MATHEMATICS

I. V. OSTROVSKII

ON A PROBLEM IN THE THEORY OF VALUE DISTRIBUTION

(Presented by Academician S. N. Bernstein, 29 I 1963)

1°. Let \(f(z)\) be a meromorphic function; \(n(r,a)\), \(N(r,a)\), \(m(r,f)\), \(T(r)\) are the quantities used in Nevanlinna theory that characterize this function. Put

\[ n(r)=n(r,0)+n(r,\infty);\qquad N(r)=N(r,0)+N(r,\infty); \]

\[ \varkappa(f)=\varliminf_{r\to\infty} N(r)[T(r)]^{-1}. \]

We shall denote by \(\rho\) the order of the function \(f(z)\), and by \(\lambda\) its lower order.

R. Nevanlinna proved \((^{3})\) that

\[ \varkappa(f)\geqslant k(\rho), \]

where \(k(\rho)\) is a quantity positive for nonintegral \(\rho\), and posed the problem of finding the best lower estimate for \(\varkappa(f)\) in terms of \(\rho\).

The first, though rather crude, estimate of \(\varkappa(f)\) for \(0\leqslant \rho\leqslant 1\) was obtained by Shah \((^{7})\), and a substantially more precise one by A. A. Gol'dberg \((^{2})\). Recently Edrei and Fuchs \((^{4,5})\) found the best estimate of \(\varkappa(f)\) for \(0\leqslant \rho\leqslant 1\) and an estimate of \(\varkappa(f)\) for \(\rho>1\) fairly close to the best possible. The result of Edrei and Fuchs may be formulated as follows:

Theorem A. Put

\[ \nu(x)= \begin{cases} 1, & 0\leqslant x\leqslant 0.5,\\ \sin \pi x, & 0.5\leqslant x\leqslant 1,\\ |\sin \pi x|\{2.2x+0.5|\sin \pi x|\}^{-1}, & 1\leqslant x\leqslant \infty. \end{cases} \tag{1} \]

The estimate

\[ \varkappa(f)\geqslant \nu(\rho) \tag{2} \]

holds.

The main result of this article is the following theorem, which generalizes Theorem A.

Theorem 1. Let \(\nu(x)\) be defined by relation (1). The estimate

\[ \varkappa(f)\geqslant \max_{\lambda\leqslant x\leqslant \rho}\nu(x) \tag{3} \]

holds.

2°. In this section we shall give two assertions (Lemmas 1 and 2) used in the proof of Theorem 1. These assertions are due to Edrei and Fuchs \((^{5})\).

Lemma 1. Let \(f(z)\), \(f(0)=1\), be a meromorphic function, and let \(q\geqslant 0\) be an integer. For any \(R>0\) and \(0\leqslant r\leqslant 0.5R\), the relation*

\[ 2T(r)-N(r)\leqslant r^{q}\int_{0}^{R} n(t)t^{-q-1}\Phi\!\left(\frac{t}{r}\right)dt +K_{1}\left(\frac{r}{R}\right)^{q+1}T(2R)+K_{2}r^{q}, \tag{4} \]

holds, where

\[ \Phi(t)=\frac{1}{2\pi}\int_{0}^{2\pi}|te^{i\theta}-1|^{-1}\,d\theta. \]

* We shall agree to denote by the letter \(K\) with indices positive quantities not depending on the variables denoted by the letters \(R,r,t,s,z\).

Lemma 1 is contained in Theorem 3b of the paper \((^5)\). Since the proof of this theorem is very cumbersome, we shall give here a comparatively simple argument which makes it possible to obtain Lemma 1.

We shall rely on the following assertion, contained implicitly in the work of Ph. Nevanlinna \((^6)\) (see also \((^1)\), p. 225).

