Reports of the Academy of Sciences of the USSR
1964. Vol. 158, No. 5

MATHEMATICS

V. P. PETRENKO

ON THE MAGNITUDES OF THE DEFECTS OF A MEROMORPHIC FUNCTION

(Presented by Academician M. A. Lavrent’ev on 15 VI 1964)

§ 1. Let \(f(z)\) be a function meromorphic in the open plane; \(T(r,f)\), \(m(r,a)\), \(n(r,a)\), \(N(r,a)\), \(\delta(a,f)\) are the quantities introduced by R. Nevanlinna, characterizing the distribution of the values of this function. We agree to denote
\[ n(r)=n(r,0)+n(r,\infty),\quad N(r)=N(r,0)+N(r,\infty), \]
by the letter \(K\) with indices absolute constants, and by the letter \(C\) with indices quantities depending only on the function under consideration.

As is known \((^{1})\), the defect quantities satisfy the relation
\[ \sum_{(a)} \delta(a)\leqslant 2 \]
(the sum is extended over all values \(a\) with \(\delta(a,f)>0\)). Put
\[ s(f)=\sum_{(a)} \sqrt{\delta(a)},\quad \sigma(\lambda)=\sup_{(f)} s(f), \]
where the supremum is taken over all meromorphic functions of lower order \(\lambda\).

B. Fuchs \((^{2})\) proved that for \(\lambda<\infty\) the quantity \(\sigma(\lambda)\) is finite and that the estimate
\[ \sigma(\lambda)\leqslant K_1\left(1+\sqrt{\lambda\,|\ln\lambda|}\right). \tag{1,1} \]
is valid.

The main result of this article is the following theorem.

Theorem 1. The inequality
\[ \sigma(\lambda)\leqslant K_2\sqrt{\lambda},\quad 0.5\leqslant \lambda<\infty. \tag{1,2} \]
is valid.

For the entire function \(((^{1}),\ p. 240)\)
\[ h_p=\int_0^z e^{t^p}\,dt,\quad p=1,2,3,\ldots, \]
one has \(\lambda=p\),
\[ s(h_p)=1+\sqrt{p}=1+\sqrt{\lambda}. \]
Therefore estimate (1,2) is sharp for large \(\lambda\) in the sense of order.

We obtain Theorem 1 as a consequence of two theorems: Theorem 2, due to Fuchs \((^{2})\), and Theorem 3, established by us.

Theorem 2 \((^{2})\). If \(f(z)\) is a meromorphic function of finite lower order \(\lambda\) and has at least two deficient values, then the relation
\[ \sum_{(a)} \sqrt{\delta(a)}\leqslant \left\{2\pi \lim_{r\to\infty}[T(r,f)]^{-1} r\,\mathfrak M\left(r,\frac{f''}{f'}\right)\right\}^{1/2}, \tag{1,3} \]
is valid, where
\[ \mathfrak M(r,g)=\frac{1}{2\pi}\int_0^{2\pi}\left|g\left(re^{i\theta}\right)\right|\,d\theta. \]

Theorem 3. If \(f(z)\) is a meromorphic function of lower order \(\lambda\) \((\lambda\geqslant 0.5)\), then
\[ \lim_{r\to\infty}[T(r,f)]^{-1}r\,\mathfrak M\left(r,\frac{f''}{f'}\right)\leqslant K_3\lambda. \tag{1,4} \]

§ 2. Auxiliary propositions.
Lemma 1. Let \(f(z)\) be a function meromorphic in the sector
\[ G_{a,R,\vartheta}=\{z:\ 0<|z|<R,\ |\arg z-\vartheta|<a\}. \]

