MATHEMATICS

A. A. GOL'DBERG

AN ESTIMATE OF THE SUM OF DEFECTS OF A MEROMORPHIC FUNCTION OF ORDER LESS THAN ONE

(Presented by Academician M. A. Lavrent'ev on 7 XII 1956)

Let (\delta(a)) denote the Nevanlinna defect at the point (a) of a function (f(z)), meromorphic in the finite (z)-plane (\left({}^{1}\right)). R. Nevanlinna showed that for meromorphic functions of nonintegral order (\rho), for any (a \ne b), (\delta(a)+\delta(b)) is less than a certain constant, which is less than (2) and depends only on (\rho), and gave estimates (\left({}^{2}\right)). For the case (0 \le \rho < 1), Shah (\left({}^{3}\right)), refining Nevanlinna’s estimates, showed that (\delta(a)+\delta(b) \le 1+\rho).

The purpose of the present note is to prove the inequality

[
\delta(a)+\delta(b)\le
\begin{cases}
1, & 0 \le \rho \le 1/3,\[4pt]
2-2^\rho \sqrt{\pi}\,\Gamma!\left(1-\dfrac{\rho}{2}\right)\Gamma^{-1}!\left(\dfrac{1-\rho}{2}\right), & 1/3<\rho<1,
\end{cases}
\tag{1}
]

which, as is not difficult to verify, is a sharper estimate than Shah’s estimate. The estimate (1) is probably not the best possible (for (1/3<\rho<1)), but, in any case, it does not deviate very greatly from it, since for every entire function of order (0 \le \rho < 1/2), (\delta(\infty)+\delta(0)=1), and for the entire function

[
\prod_{n=1}^{\infty} (1-zn^{-1/\rho})
]

of order (1/2 \le \rho < 1), (\delta(\infty)+\delta(0)=2-\sin \pi\rho). The graphs of the corresponding functions are shown in Fig. 1.

Fig. 1

Let us have a meromorphic function (w=f(z)) of order (\rho), (0 \le \rho < 1). Without loss of generality, we may restrict ourselves to estimating (\delta(0)+\delta(\infty)) and assume that (f(0)=1). Then

[
f(z)=\prod_{\nu}\left(1-za_\nu^{-1}\right)\cdot
\prod_{\mu}\left(1-zb_\mu^{-1}\right)^{-1},
]

where the series (\sum_{\nu}|a_\nu|^{-\lambda}), (\sum_{\mu}|b_\mu|^{-\lambda}) and the integrals

[
\int_{0}^{\infty} N(r,0) r^{-\lambda-1}\,dr,
]

[
\int_{0}^{\infty} N(r,\infty) r^{-\lambda-1}\,dr
]

converge for all (\lambda), (\rho<\lambda\le 1); from this will follow the convergence of all the series and integrals that we shall encounter below.

[
m(r,0)=\frac{1}{2\pi}\int_0^{2\pi}\ln^+\left|\frac{1}{f(re^{i\varphi})}\right|\,d\varphi
\leq \sum_\nu \frac{1}{2\pi}\int_0^{2\pi}\ln^+\left|\frac{a_\nu}{re^{i\varphi}-a_\nu}\right|\,d\varphi+
]
[
+\sum_\mu \frac{1}{2\pi}\int_0^{2\pi}\ln^+\left|\frac{re^{i\varphi}-b_\mu}{b_\mu}\right|\,d\varphi
=\sum_\nu u_\nu(r)+\sum_\mu U_\mu(r).
\tag{2}
]

Denote by (c^\nu) ((C^\mu)) the disk (|z-a_\nu|<|a_\nu|) ((|z-b_\mu|>|b_\mu|)); by (\gamma_r^\nu) ((\Gamma_r^\mu)) the arcs of the circle (|z|=r) lying in (c^\nu) ((C^\mu)); and by (\omega_\nu(r)) ((\Omega_\mu(r))) the radian measure of the arc of the circle (|z-a_\nu|=|a_\nu|) ((|z-b_\mu|=|b_\mu|)) lying inside the disk (|z|<r).

A simple calculation gives that (\omega_\nu(r)=4\arcsin(2^{-1}|a_\nu|^{-1}r)) for (0\leq r\leq 2|a_\nu|); (\Omega_\mu(r)=4\arcsin(2^{-1}|b_\mu|^{-1}r)) for (0\leq r\leq 2|b_\mu|); (\Omega_\mu(r)=2\pi) for (2|b_\mu|\leq r<\infty).