Let \(f(z)\), \(f(0)=1\), be a meromorphic function with zeros \(a_\mu\) and poles \(b_\nu\), and let \(q \geq 0\) be an integer. For every \(R>0\) the relation

\[ f(z)=\alpha_R(z)\omega_R(z), \]

holds, where

\[ \alpha_R(z)= \prod_{|a_\mu|<R} E\left(\frac{z}{a_\mu},q\right) \left\{ \prod_{|b_\nu|<R} E\left(\frac{z}{b_\nu},q\right) \right\}^{-1} \]

(\(E(z,q)\) is the canonical Weierstrass factor of genus \(q\)), and \(\ln\omega_R(z)\) for
\(|z|\leq 0.5R\) (\(\ln\omega_R(0)=0\)) admits the estimate

\[ |\ln\omega_R(z)|\leq K_3(rR^{-1})^{q+1}T(2R)+K_4r^q \qquad (|z|=r\leq 0.5R). \tag{5} \]

To obtain this assertion, it suffices to integrate \(q+1\) times with respect to \(z\), from \(0\) to \(z\), the expression for \(\{\ln f(z)\}^{(q+1)}\) given in the book \((^1)\) (p. 225), and to take into account the estimates for \(S_R(z)\) and \(I_R(z)\).

We borrow the subsequent arguments from Edrei and Fuchs \((^5)\). Integrating with respect to \(\theta\) from \(0\) to \(2\pi\) the known inequality

\[ \left|\ln\left|E(re^{i\theta},q)\right|\right| \leq \int_0^r s^q\left|se^{i\theta}-1\right|^{-1}\,ds, \]

we obtain the estimate

\[ m(r,E(r,q))+m(r,\{E(z,q)\}^{-1}) \leq \int_0^r s^q\Phi(s)\,ds = \int_1^\infty s^{-q-1}\Phi\left(\frac{s}{r}\right)\,ds. \]

Using this estimate, we have

\[ m(r,\alpha_R)+m(r,\alpha_R^{-1}) \leq r^q\int_0^R\left\{\int_u^\infty t^{-q-1}\Phi\left(\frac{t}{r}\right)\,dt\right\}dn(u) = \]

\[ = r^q\left\{ \int_0^R n(t)t^{-q-1}\Phi\left(\frac{t}{r}\right)\,dt + n(R)\int_R^\infty t^{-q-1}\Phi\left(\frac{t}{r}\right)\,dt \right\} \leq \]

\[ \leq r^q\int_0^R n(t)t^{-q-1}\Phi\left(\frac{t}{r}\right)\,dt + K_5\left(\frac{r}{R}\right)^{q+1}T(2R) \qquad (0\leq r\leq 0.5R). \]

But, by virtue of (5),

\[ m(r,\omega_R)+m(r,\omega_R^{-1}) \leq K_6\left(\frac{r}{R}\right)^{q+1}T(2R)+K_7r^q \qquad (0\leq r\leq 0.5R). \]

It remains to note that

\[ 2T(r)-N(r)=m(r,f)+m(r,f^{-1})\leq \]

\[ \leq m(r,\alpha_R)+m(r,\alpha_R^{-1}) + m(r,\omega_R)+m(r,\omega_R^{-1}). \]

Lemma 2. Put

\[ J(\beta)=\int_0^\infty t^{-\beta-1}\Phi(t^{-1})\,dt, \qquad 0<\beta<1. \]

The estimate is valid

\[ J(\beta)\leq 4.4\,\operatorname{cosec}\pi\beta. \tag{6} \]

We note that estimate (6) is obtained from the relation established by Edrei and Fuchs (5) *

\[ J(\beta)=\pi^2 \operatorname{cosec}\pi\beta \left\{\Gamma\left(1-\frac{\beta}{2}\right)\Gamma\left(\frac{1+\beta}{2}\right)\right\}^{-2}. \tag{7} \]

(the expression in braces attains its minimum at \(\beta=0.5\)).

\(3^\circ.\) Proof of Theorem 1. Let us first make a number of remarks simplifying our problem.

