Exceptional value

A concept in value-distribution theory. Let $f(z)$ be a meromorphic function in the whole $z$-plane and let $n(r,a,f)$ denote its number of $a$-points (counting multiplicities) in the disc $|z|\leq r$. According to R. Nevanlinna's first fundamental theorem (cf. [1]), as $r\to\infty$,

$$N(r,a,f)+m(r,a,f)=T(r,f)+O(1),$$

where $T(r,f)$ is the characteristic function, which does not depend on $a$, $N(r,a,f)$ is the counting function (the logarithmic average of $n(r,a,f)$) and $m(r,a,f)>0$ is a function expressing the average proximity of the values of $f$ to $a$ on $|z|=r$ (cf. Value-distribution theory). For the majority of values $a$ the quantities $N(r,a,f)$ and $T(r,f)$ are asymptotically equivalent, as $r\to\infty$. A (finite or infinite) number $a$ is called an exceptional value if this equivalence as $r\to\infty$ is violated. One distinguishes several kinds of exceptional values.

A number $a$ is called an exceptional value of $f$ in the sense of Poincaré if the number of $a$-points of $f$ in the whole plane is finite (cf. [1], [2]), in particular if $f(z)\neq a$ for any $z$.

A number $a$ is called an exceptional value of $f$ in the sense of Borel if $n(r,a,f)$ grows slower, in a certain sense, than $T(r,f)$, as $r\to\infty$ (cf. [1], [2]).

A number $a$ is called an exceptional value of $f$ in the sense of Nevanlinna (cf. [1]) if its defect (cf. Defective value)

$$\delta(a,f)=1-\lim_{r\to\infty}\sup\frac{N(r,a,f)}{T(r,f)}>0.$$

A number $a$ is called an exceptional value of $f$ in the sense of Valiron if

$$\Delta(a,f)=1-\lim_{r\to\infty}\inf\frac{N(r,a,f)}{(T(r,f)}>0.$$

A number $a$ for which

$$\beta(a,f)=\lim_{r\to\infty}\inf\frac{\max\limits_{|z|=r}\ln^+1/|f(z)-a|}{T(r,f)}>0$$

is also called an exceptional value for $f$. The quantity $\beta(a,f)$ (the positive deviation of $f$) characterizes the rate of the asymptotic approximation of $f(z)$ to $a$ (cf. [3]).

References

[1]	R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[2]	A.A. Gol'dberg, I.V. Ostrovskii, "Value distribution of meromorphic functions" , Moscow (1970) (In Russian)
[3]	V.P. Petrenko, "Growth of meromorphic functions of finite lower order" Math. USSR Izv. , 3 : 2 (1969) pp. 391–432 Izv. Akad. Nauk SSSR Ser. Mat. , 33 : 2 (1969) pp. 414–454

Comments

An $a$-point of $f$ is a point $z$ such that $f(z)=a$.