UDC 517:535.6

MATHEMATICS

V. P. PETRENKO

ON THE STRUCTURE OF THE SET OF POSITIVE DEVIATIONS OF MEROMORPHIC FUNCTIONS

(Presented by Academician M. A. Lavrentiev on 29 IV 1969)

§ 1. In order to study deeper asymptotic properties of functions meromorphic in (|z|<R), the papers ((^{1-3})) introduced the notion of the magnitude of deviation of a meromorphic function (f(z))

[
\beta(a,f)=\lim_{r\to R}\frac{\ln^{+} M(r,a,f)}{T(r,f)},
]

where

[
M(r,a,f)=\max_{|z|=r}\frac{1}{|f(z)-a|},\quad a\ne\infty,
]

[
M(r,\infty,f)=\max_{|z|=r}|f(z)|,
]

and (T(r,f)) is the Nevanlinna characteristic of the function (f(z)) meromorphic in (|z|<R).

The results obtained on the properties of the quantities of deviations of (f(z)) show that the distribution of values of meromorphic functions can be studied using not only the notion of the defect at the point (a) in the sense of Nevanlinna (see ((^{4})))

[
\delta(a,f)=\lim_{r\to R}\frac{m(r,a,f)}{T(r,f)}
]

and the notion of the defect of (f(z)) at the point (a) in the sense of Valiron

[
\Delta(a,f)=\overline{\lim}_{r\to R}\frac{m(r,a,f)}{T(r,f)},
]

but also the notion of the magnitude of deviation of (f(z)) at the point (a).

Let, for a function (f(z)) meromorphic in (|z|<R) and of lower order* (\lambda) (see ((^{4})), p. 267),

[
D_{R,\lambda}(f)={a:\delta(a,f)>0},
]

[
\Omega_{R,\lambda}(f)={a:\beta(a,f)>0},
]

[
V_{R,\lambda}(f)={a:\Delta(a,f)>0}.
]

The set (D_{R,\lambda}(f)) is called the set of defective values of (f(z)) in the sense of Nevanlinna; the set (\Omega_{R,\lambda}(f)) is naturally called the set of positive deviations of (f(z)), and the set (V_{R,\lambda}(f)) is called the set of defective values of (f(z)) in the sense of Valiron.

Fundamental investigations of the structure of the sets (D_{R,\lambda}(f)) and (V_{R,\lambda}(f)) were carried out in ((^{4-8})).

[
\text{* Recall that the lower order of a meromorphic function in } |z|<R \text{ is called}
]

[
\lambda=\lim_{r\to R}\frac{\ln T(r,f)}{\ln\frac{1}{R-r}},\quad R<\infty,\qquad
\lambda=\lim_{r\to\infty}\frac{\ln T(r,f)}{\ln r},\quad R=\infty.
]

This work is devoted to the study of the structure of the set (\Omega_{R,\lambda}(f)). The results obtained show that, despite possible substantial differences between the sets (\Omega_{R,\lambda}(f)), (D_{R,\lambda}(f)), and (V_{R,\lambda}(f)), nevertheless in most cases the set (\Omega_{R,\lambda}(f)), just like the sets (D_{R,\lambda}(f)) and (V_{R,\lambda}(f)), is an exceptional set for (f(z)).

§ 2. Main results of the work.

Theorem 1 (see ((^1,^2))). For (\lambda<\infty), the set (\Omega_{\infty,\lambda}(f)) is at most countable, and

[
D_{\infty,\lambda}(f)\subseteq \Omega_{\infty,\lambda}(f)\subseteq V_{\infty,\lambda}(f).
]

The following three theorems characterize the differences between the deviations of meromorphic functions (\beta(a,f)) and the values of their Nevanlinna defects (\delta(a,f)) and Valiron defects (\Delta(a,f)).

Theorem 2. The sets (\Omega_{\infty,\infty}(f)) and (\Omega_{\infty,\infty}(f)\setminus V_{\infty,\infty}(f)) may have the cardinality of the continuum.

Theorem 3. There exists a set (C) of the cardinality of the continuum and an entire function (G(z)) of infinite lower order such that (\beta(a,G)=\infty) for every (a\in C).