Further,
[
\frac{du_\nu(r)}{d\ln r}
=\frac{1}{2\pi}\int_{\gamma_r^\nu}\frac{\partial}{\partial r}\ln\left|\frac{a_\nu}{re^{i\varphi}-a_\nu}\right|\,r\,d\varphi
=-\frac{1}{2\pi}\int_{\gamma_r^\nu} d\arg(z-a_\nu).
]

Consequently,
[
\frac{du_\nu}{d\ln r}=\frac{\omega_\nu(r)}{2\pi}
\quad \text{for } 0\leq r<|a_\nu|;
]
[
\frac{du_\nu}{d\ln r}=\frac{\omega_\nu(r)-2\pi}{2\pi}=0
\quad \text{for } |a_\nu|<r\leq 2|a_\nu|;
]
[
\frac{du_\nu}{d\ln r}
\quad \text{for } 2|a_\nu|<r<\infty.
]

[
u_\nu(r)=
\begin{cases}
\displaystyle \frac{2}{\pi}\int_0^r \arcsin\frac{r}{2|a_\nu|}\,\frac{dr}{r}
-\ln^+\frac{r}{|a_\nu|}, & 0\leq r\leq 2|a_\nu|,\[1.2em]
0, & 2|a_\nu|\leq r<\infty.
\end{cases}
\tag{3}
]

[
u_\nu(r)+\ln^+\frac{r}{|a_\nu|}
=
\begin{cases}
\displaystyle \frac{2}{\pi}\int_0^r \arcsin\frac{r}{2|a_\nu|}\,\frac{dr}{r}, & 0\leq r\leq 2|a_\nu|,\[1.2em]
\displaystyle \ln\frac{r}{|a_\nu|}, & 2|a_\nu|\leq r<\infty.
\end{cases}
\tag{4}
]

Analogously we obtain
[
\frac{dU_\mu(r)}{d\ln r}
=\frac{1}{2\pi}\int_{\Gamma_r^\mu}\frac{\partial}{\partial r}\ln\left|\frac{re^{i\varphi}-b_\mu}{b_\mu}\right|\,r\,d\varphi
=\frac{1}{2\pi}\int_{\Gamma_r^\mu}d\arg(z-b_\mu)
=\frac{\Omega_\mu(r)}{2\pi},
]

[
U_\mu(r)=
\begin{cases}
\displaystyle \frac{2}{\pi}\int_0^r \arcsin\frac{r}{2|b_\mu|}\,\frac{dr}{r}, & 0\leq r\leq 2|b_\mu|,\[1.2em]
\displaystyle \ln\frac{r}{|b_\mu|}, & 2|b_\mu|\leq r<\infty.
\end{cases}
\tag{5}
]

Introduce the function (\chi(x)=\dfrac{2}{\pi}\int_0^x \arcsin\dfrac{x}{2}\,\dfrac{dx}{x}) for (0\leq x\leq 2); (\chi(x)=\ln x) for (2<x<\infty). It is not difficult to establish the continuity of (\chi(x)) for (0\leq x<\infty).

We also note that, since (u_\nu(r)>0) for (0<r<2|a_\nu|), it follows from (3) that

[
\frac{2}{\pi}\int_0^x \arcsin \frac{x}{2}\,\frac{dx}{x}>\ln^+x,\qquad 0<x<2.
\tag{6}
]

From (2), (4), (5) we obtain

[
T(r,f)=m(r,0)+N(r,0)=m(r,0)+\sum_\nu \ln^+\frac{r}{|a_\nu|}\leq
]

[
\leq \sum_\nu\left(u_\nu(r)+\ln^+\frac{r}{|a_\nu|}\right)+\sum_\mu U_\mu(r)
=\sum_\nu \chi\left(\frac{r}{|a_\nu|}\right)+\sum_\mu \chi\left(\frac{r}{|b_\mu|}\right).
\tag{7}
]

We shall now prove the following auxiliary inequality: for (0<a<\infty), (0<\lambda<1),

[
\varkappa(\lambda)=\Gamma\left(\frac{1-\lambda}{2}\right)2^{-\lambda}\pi^{-1/2}\Gamma^{-1}\left(1-\frac{\lambda}{2}\right)
]

[
\int_a^\infty \chi(x)x^{-\lambda-1}\,dx
<
\varkappa(\lambda)\int_a^\infty (\ln^+x)x^{-\lambda-1}\,dx.
\tag{8}
]

Similar auxiliary inequalities for obtaining various estimates have been used by many authors (see, for example, ((^{4-6}))). By a simple, though somewhat cumbersome, calculation one verifies that

