen, Springer, Berlin 1955. Unfortunately available only in German and Russian. But you can read the original papers. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 25, 2014 at 1:06
  • $\begingroup$ Thank you, professor. I can only read little German. You mentioned it has been generalized by Turmura, H.Selberg, Kakutani, H.Cartan and someothers. Is that easy to find the papers from the internet? $\endgroup$
    – yaoxiao
    Commented Nov 25, 2014 at 16:12
  • $\begingroup$ @yaoxiao: References are in Goldberg-Ostrovskii. How easy is to find them on Internet may depend on your location/affiliation. Just try. BTW: I maintain a list of freely available old journals on my web site. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 25, 2014 at 21:02
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I like to advocate differentiating between the singularities of the inverse function, which are denoted by $\operatorname{sing}(f^{-1})$, and the set of singular values of $f$, which I denote by $S(f)$. The latter set can be defined as the closure of $\operatorname{sing}(f^{-1})$, though it also has an intrinsic definition: it is the complement of the set of values having a neighbourhood $U$ over which $f$ is an unramified covering map. On the other hand, $\operatorname{sing}(f^{-1})$ is precisely the set of critical and asymptotic values of $f$.

Then Alex's answer says that a deficient value need not belong to $\operatorname{sing}(f^{-1})$, but must always lie in $S(f)$. The latter part should be intuitively clear, since $f$ takes any other value in a set $U$ as above the same number of times, and the set of deficient values is countable.

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  • $\begingroup$ Thanks for your response, professor. I also have this intuition for the concept of dificient value. I just find few book cover this interesting fact. I heard there are some other dificency values in Nevanlinna theory. I guess all of them should be in S(f) (your term), while it needs a careful proof. $\endgroup$
    – yaoxiao
    Commented Nov 25, 2014 at 16:05
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    $\begingroup$ @yaoxiao: I recommend that you read the papers. Selberg's extension of Collingwood's theorem says $m(r,a)=O(\log r)$, if $a\not\in S(f)$; from this follows that Valiron deficiency is also zero. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 25, 2014 at 20:59
  • $\begingroup$ @yaoxiao: Good books on Nevanlinna theory (in dimension 1) are few: Nevanlinna himself (2 books), Hayman, Goldberg-Ostrovskii and S. Lang. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 25, 2014 at 21:07
  • $\begingroup$ @Lasse Rempe-Gillen: your intuitive argument is interesting: if you can give a simple proof that all points in $U$ have "equal rights" (to contribute to deficiency) it will be rigorous:-) $\endgroup$
    – Alexandre Eremenko
    Commented Nov 25, 2014 at 21:12

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