Radial distribution and deficiencies of the values of a meromorphic function

Theorem 1. Let f(z) be a meromorphic function with positive zeros and negative poles. Received April 25, 1960. ^&lt;sup>1</sup> The research of this author was supported by the United States Air Force, Contract No. AF49(638)-571, monitored by the Office of Scientific Research. $^{^2}$ The research of this author was supported by a grant from the National Science Foundation (G 5253). Assume that (2) $$\sum_{\mu} \frac{1}{a_{\mu}} + \sum_{\nu} \frac{1}{|b_{\nu}|} = + \infty ,$$ and that $$\sum_{\mu} \frac{1}{a_{\mu}^{\epsilon}} + \sum_{\nu} \frac{1}{|b_{\nu}|^{\epsilon}} < + \infty ,$$ for some finite positive value of $\xi$ . Then $$\limsup_{r\to\infty}\frac{N\!\left(r,\frac{1}{f}\right)+N\!(r,f)}{T\!(r,f)}\leqq\frac{1}{1+A}\;,$$ where A(>0) is an absolute constant. If the condition (3) is omitted, we still have (5) $$\limsup_{r \to \infty} \frac{N\left(r, \frac{1}{f}\right) + N(r, f)}{T(r, f)} \leq 1.$$

Corollary 1.1. The assumptions of Theorem 1 imply $$\delta(0,f) \ge \frac{A}{1+A}$$ , $\delta(\infty,f) \ge \frac{A}{1+A}$ . If the condition (3) is omitted, but $$0<\alpha \leq \liminf_{r\to\infty}\frac{N(r,f)}{N\!\!\left(r,\frac{1}{f}\right)} \leq \limsup_{r\to\infty}\frac{N\!\!\left(r,f\right)}{N\!\!\left(r,\frac{1}{f}\right)} \leq \frac{1}{\beta} < + \, \infty$$ , we still have $$\delta(0,f) \ge \frac{\alpha}{1+\alpha}$$ , $\delta(\infty,f) \ge \frac{\beta}{1+\beta}$ .

Corollary 1.2. Let f(z) be an entire function with real zeros. If (6) $$\sum_{\mu} \frac{1}{|a_{\mu}|^2} = + \infty ,$$ and if $$\sum_{\mu} \frac{1}{|a_{\mu}|^{\xi}} < + \infty ,$$ for some finite positive value of $\xi$ , then $$\delta(0,f) \ge \frac{A}{1+A} ,$$ where A is the absolute constant in (4). The condition (2) of Theorem 1 cannot be omitted; we shall see that the theorem does not hold for certain meromorphic functions of finite order, with positive poles and such that $$\sum_{\mu} \frac{1}{a_{\mu}^{\kappa}} + \sum_{\nu} \frac{1}{|b_{\nu}|^{\kappa}} = + \infty ,$$ for every $\kappa$ less than one. Similarly, Corollary 1.2 does not hold for certain entire functions of finite order, with real zeros and such that $$\sum_{\mu} \frac{1}{|a_{\mu}|^{\kappa}} = + \infty ,$$ for every $\kappa$ less than two. The conditions (3) and (7) are used essentially in our proofs, but it is possible that our results hold without such restrictions. This conjecture is plausible if we observe that the assertions (4) and (8) do not contain the parameter $\xi$ . Our method gives a little more than has been stated. In the special case of entire functions it yields

Theorem 2. Let f(z) be entire. Assume that all its zeros $a_{\mu}$ lie on the radii defined by $$re^{i\omega_0}$$ , $re^{i\omega_1}$ , $\cdots$ $re^{i\omega_m}$ $(r>0)$ , where the $\omega$ 's are real. Then, there exists a positive constant K, depending only on the $\omega$ 's, and such that the condition $$\sum_{\mu} \frac{1}{|a_{\mu}|^{\kappa}} = + \infty ,$$ and the condition $$\sum_{\mu} \frac{1}{|a_{\mu}|^{\epsilon}} < + \infty ,$$ for some finite value of $\xi$ , imply $$\delta(0,f) \ge \frac{A}{1+A},$$ where A is the absolute constant in (4). All the previous theorems and corollaries assert that 0 and $\infty$ are among the deficient values of certain functions f(z). Hence, by Theorem 4 of [1], the lower order $\mu$ , of f(z) is positive. Assume now that h(z) denotes a meromorphic function which does not vanish identically, is of order less than $\mu$ , but is otherwise arbitrary. Then, by elementary inequalities of Nevanlinna's theory, $$T(r,hf) \sim T(r,f) \; , \ \frac{m(r,fh)}{T(r,fh)} = \frac{m(r,f)}{T(r,f)} + o \left(1\right), \frac{m\left(r,\frac{1}{fh}\right)}{T(r,fh)} = \frac{m\left(r,\frac{1}{f}\right)}{T(r,f)} + o \left(1\right) \; ,$$ and hence $$\delta(0, fh) = \delta(0, f)$$ , $\delta(\infty, fh) = \delta(\infty, f)$ . This shows that our theorems remain true even if infinitely many zeros and poles have unknown arguments but are sufficiently rare. It will be shown in [2] that a radial distribution of zeros and poles makes it, in general, impossible for the function to have other deficient values than 0 and $\infty$ . Combining the results of [2] with those of the present investigation, it is possible to obtain information concerning all the deficient values of certain interesting classes of functions. The following result is one of the simplest which may be obtained in this way. Let f(z) be an entire function of finite order $\lambda$ . Assume that all the zeros of f(z) are real and that $\lambda > 2$ . Then (11) holds and $$\delta(\tau, f) = 0$$ , for $\tau \neq 0$ , $\tau \neq \infty$ .