[
\int_0^\infty \chi(x)x^{-\lambda-1}\,dx
=
\varkappa(\lambda)\int_0^\infty (\ln^+x)x^{-\lambda-1}\,dx.
\tag{9}
]

From (6), (9), and the definition of (\chi(x)) it follows that (\varkappa(\lambda)>1). Let

[
\Phi(a)=\int_a^\infty [\varkappa(\lambda)\ln^+x-\chi(x)]x^{-\lambda-1}\,dx
=\int_a^\infty [-\psi(x)]x^{-\lambda-1}\,dx.
]

[
\Phi'(a)=\psi(a)a^{-\lambda-1}.
]
Obviously, (\psi(a)>0) for (0<a\leq 1). Further,
[
\psi'(a)=2(\pi a)^{-1}\arcsin(a/2)-a^{-1}\varkappa(\lambda)<0
]
for (1<a\leq2), and
[
\psi'(a)=[1-\varkappa(\lambda)]a^{-1}<0
]
for (2\leq a<\infty). Consequently, for (1<a<\infty), (\psi) is strictly monotonically decreasing, and since (\psi(1)>0) and (\psi(2)=[1-\varkappa(\lambda)]\ln 2<0), there exists a unique point (a_0), (\psi(a_0)=0), (1<a_0<2), such that (\psi(a)>0) for (0<a<a_0) and (\psi(a)<0) for (a_0<a<\infty). Hence (\Phi(a)) is strictly monotonically increasing for (0<a<a_0) and strictly monotonically decreasing for (a_0<a<\infty). But (\Phi(a)\to0) as (a\to\infty), and (\Phi(0)=0) by virtue of (9); therefore (\Phi(a)>0) for all (a>0), and this is equivalent to (8).

From (8), for (\rho<\lambda<1), it follows that

[
\int_0^\infty \left[\sum_\nu \chi\left(\frac{r}{|a_\nu|}\right)+\sum_\mu \chi\left(\frac{r}{|b_\mu|}\right)\right]r^{-\lambda-1}\,dr<
]

[
<\varkappa(\lambda)\int_a^\infty\left(\sum_\nu \ln^+\frac{r}{|a_\nu|}
+\sum_\mu \ln^+\frac{r}{|b_\mu|}\right)r^{-\lambda-1}\,dr
=
]

[
=\varkappa(\lambda)\int_a^\infty [N(r,0)+N(r,\infty)]r^{-\lambda-1}\,dr.
]

Since in (10) (a) can be chosen arbitrarily large, there exists a sequence (r_k \to \infty) such that

[
\sum_{\nu}\chi\left(\frac{r_k}{|a_\nu|}\right)+\sum_{\mu}\chi\left(\frac{r_k}{|b_\mu|}\right)<\varkappa(\lambda)\,[N(r_k,0)+N(r_k,\infty)].
]

By (7),

[
T(r_k,f)<\varkappa(\lambda)[N(r_k,0)+N(r_k,\infty)];
]

[
\overline{\lim}_{r\to\infty}[N(r,0)+N(r,\infty)]/T(r,f)\geq \varkappa^{-1}(\lambda).
]

Since (\varkappa(\lambda)) is a continuous function, letting (\lambda\to\rho) we obtain

[
\overline{\lim}_{r\to\infty}[N(r,0)+N(r,\infty)]/T(r,f)\geq \varkappa^{-1}(\rho).
]

Hence it follows that (\delta(0)+\delta(\infty)\leq 2-\varkappa^{-1}(\rho)), (0\leq \rho<1). For (1/3<\rho<1) this is our desired estimate (1). For (0\leq \rho\leq 1/3), estimate (1) follows from assertion (7): if for (0\leq \rho<1/2) (\delta(a)>1-\cos\pi\rho), then (a) is the only deficient value of (f(z)); indeed, for (0\leq \rho\leq 1/3), (\max[1,2(1-\cos\pi\rho)]=1).

Uzhgorod
State University

Received
8 IX 1956

CITED LITERATURE

(^{1}) R. Nevanlinna, Univalent Analytic Functions, 1941.
(^{2}) R. Nevanlinna, Le théorème de Picard—Borel et la théorie des fonctions méromorphes, 1929.
(^{3}) S. M. Shah, Math. Student, 12, 67 (1944).
(^{4}) E. Titchmarsh, Theory of Functions, 1951, pp. 8.74.
(^{5}) G. Valiron, Mathematica, 11, 264 (1935).
(^{6}) O. Teichmüller, Deutsche Math., 4, 163 (1939).
(^{7}) A. A. Gol’dberg, DAN, 98, 893 (1954).