We may assume that \(\lambda<\rho\), for when \(\lambda=\rho\), estimate (3) coincides with (2). Since the function \(\nu(x)\) is continuous, it suffices to prove that for every nonintegral \(x\) satisfying the condition \(\lambda<x<\rho\), the relation

\[ \varkappa(f)\geqslant \nu(x) \tag{8} \]

holds.

In the note (9) the author proved that, for \(\lambda<1\), \(\varkappa(f)\geqslant \nu(\lambda)\) holds. Since \(\nu(x)\) is nonincreasing for \(0\leqslant x\leqslant 1\), it follows that (8) holds for \(\lambda<x<\min(\rho,1)\) (and hence Theorem 1 also holds when \(\rho\leqslant 1\)). Thus our problem reduces to proving the validity of relation (8) for all nonintegral values \(x\) satisfying the condition \(\max(\lambda,1)<x<\rho\) \((>1)\).

Without loss of generality, one may assume that \(f(0)=1\). Put \(q=[x]\) in relation (4). Let \(\varkappa'\) be an arbitrary number satisfying \(\varkappa'>\varkappa(f)\). Then, for \(r\geqslant r_0\), we shall have \(N(r)\leqslant \varkappa' T(r)\), and hence

\[ (2-\varkappa')T(r)\leqslant r^q\int_0^R n(t)t^{-q-1}\Phi\left(\frac{t}{r}\right)\,dt +K_8\left(\frac{r}{R}\right)^{q+1}T(2R)+K_9 r^q \]

\[ (r_0\leqslant r\leqslant 0.5R). \]

Multiplying both sides of this relation by \(r^{-x-1}\) and integrating with respect to \(r\) from \(r_0\) to \(0.5R\), we obtain

\[ (2-\varkappa')\int_{r_0}^{0.5R} T(r)r^{-x-1}\,dr\leqslant \]

\[ \leqslant \int_{r_0}^{0.5R} r^{q-x-1} \left\{\int_0^R n(t)t^{-q-1}\Phi\left(\frac{t}{r}\right)\,dt\right\}\,dr +K_{10}T(2R)R^{-x}+K_{11}. \tag{9} \]

Denote the repeated integral standing on the right-hand side by \(I\). We have

\[ I=\int_0^R n(t)t^{-q-1} \left\{\int_{r_0}^{0.5R} r^{q-x-1}\Phi\left(\frac{t}{r}\right)\,dr\right\}\,dt= \]

\[ =\int_0^R n(t)t^{-x-1} \left\{\int_{r_0/t}^{0.5R/t} u^{q-x-1}\Phi(u^{-1})\,du\right\}\,dt \leqslant \int_0^R n(t)t^{-x-1} \left\{\int_0^\infty u^{q-x-1}\Phi(u^{-1})\,du\right\}\,dt= \]

\[ =J(x-q)\int_0^R n(t)t^{-x-1}\,dt =J(x-q)\left\{x\int_0^R N(t)t^{-x-1}\,dt+N(R)R^{-x}\right\}= \]

* Relation (7) is also obtained in the following way. \(J(\beta)\) can be represented in the form

\[ J(\beta)=\frac{1}{\pi}\int_0^1 (t^{\beta-1}+t^{-\beta}) \left\{\int_0^\pi (t^2+1-2t\cos\theta)^{-0.5}\,d\theta\right\}\,dt. \]

Then, expanding \((t^2+1-2t\cos\theta)^{-0.5}\) in a series in Legendre polynomials \(P_n(\cos\theta)\) and carrying out the integration, we find for \(J(\beta)\) a representation in the form of a series. This series (see (8), vol. 1, p. 8, formula (4)) is the Mittag-Leffler series of the function

\[ \frac{\Gamma\left(\frac{1-\beta}{2}\right)\Gamma\left(\frac{\beta}{2}\right)} {2\Gamma\left(1-\frac{\beta}{2}\right)\Gamma\left(\frac{1+\beta}{2}\right)}, \]

which coincides with the right-hand side of (7).

