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)