Lemma 1. Let f(z) be meromorphic with zeros $\{a_{\mu}\}$ and poles $\{b_{\nu}\}$ . Assume $$|\arg a_{\mu}| \leq \gamma < \frac{\pi}{2} \qquad (\mu = 1, 2, 3, \cdots);$$ (1.2) $$|\arg b_{\nu} - \pi| \leq \gamma < \frac{\pi}{2}$$ $(\nu = 1, 2, 3, \cdots);$ ^&lt;sup>8</sup> A direct study of the lower order of our functions will be found in [2]. For the functions in Theorem 1 and its Corollaries, this study yields "best possible" bounds for $\mu$ . (1.3) $$\sum_{\mu} \frac{1}{|a_{\mu}|} + \sum_{\nu} \frac{1}{|b_{\nu}|} = + \infty.$$ Then, for r large enough, $$(1.4) \qquad \frac{1}{2\pi} \int_0^{2\pi} \log |f(re^{i\theta})| \cos \theta d\theta \geq \cos \gamma \left\{ N\!\!\left(r,\frac{1}{f}\right) + N\!\!\left(r,f\right) \right\}.$$

Lemma 2. Let g(z) be an absolutely convergent product of primary factors of genus 2. Assume that the zeros of g(z) lie in the sector $\Delta(\varepsilon)$ defined by (2.1) $$|\arg z| \leq \frac{\pi}{6} - \varepsilon \qquad \left(0 < \varepsilon \leq \frac{\pi}{6}\right).$$ Then $$(2.2) \qquad \int_{\frac{(\pi/3)-(\varepsilon/2)}{(\pi/3)-(\varepsilon/2)}}^{\frac{(\pi/3)+(\varepsilon/2)}{2}} \log^+ \left| \frac{g(re^{i\theta})}{g(-re^{i\theta})} \right| d\theta \ge 2\varepsilon \sin \frac{\varepsilon}{2} \, r^3 \! \int_0^\infty \! \frac{n\! \left(t,\frac{1}{g}\right)}{t^2(t^2+r^2)} \, dt \; .$$ Proof. Let $$E(u, q) = (1 - u) \exp\left(u + \frac{u^2}{2} + \cdots + \frac{u^q}{q}\right)$$ , denote the primary factor of genus q; we write E(u) instead of E(u, 2). It follows from the definitions that $$\log \frac{E(u)}{E(-u)} = \log \left(\frac{1-u}{1+u}\right) + 2u = 2 \int_0^u \frac{t^2}{t^2-1} dt$$ , and hence, if $$(2.3) u = re^{i\phi}, \phi \not\equiv 0 \pmod{\pi},$$ $$\log \left| \frac{E(re^{i\phi})}{E(-re^{i\phi})} \right| = 2 \int_0^r \frac{x^4 \cos \phi - x^2 \cos 3\phi}{x^4 - 2x^2 \cos 2\phi + 1} dx.$$ Let $\{c_{\nu}\}$ be the sequence of the zeros of g(z); putting $$\theta_{\nu} = \arg c_{\nu}$$ , we have, by assumption $$(2.4)$$ If $z(=re^{i\theta})$ is confined to the sector $$\left|\theta - \frac{\pi}{3}\right| \le \frac{\varepsilon}{2} ,$$ (2.3), (2.4) and (2.5) yield $$\begin{split} \log\left|\frac{E\left(\frac{z}{c_{\scriptscriptstyle \nu}}\right)}{E\left(\frac{-z}{c_{\scriptscriptstyle \nu}}\right)}\right| &= 2\int_{\scriptscriptstyle 0}^{\scriptscriptstyle r/\mid c_{\scriptscriptstyle \nu}\mid} \frac{x^4\cos\left(\theta-\theta_{\scriptscriptstyle \nu}\right)-x^2\cos3(\theta-\theta_{\scriptscriptstyle \nu})}{x^4-2x^2\cos2(\theta-\theta_{\scriptscriptstyle \nu})+1}\,dx \ &\geq 2\sin\frac{\varepsilon}{2}\int_{\scriptscriptstyle 0}^{\scriptscriptstyle r/\mid c_{\scriptscriptstyle \nu}\mid} \frac{x^2}{1+x^2}\,dx \;. \end{split}$$ Hence, in the region defined by (2.5) $$\begin{aligned} \log^{+}\left|\frac{g(re^{i\theta})}{g(-re^{i\theta})}\right| &\geq 2\sin\frac{\varepsilon}{2}\sum_{\nu=1}^{\infty}\int_{0}^{r/|c_{\nu}|}\frac{x^{2}}{1+x^{2}}\,dx \\ &= 2\sin\frac{\varepsilon}{2}\,r^{3}\!\!\int_{0}^{\infty}\!\!\frac{n\!\left(t,\frac{1}{g}\right)}{t^{2}\!\left(t^{2}+r^{2}\right)}\,dt \end{aligned}$$ and this clearly implies (2.2).

