UDC 517.535.4

MATHEMATICS

N. U. ARAKELYAN

ENTIRE FUNCTIONS OF FINITE ORDER WITH AN INFINITE SET OF DEFICIENT VALUES

(Presented by Academician M. V. Keldysh on 11 VII 1966)

A well-known conjecture of R. Nevanlinna states \((^{1,2})\) that an entire function of finite order \(\rho\) has only a finite number (not more than \([2\rho]+1\)) of deficient values. For the case \(\rho \leq 1/2\) this conjecture has been confirmed. Edrei and Fuchs showed, in particular \((^3)\), that an entire function of order \(\rho \leq 1/2\) cannot have finite deficient values. It turns out that in the case \(\rho > 1/2\) R. Nevanlinna’s conjecture is false. Moreover, the following is true.

Theorem. For an arbitrary sequence of complex numbers \(\{a_k\}_1^\infty\) and for any \(\rho > 1/2\), there exists an entire function of order \(\rho\) and of normal type for which the numbers \(a_k\), \(k=1,2,\ldots\), are deficient.

\(1^\circ\). The proof of this theorem is based on the following two lemmas from approximation theory.

Lemma 1. Let \(L\) denote a rectifiable curve joining the points \(a\) and \(b\). Then for arbitrary numbers \(\varepsilon > 0\) and \(d > 0\) there exists a polynomial \(P(z)\) such that

\[ \left|\frac{1}{a-z}-P\left(\frac{1}{z-b}\right)\right|<\varepsilon \tag{1} \]

for all points \(z\) lying outside the \(d\)-neighborhood of the curve \(L\), and

\[ \left|P\left(\frac{1}{z-b}\right)\right|< \exp\left[(1+|\ln \varepsilon d|)\exp\frac{5\,\operatorname{length} L}{d}\right] \tag{2} \]

outside the circle \(|z-b|<d\).

Lemma 2. Let \(1/2<\rho\leq 1\), \(0<\alpha<\dfrac{\pi}{\rho}(\rho-1/2)\), and let the function \(F(z)\) be analytic in the angle \(|\arg z|\leq \beta/2\) \((\beta>\alpha)\) and satisfy \(|F(z)|<\exp[\operatorname{const}(|z|+1)^\rho]\). Then for any \(\varepsilon>0\) there exists an entire function \(G(z)\) of order \(\rho\) and of normal type satisfying the inequality \(|F(z)-G(z)|<\varepsilon\exp(-|z|^\rho)\), \(|\arg z|\leq \alpha/2\).

We note that Lemma 1 is a special case of an important lemma of M. V. Keldysh \((^4)\), while Lemma 2 is an immediate consequence of a theorem of M. V. Keldysh \((^5)\) on approximation in an angle by entire functions of restricted growth.

\(2^\circ\). In proving the theorem it is sufficient to restrict ourselves to the case when \(1/2<\rho\leq 1\). Indeed, if \(\rho>1\), choose a natural number \(N\) such that \(\rho\leq N<2\rho\). Having constructed an entire function \(G(z)\) of order \(\rho/N\) with deficient values \(\{a_k\}_1^\infty\), we may then consider the entire function \(G(z^N)\) of order \(\rho\) with the same defects.

\(3^\circ\). Now take an arbitrary number \(\alpha\), \(0<\alpha<\dfrac{\pi}{\rho}(\rho-1/2)\), and a sequence \(\{\theta_k\}_{-\infty}^{+\infty}\) such that \(\theta_{-k}=-\theta_k\), \(\theta_k\uparrow+\infty\) as \(k\uparrow+\infty\). Denote by \(\Delta_k(\varepsilon)\) the angle \(|\arg z-\theta_k|\leq \varepsilon\alpha_k\), where \(\alpha_k=\min(\theta_{k+1}-\theta_k,\theta_k-\theta_{k-1})\), by \(\sigma_n(\delta)\) \((\delta\geq0)\) the annulus \((1-\delta)2^n\leq |z|\leq(1+\delta)2^{n+1}\), and put \(E_{k,n}(\varepsilon,\delta)=\Delta_k(\varepsilon)\cap\sigma_n(\delta)\). Let now

now \(\{n_k\}_1^\infty\) is some sequence of even natural numbers, \(n_k \uparrow +\infty\) as \(k \uparrow +\infty\). Setting \(n_{-k}=n_k+1\), consider the closed sets

\[ E(\varepsilon,\delta)=\bigcup_{|k|=1}^{+\infty}\bigcup_{m=0}^{+\infty}E_{k,n_k+2m}(\varepsilon,\delta), \]

\[ E_0=E(1/4,0),\qquad E_1=E(1/2,1/8),\qquad E_2=E(3/4,1/4), \]

