scholarly journals On the distribution of values of meromorphic functions of bounded characteristic

1954 ◽  
Vol 91 (0) ◽  
pp. 87-112 ◽  
Author(s):  
Olli Lehto
Keyword(s):  
Meromorphic Functions ◽  
Distribution Of Values ◽  
Bounded Characteristic

scholarly journals URS AND URSIMS FOR P-ADIC MEROMORPHIC FUNCTIONS INSIDE A DISC

2001 ◽  
Vol 44 (3) ◽  
pp. 485-504 ◽  
Author(s):  
Abdelbaki Boutabaa ◽  
Alain Escassut
Keyword(s):  
Characteristic Function ◽  
Analytic Functions ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Closed Field ◽  
Open Disc ◽  
Absolute Value ◽  
Unique Range Set ◽  
Bounded Characteristic ◽  
Primary 12H25

AbstractLet $K$ be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We show that the $p$-adic main Nevanlinna Theorem holds for meromorphic functions inside an ‘open’ disc in $K$. Let $P_{n,c}$ be the Frank–Reinders’s polynomial$$ (n-1)(n-2)X^n-2n(n-2)X^{n-1}+ n(n-1)X^{n-2}-c\qq (c\neq0,\ c\neq1,\ c\neq2) $$and let $S_{n,c}$ be the set of its $n$ distinct zeros. For every $n\geq 7$, we show that $S_{n,c}$ is an $n$-points unique range set (counting multiplicities) for unbounded analytic functions inside an ‘open disc’, and for every $n\geq10$, we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for the same set of functions. Similar results are obtained for meromorphic functions whose characteristic function is unbounded: we obtain unique range sets ignoring multiplicities of $17$ points. A better result is obtained for an analytic or a meromorphic function $f$ when its derivative is ‘small’ comparatively to $f$. In particular, for every $n\geq5$ we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for unbounded analytic functions with small derivative. Actually, in each case, results also apply to pairs of analytic functions when just one of them is supposed unbounded. The method we use is based upon the $p$-adic Nevanlinna Theory, and Frank–Reinders’s and Fujimoto’s methods used for meromorphic functions in $\mathbb{C}$. Among other results, we show that the set of functions having a bounded characteristic function is just the field of fractions of the ring of bounded analytic functions in the disc.AMS 2000 Mathematics subject classification: Primary 12H25. Secondary 12J25; 46S10

The distribution of values of certain entire and meromorphic functions

1981 ◽  
Vol 178 (4) ◽  
pp. 509-525 ◽  
Author(s):  
James M. Anderson ◽  
Irvine N. Baker ◽  
James G. Clunie
Keyword(s):  
Meromorphic Functions ◽  
Distribution Of Values

scholarly journals Some unsolved problems on meromorphic functions of uniformly bounded characteristic

1985 ◽  
Vol 8 (3) ◽  
pp. 477-482 ◽  
Author(s):  
Shinji Yamashita
Keyword(s):  
Characteristic Function ◽  
Meromorphic Functions ◽  
Holomorphic Functions ◽  
Bounded Mean Oscillation ◽  
Open Problems ◽  
Mean Oscillation ◽  
The Family ◽  
Bounded Characteristic ◽  
Uniformly Bounded

The familyUBC(R)of meromorphic functions of uniformly bounded characteristic in a Rieman surfaceRis defined in terms of the Shimizu-Ahlfors characteristic function. There are some natural parallels betweenUBC(R)andBMOA(R), the family of holomorphic functions of bounded mean oscillation inR. After a survey some open problems are proposed in contrast withBMOA(R).

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