Lemma 3. Let f(z) be a meromorphic function of genus not greater than 2. Assume - (i) that its zeros $\{a_{\mu}\}\ lie\ in\ the\ region\ \Delta(\varepsilon)\ defined\ by\ (2.1);$ - (ii) that its poles $\{b_n\}$ lie in the region $\Delta^*(\varepsilon)$ defined by $$|\arg z - \pi| \le \frac{\pi}{6} - \varepsilon$$ $\left(0 < \varepsilon \le \frac{\pi}{6}\right);$ (iii) $$\sum_{\mu} \frac{1}{|a_{\mu}|} + \sum_{\nu} \frac{1}{|b_{\nu}|} = + \infty$$ . Then $$(2.7) \qquad \frac{1}{2\pi} \int_{(\pi/3)-(\varepsilon/2)}^{(\pi/3)+(\varepsilon/2)} \log^{+} \left| \frac{f(re^{i\theta})}{f(-re^{i\theta})} \right| d\theta$$ $$\geq \frac{(1-\eta(r))}{2\pi} \varepsilon \sin \frac{\varepsilon}{2} \left\{ N\!\left(r, \frac{1}{f}\right) + N\!(r, f) \right\},$$ where $\eta(r) \to 0$ as $r \to +\infty$ .

Lemma 4. If, in Lemma 3, we restrict the value of the parameter $\varepsilon$ by the inequalities $$(2.13) \frac{9}{10} \frac{\pi}{6} \le \varepsilon \le \frac{\pi}{6} ,$$ then, for all sufficiently large values of r, $$(2.14) T(r,f) \geq (1+A) \Big\{ N\!\Big(r,\frac{1}{f}\Big) + N\!(r,f) \Big\} ,$$ where A(>0) is an absolute constant. The inequality (2.14) still holds if f(z) is replaced by F(z): (2.15) $$F(z) = e^{S(z)} f(z)$$ where S(z) is an entire function (which may reduce to a polynomial).

Lemma 5. Let f(z) be meromorphic. Assume that there exists an integer $q(\ge 1)$ such that (3.1) $$\sum_{\mu} \frac{1}{|a_{\mu}|^{q}} + \sum_{\nu} \frac{1}{|b_{\nu}|^{q}} = + \infty.$$ (3.2) $$\sum_{\mu} \frac{1}{|a_{\mu}|^{q+1}} + \sum_{\nu} \frac{1}{|b_{\nu}|^{q+1}} < + \infty.$$ Let p be an odd integer $$(3.3) 1 \leq p \leq q.$$ Consider the sectors $\Delta_k$ defined by (3.4) $$\left| \arg z - \frac{2\pi k}{p} \right| \le \frac{\pi}{60q}$$ $(k = 0, 1, 2, \dots, p - 1).$ and the sectors $\Delta_k^*$ defined by (3.5) $$\left|\arg z - \pi - \frac{2\pi k'}{p}\right| \leq \frac{\pi}{60q} \quad (k' = 0, 1, 2, \dots, p-1).$$ If every zero of f(z) lies in one of the sectors $\Delta_k$ and every pole in one of the sectors $\Delta_{k'}^*$ , then (3.6) $$\limsup_{r \to \infty} \frac{N\left(r, \frac{1}{f}\right) + N(r, f)}{T(r, f)} \le \frac{1}{1 + A},$$ where A is the absolute constant in Lemma 4.

