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