For any real \(\vartheta\), any \(\alpha\) \((0<\alpha<\pi)\), and \(r\) \((0<r_0<r<0.5R)\), the inequality holds

\[ \begin{aligned} \frac{r}{2\pi}\int_{-\alpha/2}^{\alpha/2} \left|\frac{f'\left(re^{i(\varphi+\vartheta)}\right)} {f\left(re^{i(\varphi+\vartheta)}\right)}\right|\,d\varphi &\le \alpha^{-1}\int_{r_0}^{R} \left\{\left|\ln\left|f\left(te^{i(\vartheta+\alpha)}\right)\right|\right| +\left|\ln\left|f\left(te^{i(\vartheta-\alpha)}\right)\right|\right|\right\} P(t,r,\alpha)\,dt \\ &\quad +K_4\alpha^{-1}\int_{0}^{R} \left\{\left|\ln\left|f\left(te^{i(\vartheta+\alpha)}\right)\right|\right| +\left|\ln\left|f\left(te^{i(\vartheta-\alpha)}\right)\right|\right|\right\} \left(\frac{r}{R^2}\right)^x t^{x-1}\,dt \\ &\quad +K_5\alpha^{-1}\int_{-\alpha}^{\alpha} \left|\ln\left|f\left(Re^{i(\vartheta+\theta)}\right)\right|\right| \left(\frac{r}{R}\right)^x\,d\theta \\ &\quad +2\sum_{c_m\in G_{\alpha,R,0}} \left(\frac{r}{|c_m|}\right)^x \Phi\left[\left(\frac{r}{|c_m|}\right)^x\right] \\ &\quad +K_6\sum_{c_m\in G_{\alpha,R,0}}' \left(\frac{r}{R^2}\right)^x |c_m|^x+C_1, \end{aligned} \tag{2.1} \]

where

\[ x=x(\alpha)=\pi(2\alpha)^{-1},\qquad P(t,r,\alpha)=t^{x-1}r^x(t^x+r^x)^{-1},\qquad \Phi(u)=\frac1{2\pi}\int_{0}^{2\pi}\frac{d\theta}{|ue^{i\theta}-1|}, \]

\(c_m=c_m(\vartheta)\) are the zeros and poles of the meromorphic function \(f(ze^{i\vartheta})\).

The proof is based on the representation of \(\ln f(re^{i\theta})\) in the sector \(G_{\alpha,R,0}\), analogous to the Schwarz–Nevanlinna formula ((1), p. 165).

Lemma 2. Let \(f(z)\) be a function meromorphic in \(|z|\le R<\infty\). For \(0<r_0<r<0.5R\), \(0<\alpha<\pi\), the estimate holds

\[ \begin{aligned} r\mathfrak m\left(r,\frac{f'}{f}\right) &\le K_7\alpha^{-2}\int_{r_0}^{R}\{m(t,0)+m(t,\infty)\}P(t,r,\alpha)\,dt \\ &\quad +K_8\alpha^{-1}\left(\frac{r}{R}\right)^x T(R,f) +K_9\sum_{|c_m|<R} \left(\frac{r}{|c_m|}\right)^x \Phi\left[\left(\frac{r}{|c_m|}\right)^x\right] \\ &\quad +K_{10}\left(\frac{r}{R}\right)^x n(R)+C_2\alpha^{-1}. \end{aligned} \tag{2.2} \]

Proof. Let \(\vartheta_k=\beta+k\alpha\), where \(0\le\beta<2\pi\), and \(k\) takes the values \(0,1,\ldots,q=[4x]\). Putting \(\vartheta=\vartheta_k\) \((k=0,1,\ldots,q)\) in inequality (2.1), we obtain \(q+1\) inequalities. Adding these inequalities over \(k\) from \(0\) to \(q\), we shall have