\[ =J(x-q)\left\{x\int_{r_0}^{0.5R} N(t)t^{-x-1}\,dt+x\int_0^{r_0} N(t)t^{-x-1}\,dt+x\int_{0.5R}^{R} N(t)t^{-x-1}\,dt+\right. \]

\[ \left.+N(R)R^{-x}\right\}\leqslant J(x-q)xx'\int_{r_0}^{0.5R}T(t)t^{-x-1}\,dt+K_{12}T(R)R^{-x}+K_{13}. \]

Substituting this estimate into (9), we obtain the inequality

\[ (2-x'-J(x-q)xx')\int_{r_0}^{0.5R}T(t)t^{-x-1}\,dt\leqslant K_{14}T(2R)R^{-x}+K_{15}. \tag{10} \]

Since \(x<\rho\), as \(R\to\infty\)

\[ \int_{r_0}^{0.5R}T(t)t^{-x-1}\,dt\to\infty. \]

Since \(x>\lambda\), there is a sequence \(R_k\uparrow\infty\) such that

\[ T(2R_k)R_k^{-x}\to 0\qquad (k\to\infty). \]

Consequently, letting \(R\to\infty\) in (10) along this sequence \(R_k\), we conclude that

\[ 2-x'-J(x-q)xx'\leqslant 0, \]

whence

\[ x'\geqslant 2\{1+xJ(x-q)\}^{-1}. \]

Since the last relation holds for any \(x'>\varkappa(f)\), it also holds for \(x'=\varkappa(f)\). To obtain (8), it remains to use relation (6) with \(\beta=x-q\).

\(4^\circ\). Theorem 1 makes it possible to shorten substantially the path to one important result (Theorem 6 of paper \((^5)\)), which is obtained in a strengthened form.

Theorem 2. Let \(f(z)\) be a meromorphic function of finite lower order \(\lambda\), and let \(p\) be an integer determined by the condition

\[ p-0.5\leqslant \lambda<p+0.5. \]

If

\[ \varkappa(f)<\beta(5e(p+1))^{-1},\qquad 0<\beta\leqslant 5, \tag{11} \]

then \(p\geqslant 1\) and

\[ p-0.1\beta<\lambda\leqslant p<p+0.1\beta. \]

In Theorem 6 of paper \((^5)\) it is asserted that, when condition (11) is fulfilled with \(0<\beta\leqslant 0.5\), the relations

\[ |\rho-p|<0.1\gamma,\qquad p-\beta\leqslant \lambda<p+0.1\beta \]

hold.

To obtain Theorem 2, it suffices to note that the set
\(Q_p=\{x:\nu(x)<\beta(5e(p+1))^{-1}\}\) contains no points of the interval
\(\{x:0\leqslant x\leqslant 0.5\}\), and to estimate the length of the interval
\(Q_p\cap\{x:p-0.5\leqslant x\leqslant p+0.5\}\).

I express my gratitude to A. A. Goldberg for his attention to the work and for valuable comments.

Kharkov State University
named after A. M. Gorky

Received
24 I 1963

References

\(^{1}\) R. Nevanlinna, Single-Valued Analytic Functions, Moscow–Leningrad, 1941.
\(^{2}\) A. A. Goldberg, DAN, 114, No. 2, 245 (1957).
\(^{3}\) R. Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes, Paris, 1929.
\(^{4}\) A. Edrei, W. H. J. Fuchs, Duke Math. J., 27, No. 2, 233 (1960).
\(^{5}\) A. Edrei, W. H. J. Fuchs, Trans. Am. Math. Soc., 93, No. 2, 292 (1959).
\(^{6}\) F. Nevanlinna, Soc. Sci. Fenn. Comm. Phys.-Math., 2, No. 4, 1 (1923).
\(^{7}\) S. M. Shah, Math. Student, 12, 67 (1944).
\(^{8}\) H. Bateman, Higher Transcendental Functions, N. Y., 1953.
\(^{9}\) I. V. Ostrovskii, DAN, 150, No. 1 (1963).