UDC 517.535.6

MATHEMATICS

V. P. PETRENKO

MAGNITUDES OF DEVIATIONS OF MEROMORPHIC FUNCTIONS OF LOWER ORDER LESS THAN ONE

(Presented by Academician M. A. Lavrent’ev on 29 XI 1968)

§ 1. Consider a function \(f(z)\), meromorphic for \(z \ne \infty\), of finite lower order \(\lambda\) and order \(\rho\). In [1] the properties were considered of the quantities

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

where \(a\) is an arbitrary complex number from the extended complex plane,

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

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

and \(T(r,f)\) is the Nevanlinna characteristic of \(f(z)\).

It turned out (see [1]) that the quantities \(\beta(a,f)\) for meromorphic functions of finite lower order possess a number of properties analogous to the properties of the deficiency quantities of R. Nevanlinna [2].

In particular, the following assertion is valid:

Theorem A [1]. For an arbitrary meromorphic function \(f(z)\) of finite lower order \(\lambda\), the set

\[ \Omega(f)=\{a:\ \beta(a,f)>0\} \]

is at most countable and

\[ \sum_{(a)} \beta(a,f)\le K_1(\lambda+1), \]

where \(K_1\) is an absolute constant.

Definition. We shall call the quantity \(\beta(a,f)\) the magnitude of deviation of the meromorphic function \(f(z)\) from the number \(a\). If for some number \(a\), \(\beta(a,f)>0\), then we shall say that \(f(z)\) has a positive deviation from the number \(a\).

It follows from Theorem A that the magnitudes of deviations of meromorphic functions of finite lower order are equal to zero for almost all numbers \(a\) from the extended complex plane.

§ 2. The deficiency quantities and other asymptotic properties of meromorphic functions of lower order \(\lambda<1\) have been studied by many authors [3–6].

We note the following important assertion [3, 4].

Theorem B. If \(f(z)\) is a meromorphic function of lower order \(0\le \lambda<0.5\) and, for some \(a\),

\[ \delta(a,f)>1-\cos \pi\lambda, \]

then

\[ \lim_{r\to\infty}\mu(r,a,f)=\infty, \]

\[ \mu(r,a,f)=\min_{|z|=r}\frac{1}{|f(z)-a|}. \]

It follows from this theorem that if the lower order of \(f(z)\) is \(\lambda<0.5\) and \(f(z)\) has no fewer than two deficient values, then all the magnitudes of the deficiencies are sufficiently small. Stronger results in this direction are known (see \((^3,^5)\)).

In the present work we obtain the following assertions on the quantities \(\beta(a,f)\) for meromorphic functions of finite lower order \(\lambda<1\), which are certain analogues of the preceding results.

Theorem 1. If, for a meromorphic function \(f(z)\) of finite lower order \(0\leq\lambda<0.5\), for some \(a\)

\[ \beta(a,f)>\pi\lambda\sin\pi\lambda \]

or

\[ \beta(a,f)>\pi\lambda\tan\frac{\pi\lambda}{2}\cdot(2-\delta(a,f)), \]

then

\[ \overline{\lim}_{r\to\infty}\mu(r,a,f)=\infty, \]

and, consequently,

\[ \beta(b,f)=0 \ \text{and}\ \delta(b,f)=0 \]

for all \(b\ne a\).

Remark. The deviations of \(f(z)\) from all numbers \(b\ne a\) are also equal to zero if the weaker relations \((0\leq\lambda\leq0.5)\)

\[ \beta(a,f)\geq\pi\lambda\sin\pi\lambda \]

or

\[ \beta(a,f)\geq\pi\lambda\tan\frac{\pi\lambda}{2}\cdot[2-\delta(a,f)] \]

are satisfied.

Theorem 2. If a meromorphic function \(f(z)\) of finite lower order \(\lambda<1\) has positive deviations from at least two points \(a\) and \(b\), then

\[ \beta(a,f)+\beta(b,f)\leq \pi\lambda\tan\frac{\pi\lambda}{2}\cdot[2-\delta(a,f)-\delta(b,f)], \]

\[ \beta(a,f)+\cos\pi\lambda\,\beta(b,f)\leq \pi\lambda\sin\pi\lambda\cdot[1-\delta(b,f)] \quad (0\leq\lambda<0.5). \]

The example analyzed in the work of A. Edrei and W. Fuchs \((^5)\) shows that the assertions of Theorems 1 and 2 cannot be strengthened.