\[ \begin{aligned} r\mathfrak m\left(r,\frac{f'}{f}\right) &\le \alpha^{-1}\sum_{k=0}^{q}\int_{r_0}^{R} \left\{ \left|\ln\left|f\left(te^{i(\beta+(k+1)\alpha)}\right)\right|\right| + \left|\ln\left|f\left(te^{i(\beta+(k-1)\alpha)}\right)\right|\right| \right\} P(t,r,\alpha)\,dt \\ &\quad +K_4\alpha^{-1}\sum_{k=0}^{q} \left\{ \left|\ln\left|f\left(te^{i(\beta+(k+1)\alpha)}\right)\right|\right| + \left|\ln\left|f\left(te^{i(\beta+(k-1)\alpha)}\right)\right|\right| \right\} \left(\frac{r}{R^2}\right)^x t^{x-1}\,dt \\ &\quad +K_5\alpha^{-1}\sum_{k=0}^{q}\int_{-\alpha}^{\alpha} \left|\ln\left|f\left(Re^{i(\theta+\beta+k\alpha)}\right)\right|\right| \left(\frac{r}{R}\right)^x\,d\theta \\ &\quad +2\sum_{k=0}^{q}\sum_{c_m(\vartheta_k)\in G_{\alpha,R,0}} \left(\frac{r}{|c_m|}\right)^x \Phi\left[\left(\frac{r}{|c_m|}\right)^x\right] \\ &\quad +K_6\sum_{k=0}^{q}\sum_{c_m(\vartheta_k)\in G_{\alpha,R,0}} \left(\frac{r}{R^2}\right)^x |c_m|^x C_2\alpha^{-1}. \end{aligned} \tag{2.3} \]

Obviously, the inequalities hold

\[ 2\sum_{k=0}^{q}\sum_{c_m(\vartheta_k)\in G_{\alpha,R,0}} \left(\frac{r}{|c_m|}\right)^x \Phi\left[\left(\frac{r}{|c_m|}\right)^x\right] \le K_9\sum_{|c_m|\le R} \left(\frac{r}{|c_m|}\right)^x \Phi\left[\left(\frac{r}{|c_m|}\right)^x\right], \tag{2.4} \]

\[ \sum_{k=0}^{q}\int_{-\alpha}^{\alpha}\left|\ln \left|f\left(Re^{i(\theta+\beta+k\alpha)}\right)\right|\right|\,d\theta \leqslant K_8 T(R,f) \tag{2,5} \]

\[ \sum_{k=0}^{q}\sum_{c_m(\vartheta_k)\in G_{\alpha,R,0}} \left(\frac{r}{R^2}\right)^{\chi}|c_m|^\chi \leqslant K_{11}\sum_{|c_m|\leq R} \left(\frac{r}{R^2}\right)^{\chi}|c_m|^\chi . \tag{2,6} \]

Replacing in inequality (2,3) the expressions occurring on the left-hand sides of (2,4), (2,5), and (2,6) by the expressions occurring on the right-hand sides, and integrating the resulting inequality with respect to \(\beta\) from \(0\) to \(2\pi\), we obtain inequality (2,2).

§ 3. Proof of Theorem 3. Let \(f(z)\) have lower order \(\lambda\) and order \(\rho\). We shall carry out the proof under the assumption that \(\lambda<\rho\). Choose \(\gamma\) so that \(\lambda<\gamma<\rho\), and take \(\alpha<\pi(2\gamma)^{-1}\). Divide inequality (2,2) by \(r^{\gamma+1}\) and integrate it with respect to \(r\) from \(r_0\) to \(0.5R\); we obtain

\[ \begin{aligned} \int_{r_0}^{0.5R} r^{-\gamma-1}\left\{r\mathfrak{M}\left(r,\frac{f'}{f}\right)\right\}\,dr &\leqslant K_7\alpha^{-2} \int_{r_0}^{0.5R}\{m(t,0)+m(t,\infty)\}\times \\ &\quad\times \int_{r_0}^{0.5R} r^{-\gamma-1}P(t,r,\alpha)\,dr\,dt +K_7\alpha^{-2}\int_{0.5R}^{R}\{m(t,0)+m(t,\infty)\}\times \\ &\quad\times \int_{r_0}^{0.5R} r^{-\gamma-1}P(t,r,\alpha)\,dr\,dt +K_{11}(\pi-2\alpha\gamma)^{-1}R^{-\gamma}T(R,t)+ \\ &\quad+K_9\sum_{r_0<|c_m|<0.5R}\int_{r_0}^{0.5R} r^{-\gamma-1}\left\{\left(\frac{r}{|c_m|}\right)^\chi \Phi\left[\left(\frac{r}{|c_m|}\right)^\chi\right]\right\}\,dr \\ &\quad+K_{12}(\pi-2\alpha\gamma)^{-1}R^{-\gamma}n(R)+ \\ &\quad+K_9\sum_{0.5R\leq |c_m|<R}\int_{r_0}^{0.5R} r^{-\gamma-1}\left\{\left(\frac{r}{|c_m|}\right)^\chi \Phi\left[\left(\frac{r}{|c_m|}\right)^\chi\right]\right\}\,dr +C_4\alpha^{-1}. \end{aligned} \tag{3,1} \]

