Defective value

2020 Mathematics Subject Classification: Primary: 30D35 [MSN][ZBL]

of a meromorphic function $ f $

A complex number $ a $( finite or infinite) whose defect $ \delta ( a , f ) $( see below) is positive. Let the function $ f $ be defined in the disc $ | z | < R \leq \infty $ of the complex plane $ \mathbf C $. The defect (or deficiency) of the value $ a $ is

$$ \delta ( a , f ) = 1 - \overline{\lim\limits}\; _ {r \rightarrow R } \ \frac{N ( r , a , f ) }{T ( r , f ) } , $$

where $ T ( r , f ) $ is Nevanlinna's characteristic function representing the growth of $ f $ for $ r \rightarrow R $, and

$$ N ( r , a , f ) = \int\limits _ { 0 } ^ { r } [ n ( t , a ) - n ( 0 , a ) ] d \mathop{\rm ln} t + n ( 0 , a ) \mathop{\rm ln} r $$

is the counting function; here, $ n ( t , a ) $ is the number of solutions of the equation $ f ( z) = a $ in $ | z | \leq t $( counted with multiplicity). If $ T ( r) \rightarrow \infty $ as $ r \rightarrow R $, then $ 0 \leq \delta ( a , f ) \leq 1 $ for all $ a \in \mathbf C \cup \{ \infty \} $. If $ f ( z) \neq a $ for any $ z $, then $ \delta ( a , f )= 1 $ and $ a $ is a defective value; this equality also holds in some other cases (e.g. $ f ( z) = ze ^ {z} $, $ R = \infty $ and $ a = 0 $).

If

$$ \overline{\lim\limits}\; \frac{T ( r , f ) }{ \mathop{\rm ln} ( 1 / ( R - r ) ) } = \infty $$

(or $ f \not\equiv \textrm{ const } $ is meromorphic throughout the plane), then $ \sum _ {a} \delta ( a , f ) \leq 2 $( the defect, or deficiency, relation), and the number of defective values for such $ f $ is at most countable. Otherwise, the set of defective values may be arbitrary; thus, for any sequences $ \{ a _ \nu \} \subset \mathbf C $ and $ \{ \delta _ \nu \} \subset \mathbf R ^ {+} $, $ \sum _ \nu \delta _ \nu \leq 1 $, it is possible to find an entire function $ f $ such that $ \delta ( a _ \nu , f ) = \delta _ \nu $ for all $ \nu $ and there are no other defective values of $ f $. Limitations imposed on the growth of $ f $ entail limitations on the defective values and their defects. For instance, a meromorphic function of order zero or an entire function of order $ < 1 / 2 $ cannot have more than one defective value.

The number

$$ \Delta ( a , f ) = 1 - \lim\limits _ {\overline{ {r \rightarrow \infty }}\; } \ \frac{N ( r , a , f ) }{T ( r , f ) } $$

( $ f $ meromorphic in $ \mathbf C $) is known as the defect in the sense of Valiron. The set of numbers $ a $ for which $ \Delta ( a , f ) > 0 $ may have the cardinality of the continuum, but always has logarithmic capacity zero.

See also Exceptional value; Value-distribution theory.

References

[1]	R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[2]	W.K. Hayman, "Meromorphic functions" , Oxford Univ. Press (1964)
[3]	A.A. Gol'dberg, I.V. Ostrovskii, "Value distribution of meromorphic functions" , Moscow (1970) (In Russian)