so that \(E_0\subset E_1\subset E_2\). Setting further
\[ \varepsilon_k=\exp(-2^{12}a_k^{-1}),\qquad k=\pm1,\pm2,\ldots, \]
we shall assume that the \(n_k\) increase so rapidly that

\[ \prod_{|k|=1}^{+\infty}\prod_{n=n_k}^{+\infty} \frac{1+\delta_{k,n}}{1-\delta_{k,n}}<{}^{16}/_{15}, \qquad \text{where }\delta_{k,n}=\exp(-\varepsilon_k2^{n\rho}), \tag{3} \]

\[ \max(a_k^{-1},2^{n_k},|a_k|)<\exp(\varepsilon_k2^{n_k\rho-1}), \qquad k=1,2,\ldots, \tag{4} \]

\[ \int_{\partial E_1}|\zeta^{-2}d\zeta|<1. \tag{5} \]

\(4^\circ\). We now denote by \(\gamma^i_{k,n}\) \((i=1,2;\ k=\pm1,\pm2,\ldots;\ n=n_k+2m;\ m=1,2,\ldots)\) the boundary of the set \(E_i\cap E_{k,n}(3/4,1/4)\). The curve
\[ l^i_{k,n}:\ z=(1+2^{i-4})2^{n+1}e^{i\theta},\quad (-1)^{n_k}\theta_n\le (-1)^{n_k}\theta\le \pi, \]
joins the curve \(\gamma^i_{k,n}\) with the point \(-(1+2^{i-4})2^{n+1}\). By \(D^i_{k,n}\) we denote the \(d\)-neighborhood of the curve \(\gamma^i_{k,n}\cup l^i_{k,n}\), where \(d=a_k2^{n-4}\). It is easy to verify that \(E_{i-1}\subset C D^i_{k,n}\), and that the length \(\gamma^i_{k,n}\cup l^i_{k,n}<2^{n+4}\). Take an arbitrary point \(\zeta\in\gamma^i_{k,n}\) and apply Lemma 1 to the domain \(D^i_{k,n}\), putting
\[ a=\zeta,\qquad b=-(1+2^{i-4})2^{n+1},\qquad \varepsilon=4^{-n-2}\exp(-5\varepsilon_k2^{n\rho}). \]
We may suppose that the function
\[ Q_i(\zeta,z)\equiv P(1/(z-b)) \]
is analytic in \(z\) in the half-plane \(\operatorname{Re}z>-1\), is defined and piecewise constant in \(\zeta\) (uniformly with respect to \(z\)) on the set \(\partial E_i\), and satisfies, by virtue of (1), (2), and (4), the inequalities

\[ |Q_i(\zeta,z)-1/(\zeta-z)|<4^{-n-2}\exp(-5\varepsilon_k2^{n\rho}) \quad\text{for }\zeta\in\gamma^i_{k,n},\ z\in C D^i_{n,k}, \tag{6} \]

\[ |Q_i(\zeta,z)|<\exp(\varepsilon_k^{1/2}2^{n\rho+4}) \quad\text{for }\zeta\in\gamma^i_{k,n},\ \operatorname{Re}z>-1. \tag{7} \]

\(5^\circ\). We now define two functions \(\varphi(z)\) and \(\psi(z)\), analytic on the set \(E_2\), by putting

\[ \varphi(z)=a_{|k|},\quad \psi(z)=\exp(-\varepsilon_k z^\rho), \qquad \text{when } z\in\Delta_k(3/4),\ k=\pm1,\pm2,\ldots. \tag{8} \]

Lemma 3. There exists a function \(\omega(z)\), analytic in the half-plane \(\operatorname{Re}z>-1\), such that

\[ 1<|\omega(z)/\psi(z)|<2 \quad\text{for }z\in E_1, \tag{9} \]

\[ |\omega(z)|<\exp[(|z|+1)^\rho] \quad\text{for }\operatorname{Re}z>-1. \tag{10} \]

Proof. Consider the function

\[ \omega_{k,n}(z)=1+\frac{1}{2\pi i}\int_{\gamma^2_{k,n}}[\psi(\zeta)-1]Q_2(\zeta,z)\,d\zeta, \tag{11} \]

analytic in the half-plane \(\operatorname{Re}z>-1\). Observing now that, by the Cauchy formula,

\[ \psi_{k,n}(z)=1+\frac{1}{2\pi i}\int_{\gamma^2_{k,n}} \frac{\psi(\zeta)-1}{\zeta-z}\,d\zeta = \begin{cases} \psi(z), & \text{for } z\in E_{k,n}(1/2,1/8),\\ 1, & \text{for } z\in C E_{k,n}(3/4,1/4), \end{cases} \]

and taking into account that the length of \(\gamma^2_{k,n}<2^{n+3}\), from (6) and (11) we obtain

\[ |\omega_{k,n}(z)-\psi_{k,n}(z)|<{}^{1}/_{4}\exp(-5\varepsilon_k2^{n\rho}) \quad\text{for }z\in C D^i_{k,n}, \]
whence

\[ {}^{3}/_{4}<|\omega_{k,n}(z)/\psi(z)|<{}^{5}/_{4} \quad\text{for }z\in E_{k,n}(1/2,1/8), \tag{12} \]

\[ |\omega_{k,n}(z)-1|<\exp(-\varepsilon_k2^{n\rho}) \quad\text{for }z\in C D^2_{k,n}\cap C E_{k,n}(3/4,1/4). \tag{13} \]