Using the relations (see \((^{10},\, ^3)\))

\[ \int_{0}^{\infty} r^{-\gamma-1}P(t,r,\alpha)\,dr = t^{-\gamma-1}\alpha\sec\alpha\gamma, \qquad \int_{0}^{\infty} r^{-\sigma}\Phi(r)\,dr \leqslant 4.4\cosec \pi\sigma \quad (0<\sigma<1), \tag{3,2} \]

from (3,1) we obtain

\[ \begin{aligned} \sin 2\alpha\gamma \int_{r_0}^{0.5R} r^{-\gamma-1} \left\{r\mathfrak{M}\left(r,\frac{f'}{f}\right)\right\}\,dr &\leqslant 2K_7\alpha^{-1}\sin\alpha\gamma \int_{r_0}^{0.5R} r^{-\gamma-1}\{m(r,0)+m(r,\infty)\,dr\} + \\ &\quad+2K_{13}\alpha^{-1}\sin\alpha\gamma\,R^{-\gamma}T(R,t) +K_{11}(\pi-2\alpha\gamma)^{-1}\sin 2\alpha\gamma\,R^{-\gamma}T(R,f) + \\ &\quad+K_{14}\alpha\gamma^{2}\int_{r_0}^{0.5R} r^{-\gamma-1}N(r)\,dr +C_5R^{-\gamma}T(2R,f)+C_6 . \end{aligned} \tag{3,3} \]

Choose in this inequality \(\alpha=\pi(4\gamma)^{-1}\). Applying it then to \(f'(z)\) instead of \(f(z)\) (this can be done, since the order and lower order of \(f(z)\) and \(f'(z)\) coincide) (\((^4)\), p. 52) and taking into account the relation (\((^5)\), p. 61)

\[ T(r,f')\leqslant 2T(r,f)+4\ln^+T(2r,f)+4\ln^+ r+K_{15} \qquad (0<r_0<r), \]

from (3,3) we find

\[ \int_{r_0}^{0.5R} r^{-\gamma-1} \left\{r\mathfrak{M}\left(r,\frac{f''}{f'}\right)\right\}\,dr \leqslant K_{16}\gamma\int_{r_0}^{0.5R} r^{-\gamma-1}T(r,f)\,dr+ \]

\[ +C_7\int_{r_0}^{0.5R} r^{-\gamma-1}\ln^+T(r,f)\,dr +C_8R^{-\gamma}\{T(4R,f)+\ln^+T(4R,f)\}+C_8 . \]

From this inequality, by means of arguments analogous to those used in \((^9,{}^7,{}^{10})\), we obtain

\[ \lim_{r\to\infty}[T(r,f)]^{-1}\left\{r\mathfrak{M}\left(r,\frac{f''}{f'}\right)\right\}\leqslant K_{16}\gamma . \]

Letting now \(\gamma\) tend to \(\lambda\), we obtain the assertion of the theorem.

