| CHUANG CHI-TAI.[J].数学年刊A辑,1980,1(1):90~114 |
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| ON THE DISTRIBUTION OF THE VALUES OFMEROMORPHIC FUNCTIONS |
| Received:August 30, 1978 |
| DOI： |
| 中文关键词: |
| 英文关键词: |
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| 英文摘要: |
| In the theory of meromorphio functions the importance of the second fundamental
theorem
\[(q - 2)T(r,f) < \sum\limits_{j = 1}^q {N(r,{a_j}} ) - {N_1}(r) + s(r)\]
is well known. In 1929, R. Nevanlinna proposed to generalize this theorem in replacing
the values aj（j=1,2,...q)by meromorphio functions \[{\varphi _j}(z)(j = 1,2,...,q)\]satisfying
the condition:
\[T(r,{\varphi _j}) = o[T(r,f)]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (j = 1,2,...q)\](1)
R. Nevanlinna himself solved this problem for the case q= 3. For general value of q,
it was treated by J. Dnfresnoy in the particular case when \[{\varphi _j}{\kern 1pt} {\kern 1pt} (j = 1,2,...q)\]are
polynomials, and by the author in the general case when \[{\varphi _j}{\kern 1pt} {\kern 1pt} (j = 1,2,...q)\]satisfy
the condition (1).
The object of this paper is to study the same problem under weaker conditions on
\[{\varphi _j}{\kern 1pt} {\kern 1pt} (j = 1,2,...q)\], namely,
\[T(r,{\varphi _j}) = {o_*}[T(r,f)]{\kern 1pt} \]or\[T(r,{\varphi _j}) = o[U(r)]{\kern 1pt} \]
where the first condition means that there exists a set s of values of r of finite exterior
measure such that
\[\mathop {\lim }\limits_{\scriptstyler \to + \infty \hfill\atop
\scriptstyler \notin s\hfill} \frac{{T(r,{\varphi _j})}}{{T(r,f)}} = 0\]
and in the second condition U(r) is type-function associated to the function f(z), In
this way, generalizations of the second fundamentaJ theorem are obtained, which have
various applications. |
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