Inequality (13) holds, in particular, for \(z\in E_1\setminus E_{k,n}(1/2,1/8)\).

The growth of the function \(\omega_{k,n}(z)\) is bounded, by virtue of (7), (11), and (4), by the inequality

\[ |\omega_{k,n}(z)|<\exp\left(\varepsilon_k^{1/2}\cdot 2^{n\rho+5}\right),\qquad \operatorname{Re} z>-1. \tag{14} \]

Now define the desired function \(\omega(z)\) by the formula

\[ \omega(z)= {}^3/_2 \prod_{|k|=1}^{+\infty}\prod_{m=0}^{+\infty}\omega_{k,n_k+2m}(z). \]

The convergence of this product follows from (13) and (3). To obtain estimate (9), it suffices to use (12), (13), and (3). Finally, to estimate the growth of \(\omega(z)\), take an arbitrary number \(z\), \(\operatorname{Re} z>-1\). Let \(z\in \sigma_N(0)\). Then \(z\in CD_{k,n}^2\cap CE_{k,n}(3/4,1/4)\) for all \(n\), except, possibly, for the cases \(n=N-1,N,N+1\). Taking (13), (14), and (3) into account, we obtain:

\[ |\omega(z)|<2\prod_{|k|=1}^{+\infty}\prod_{n=N-1}^{N+1}|\omega_{k,n}(z)|\exp(2^{N\rho})<\exp[(|z|+1)^\rho]. \]

Lemma 4. There exists a function \(F(z)\), analytic in the angle \(|\arg z|\leq \pi/2\), such that

\[ |\varphi(z)-F(z)|<{}^1/_2|\psi(z)|\qquad \text{for } z\in E_0, \]

\[ |F(z)|<\exp[\mathrm{const}\cdot(|z|+1)^\rho]\qquad \text{for } |\arg z|\leq \pi/2. \]

Proof. We note that from (6) the estimate follows

\[ |\theta_1(\xi,z)-1/(\xi-z)|<|\psi(\xi)|/|\xi|^2,\qquad \xi\in \partial E_1, \tag{15} \]

valid for \(z\in E_0\), and also in the disk \(|z|\leq {}^1/_4|\xi|\). The growth of the function \(Q_1(\xi,z)\) is bounded, by virtue of estimate (7), by the inequality

\[ |Q_1(\xi,z)|<\exp(|\xi|^\rho)\qquad \text{for } \xi\in \partial E_1,\ \operatorname{Re} z>-1. \tag{16} \]

Now denote

\[ \Gamma_n=\bigcup_{\substack{(k,m)\\ n_k+2m\leq n}}\gamma_{k,n_k+2m}\qquad \text{for } n\geq n_1 \]

(so that \(\partial E_1=\bigcup_{n=n_1}^{+\infty}\Gamma_n\)) and consider the sequence of functions analytic for \(\operatorname{Re} z>-1\),

\[ H_n(z)=\frac{1}{2\pi i}\int_{\Gamma_n}\frac{\varphi(\xi)}{\omega(\xi)}Q_1(\xi,z)\,d\xi, \tag{17} \]

where \(\omega(z)\) is the function constructed in Lemma 3. From inequalities (4) and (10) (taking account of (8)) it follows that

\[ |\varphi(\xi)/\omega(\xi)|<1/|\psi(\xi)|^2,\qquad \xi\in E_1. \tag{18} \]

Take a natural number \(N\), and let \(|z|\leq 2^N,\ m>n>N+1\). Then

\[ 0\equiv \frac{1}{2\pi i}\int_{\Gamma_m\setminus\Gamma_n}\frac{\varphi(\xi)}{\omega(\xi)}\frac{d\xi}{\xi-z}, \]

whence, taking (15), (17), (18), and (5) into account, we obtain:

\[ |H_m(z)-H_n(z)|< \frac{1}{2\pi}\int_{\partial E_1\setminus\Gamma_n}|\xi^{-2}\,d\xi|\to 0 \qquad \text{as } n\to\infty . \tag{19} \]