For \(\lambda=\rho\) the proof of Theorem 3 is carried out as follows. Divide inequality (2.2) with \(\alpha=\pi(4\rho)^{-1}\) by \(r^{\rho(r)+1}\), where \(\rho(r)\) is the refined order of the meromorphic function \(f(z)\), and integrate with respect to \(r\) from \(r_0\) to \(0.5R\). Arguing further analogously to the case \(\lambda<\rho\), but in estimating the integrals (3.2) using Lemma 2 from \((^6)\) (p. 78), we arrive at the relation

\[ \int_{r_0}^{0.5R} r^{-\rho(r)-1} \left\{r\mathfrak{M}\left(r,\frac{f''}{f'}\right)\right\}\,dr \leqslant K_{18}\rho\int_{r_0}^{0.5R} r^{-\rho(r)-1}T(r,f)\,dr+ \]

\[ +\,C_{10}\int_{r_0}^{0.5R} r^{-\rho(r)-1}\ln^+T(r,f)\,dr +C_{11}R^{-\rho(R)}\{T(4R,f)+\ln^+T(4R,f)\}+C_{12}. \]

Taking into account the properties of the refined order, we have

\[ \lim_{R\to\infty} R^{-\rho(R)}T(R,f)=1,\qquad \lim_{R\to\infty}\int_{r_0}^{0.5R} r^{-\rho(r)-1}T(r,f)\,dr=\infty . \]

Using these relations and arguing analogously (cf. \((^9,{}^7,{}^{10})\)) to the case \(\lambda<\rho\), we obtain the assertion of the theorem.

§ 4. Theorem 4. Let \(\Delta\) be the sum of the deficiencies of a meromorphic function \(f(z)\) of finite lower order \(\lambda\). Then \(f(z)\) has at least one deficiency satisfying the condition

\[ \delta(a)> \Delta^2(4K_2^2\lambda)^{-1}. \]

For the proof we use Theorem 1 and arguments analogous to \((^2)\) (p. 209).

§ 5. Remark. In \((^7)\) we obtained the following result, supplementing Theorem 1.

Theorem 5. The estimate

\[ \sigma_1(\lambda)\leqslant K_{19}\sqrt{\lambda},\qquad 0<\lambda<0.5, \tag{5.1} \]

is valid, where \(\sigma_1(\lambda)=\sup_{(f)} S(f)\), the supremum being taken over all meromorphic functions of lower order \(\lambda\) having at least two deficient values.

With the aid of the method of \((^2)\), from this theorem we obtain the following result.

Theorem 6. If a meromorphic function \(f(z)\) of lower order \(\lambda\) \((\lambda<0.5)\) has at least two deficient values, then

\[ \sum_{(a)}\delta(a)\leqslant K_{20}\lambda^{3/2}. \tag{5.2} \]

This theorem strengthens the result of Edrei and Fuchs \((^8)\). We have not been able to establish the sharpness of the estimates (5.1) and (5.2). We suppose that in (5.2) the exponent \(3/2\) can be replaced by \(2\).

Kharkov State University
named after A. M. Gorky

Received
8 VI 1964

CITED LITERATURE

1. R. Nevanlinna, Single-Valued Analytic Functions, Moscow–Leningrad, 1941.
2. W. H. J. Fuchs, Ann. Math., 68, No. 2, 203 (1958).
3. A. Edrei, W. H. J. Fuchs, Trans. Am. Math. Soc., 93, No. 2, 292 (1959).
4. G. Wittich, Recent Investigations on Single-Valued Analytic Functions, Moscow, 1960.
5. R. Nevanlinna, Le théorème de Picard—Borel et la théorie des fonctions méromorphes, Paris, 1929.
6. M. A. Evgrafov, Asymptotic Estimates and Entire Functions, 2nd ed., Moscow, 1962.
7. V. P. Petrenko, Izv. AN ArmSSR, ser. phys.-mat. sciences, 17, No. 1 (1964).
8. A. Edrei, W. H. J. Fuchs, Duke Math. J., 27, No. 3, 233 (1960).
9. I. V. Ostrovskii, DAN, 151, No. 1, 34 (1963).
10. V. P. Petrenko, DAN, 155, No. 2 (1964).