§ 3. The proofs of Theorems 1 and 2 are carried out by means of the method used by us in \((^1)\). Indeed, let, for fixed \(r\), \(\theta(r)\) and \(\theta^*(r)\) be defined by the condition \((0\leq\theta(r)<2\pi,\ 0\leq\theta^*(r)<2\pi)\)

\[ |f(re^{i\theta(r)})|=M(r,\infty,f),\qquad \frac{1}{|f(re^{i\theta^*(r)})|}=M(r,0,f). \]

For fixed \(r\), together with the meromorphic function \(f(z)\) \((f(0)=1)\), consider the meromorphic functions

\[ F_r(z)=f(r^{i\theta(r)}z),\qquad F_r^*(z)=\frac{1}{f(e^{i\theta^*(r)}z)} \quad (z=se^{i\theta}). \]

For each fixed \(r\) \((r_0\leq r\leq 0.5R)\) we have (see \((^1)\))

\[ \begin{aligned} \ln M(r,\infty,f)+\ln M(r,0,f) &\le (2x)^2 r^{2x}\int_0^R \left\{\frac{1}{2\pi}\int_{-\alpha}^{\alpha}\ln |F_r(te^{i\theta})|\,d\theta+\right.\\ &\quad \left.+\frac{1}{2\pi}\int_{-\alpha}^{\alpha}\ln |F_r^*(te^{i\theta})|\,d\theta\right\} \frac{t^{2x-1}\,dt}{(t^{2x}+r^{2x})^2} +\sum_{|b_k|\le 2R}\ln\left|\frac{r^{2x}+|b_k|^{2x}}{r^{2x}-|b_k|^{2x}}\right|+\\ &\quad +\sum_{|a_k|\le 2R}\ln\left|\frac{r^{2x}+|a_k|^{2x}}{r^{2x}-|a_k|^{2x}}\right| +C_1\left(\frac{r}{R}\right)^{2x}\{T(4R,f)+T_1(4R,f)\}; \end{aligned} \tag{1} \]

where \(x=\pi/2\alpha\), \(\pi-\varepsilon<\alpha=\alpha(r)<\pi\) \((0<\varepsilon<\pi/2)\), \(a_k\) are the zeros of the meromorphic function \(f(z)\), and \(b_k\) are its poles (\(C_1\) is a positive constant).

Moreover \((0\le t\le R)\),

\[ \int_{-\pi}^{\pi}\ln |F_r(te^{i\theta})|\,d\theta+ \int_{-\pi}^{\pi}\ln |F_r^*(te^{i\theta})|\,d\theta=0. \tag{2} \]

Theorem 2 follows from estimate (1), if one uses relation (2) and carries out arguments analogous to those given in papers \((^1,^7,^8)\). Theorem 1 is proved analogously.

Theorems 1 and 2 can also be obtained by means of the method used in papers \((^3,^9)\).

Kharkov State University
named after A. M. Gorky

Received
12 XI 1968

REFERENCES

\(^{1}\) V. P. Petrenko, DAN, 184, No. 5 (1969).
\(^{2}\) R. Nevanlinna, Single-valued analytic functions, Moscow–Leningrad, 1941.
\(^{3}\) I. V. Ostrovskii, DAN, 150, No. 1, 32 (1963).
\(^{4}\) A. A. Gol’dberg, DAN, 98, No. 3, 893 (1954).
\(^{5}\) A. Edrei, W. H. J. Fuchs, Duke Math. J., 27, No. 3, 233 (1960).
\(^{6}\) G. Valiron, C. R., 230, 40 (1950).
\(^{7}\) V. P. Petrenko, DAN, 155, No. 2 (1964).
\(^{8}\) V. P. Petrenko, DAN, 158, No. 5, 1030 (1965).
\(^{9}\) A. A. Gol’dberg, I. V. Ostrovskii, Notes of the Mathematical Department of Kharkov State University and the Kharkov Mathematical Society, 27, ser. 4, 3 (1961).