It follows that the sequence \(H_n(z)\) converges uniformly in every disk to some function \(H(z)\) analytic for \(\operatorname{Re} z>-1\). Similarly, taking into account the formula

\[ \frac{\varphi(z)}{\omega(z)} =\frac{1}{2\pi i}\int_{\Gamma_n}\frac{\varphi(\xi)}{\omega(z)}\frac{d\xi}{\xi-z}, \qquad z\in E_0,\quad |z|\leq 2^n, \]

for an arbitrary point \(z\in E_0\) we have the estimate

\[ \left|H(z)-\frac{\varphi(z)}{\omega(z)}\right| =\lim_{n\to\infty}\left|H_n(z)-\frac{\psi(z)}{\omega(z)}\right| \leq \frac{1}{2\pi}\int_{\partial E_1}|\xi^{-2}\,d\xi|<{}^1/_4. \tag{20} \]

Let us now estimate the growth of the function \(H(z)\). Let \(z \in \sigma_{N-1}(0)\). If \(\zeta \in \Gamma_{N+2}\), then \(|\zeta| \leqslant 18|z|\). Taking now (19), (17), (18), (16), and (15) into account, we obtain

\[ |H(z)|<1+|H_{N+2}(z)|<1+\max_{\zeta\in\Gamma_{N+2}} \left|\frac{\zeta}{\psi(\zeta)}\right|^{2}\cdot |Q_1(\zeta,z)|< \]

\[ <1+\max_{|\zeta|\leqslant 18|z|}|\zeta|^2\exp(2|\zeta|^\rho) <\exp[\mathrm{const}\cdot(|z|+1)^\rho]. \tag{21} \]

By virtue of (20), (9), and (21), the function \(F(z)=H(z)\omega(z)\) satisfies all the conditions of Lemma 4.

\(6^\circ\). From Lemmas 4 and 2 there follows the existence of an entire function \(G(z)\) of order \(\rho\) and of normal type, satisfying the inequality

\[ |G(z)-\varphi(z)|<|\psi(z)|,\qquad z\in E_0. \tag{22} \]

The function \(G(z)\) is the required function with an infinite set of deficient values \(\{a_k\}_1^\infty\). Indeed, by virtue of the definition of the set \(E_0\) and of the function \(\varphi(z)\), for any \(k=1,2,\ldots\), when \(r\geqslant 2^{n_k}\), there exists an arc \(\lambda_{k,r}\) of the circumference \(|z|=r\), \(\lambda_{k,r}\subset E_0\), such that \(\varphi(z)=a_k\) for \(z\in\lambda_{k,r}\), and the length \(\lambda_{k,r}=\dfrac{\alpha_k}{2}r\). Then from (22) we obtain that
\[ |G(z)-a_k|<|\psi(z)|<\exp\left(-\frac{\varepsilon_k}{2}r^\rho\right), \qquad z\in\lambda_{k,r},\ r\geqslant 2^{n_k}. \]

Hence, in the generally accepted notation, we have

\[ m(r,a_k,G)=\frac{1}{2\pi}\int_0^{2\pi} \ln^+\frac{1}{|G(re^{i\theta})-a_k|}\,d\theta > \frac{\alpha_k\varepsilon_k}{8\pi}r^\rho > \varepsilon_k^2 r^\rho \quad \text{for } r\geqslant 2^{n_k}, \]

and since \(T(r,G)\leqslant \ln^+ M(r,G)<\sigma r^\rho\) for \(r\geqslant 1\), we finally obtain

\[ \delta(a_k,G)=\lim_{r\to\infty}\,[m(r,a_k,G)/T(r,G)] \geqslant \varepsilon_k^2/\sigma>0. \]

The theorem is completely proved.

Remark. Analysis of the example constructed above gives some grounds to suppose that, for an arbitrary entire function \(G(z)\) of finite order, the condition

\[ \sum_{0<\delta(a,G)<1}\frac{1}{\ln\delta^{-1}(a,G)}<+\infty \]

is satisfied.

This condition cannot be further strengthened, since in our example
\(1/\ln\delta^{-1}(a_k,G)>\mathrm{const}\cdot\alpha_k>0\), where of the numbers \(a_k\) it is required only that the series
\(\sum_{k=1}^{\infty}\alpha_k\) converge.

Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR

Received
8 VII 1966

REFERENCES

\(^{1}\) R. Nevanlinna, La théorème Picard—Borel et la théorie des fonctions meromorphes, Paris, 1929.
\(^{2}\) R. Nevanlinna, Single-Valued Analytic Functions, Moscow–Leningrad, 1941.
\(^{3}\) W. K. Hayman, Meromorphic Functions, Oxford, 1964.
\(^{4}\) S. N. Mergelyan, UMN, 8, no. 4 (1953).
\(^{5}\) S. N. Mergelyan, UMN, 7, no. 2 (1952).