Theorem 4. For every (\rho), (0\leq \rho\leq\infty), there exists a set (C) of the cardinality of the continuum and a function (g_\rho(z)), meromorphic for (|z|<1), of order (\rho), such that (\beta(a,g_\rho)=\infty) for every (a\in C).

Corollary 1. For any (\rho\geq 0), the set (\Omega_{1,\rho}(f)) may have the cardinality of the continuum.

Let us note that for (\rho=0) Theorem 4 follows from the consideration of functions of bounded type (see ((^4)), p. 210).

Next, for a function (f(z)) meromorphic in (|z|<R), for (0<\alpha\leq 1), put

[
\beta_{\alpha}(a,f)=\lim_{r\to R}\frac{\ln^{+} M(r,a,f)}{T^{\alpha}(r,f)}
]

[
\Omega_{R,\lambda}^{(\alpha)}(f)={a:\beta_{\alpha}(a,f)>0}.
]

The following assertions are certain analogues of the well-known Ahlfors–Nevanlinna theorem (see ((^5)), p. 17; ((^4)), p. 281) on the structure of the set (V_{R,\lambda}(f)). In what follows, by (E) we denote a bounded set in the (a)-plane having positive capacity (C(E)), and by (\mu(a)) the distribution of unit mass on (E) solving Robin’s problem for (E) (see ((^4)), pp. 123, 135).

Theorem 5. For a function (f(z)) meromorphic at (z\ne\infty), the following assertions are valid:

1) For every (\alpha>1/2)

[
\int_E \beta_{\alpha}(a,f)\,d\mu(a)=0;
]

2) if (E\subset {|a|\leq 1}), (C(E)) is the capacity of the set (E), and (\lambda) is the lower order of (f(z)) ((f(0)=1)), then

[
\int_E \beta_{1/2}(a,f)\,d\mu(a)\leq K_1(1+\lambda^2)\left[1+\ln^{+}\frac{1}{C(E)}\right].
]

Corollary 2. For every (\alpha>1/2), the set (\Omega_{\infty,\lambda}^{(\alpha)}(f)) has inner capacity zero ((\Omega_{\infty,\lambda}^{(1)}(f)=\Omega_{\infty,\lambda}(f),\ \lambda\geq 0)).

Theorem 6. If (f(z)) is meromorphic for (|z|<1) and has lower order (\lambda), then for every (\alpha>1/2+3/\lambda) ((\lambda>0))

[
\int_E \beta_{\alpha}(a,f)\,d\mu(a)=0.
]

Corollary 3. If $\alpha > 1/2 + 3/\lambda$, then the set $\Omega_{1,\lambda}^{(\alpha)}(f)$ has inner capacity zero.

§ 3. Theorem 1 was proved in works (¹, ²).

Theorems 2, 3, and 4 are proved by means of a modification of known constructions (⁶–¹⁰).

Theorems 5 and 6 are proved by means of a modification of a method previously used by the author (¹, ²).

Kharkov State University
named after A. M. Gorky

Received
22 IV 1969

REFERENCES

¹ V. P. Petrenko, Izv. Akad. Nauk SSSR, Ser. Mat., 33, No. 2 (1969).
² V. P. Petrenko, DAN, 184, No. 5 (1969).
³ V. P. Petrenko, DAN, 187, No. 1 (1969).
⁴ R. Nevanlinna, Univalent Analytic Functions, Moscow, 1941.
⁵ L. Ahlfors, Soc. Sci. Fenn. Comm. Phys.-Math., 5, No. 16, 1 (1931).
⁶ A. A. Gol'dberg, DAN, 98, 893 (1954).
⁷ A. A. Gol'dberg, Ukr. Mat. Zh., 11, No. 4, 438 (1959).
⁸ G. Valiron, Rend. Circolo Mat. Palermo, 43, 255 (1919).
⁹ V. K. Hayman, Meromorphic Functions, Moscow, 1966.
¹⁰ W. H. J. Fuchs, W. K. Hayman, Studies in Mathematical Analysis and Related Topics, Essays in Honour of George Pólya, Stanford, 1962.