Lemma 6. Let f(z) be entire. Modify the assumptions of Lemma 5 by: - (i) omitting all reference to poles; - (ii) omitting the restriction that p be odd (p may be any integer satisfying the inequality (3.3)). Then (3.6) still holds. The proof of Lemma 5 also yields Lemma 6 provided the integer s (even or odd) is defined by $$s \leq \frac{q}{p} < s+1$$ , instead of (3.7). The definitions (3.8) remain unchanged and (3.9) takes the sharper form $$l \leq q < 2l$$ . The other changes in the proof are obvious and need not be mentioned here. We obtain Corollary 1.2 by taking p=2, in Lemma 6. 5. Best possible character of the conditions (2) and (6). Let $$(5.1) s_1, s_2, s_3, \cdots$$ be a sequence of integers such that $$(5.2) s_1 \geq 2, s_{\lambda+1} > 2s_{\lambda} (\lambda = 1, 2, 3, \cdots).$$ Consider the entire function $$f(z) = \prod_{\lambda=1}^{\infty} \prod_{m=s_{\lambda}}^{2s_{\lambda}} \left(1 - \frac{z}{m(\log m)^2}\right).$$ Denoting by $\{a_{\mu}\}$ the sequence of the zeros of f(z), elementary estimates yield $$\sum_{\mu} \frac{1}{\alpha_{\mu}} < + \infty , \qquad \sum_{\mu} \frac{1}{\alpha_{\mu}^{\kappa}} = + \infty \qquad (\kappa < 1).$$ These relations hold for every choice of the sequence (5.1). Hence we may take the ratios $s_{\lambda+1}/s_{\lambda}$ to be rapidly increasing with $\lambda$ and, using the well-known formula [4; p. 271]: $$\log M(r,f) = r \int_0^\infty \frac{n\left(t, \frac{1}{f}\right)}{t(t+r)} dt ,$$ choose (5.1) so that (5.4) $$\lim_{r \to \infty} \inf \frac{\log T(r, f)}{\log r} \leq \lim_{r \to \infty} \inf \frac{\log \log M(r)}{\log r} = 0.$$ It is sufficient to choose the sequence (5.1) in such a way that, for some arbitrarily large u, n(t, 1/f) is constant in $u \le t \le e^u$ . Hence, putting $$F(z) = \frac{f(z)}{f(-z)} ,$$ (5.5) $$\liminf_{r\to\infty} \frac{\log T(r, F(z))}{\log r} = 0.$$ It has been shown elsewhere [1; p. 297, Theorem 4] that the condition (5.5) implies $$\delta(\tau, F(z)) = 0$$ , except possibly for a single value of $\tau$ , finite or infinite. Hence the inequalities $$\delta(0, F(z)) > 0$$ , $\delta(\infty, F(z)) > 0$ , are both impossible since one of them would imply the other one. We thus have $$1=\limsup_{r\to\infty}\frac{N(r,F)}{T(r,F)}=\frac{1}{2}\limsup_{r\to\infty}\frac{N\Big(r,\frac{1}{F}\Big)+N(r,F)}{T(r,F)}$$ , although F(z) satisfies all the conditions of Theorem 1 except (2) which is replaced by the weaker condition (5.3). Similarly, (5.4) and Theorem 4 of [1] yield $$\delta(\tau, f(z)) = 0 \qquad (\tau \neq \infty)$$ and hence, putting $$F^*(z) = f(z^2)$$ we have $$\delta(\tau, F^*(z)) = 0 \qquad (\tau \neq \infty).$$ In particular $\delta(0, F^(z)) = 0$ , although $F^(z)$ satisfies all the conditions of Corollary 1.2 except (6) which is replaced by a weaker condition analogous to (5.3). 6. Proof of Theorem 2. Our proof is a straightforward consequence of Lemma 6 and of a classical theorem of H. Weyl [5; p. 335, Satz 16]. We consider the arguments $\omega_j$ of the radii carrying the zeros of f(z) and assume $$\omega_0 = 0$$ ; this is clearly no restriction. Let k+1 $(0 \le k \le m)$ be the maximum number of linearly independent elements among $$(6.1) 2\pi, \omega_1, \omega_2, \cdots \omega_m.$$ Renumbering, if necessary, the $\omega$ 's we may assume: (i) that a relation such as (6.2) $$\mu_0 2\pi + \sum_{j=1}^k \mu_j \omega_j = 0 ,$$ is impossible for integral values of the $\mu$ 's, not all zero; (ii) if k < m, there exist integers $n_{ij}$ and $\sigma(>0)$ such that (6.3) $$\sigma \omega_{l} = 2\pi n_{l0} + \sum_{i=1}^{k} n_{lj} \omega_{j} \qquad (l = k+1, \dots, m).$$ Put $$M_i = \sum\limits_{j=1}^k \mid n_{i_j} \mid$$ and (6.4) $$M = \sup \{\sigma, M_{k+1}, M_{k+2}, \dots, M_m\}$$ . Since no relation such as (6.2) is possible, Weyl's theorem asserts the existence of a sequence $$(6.5) \lambda_1, \lambda_2, \lambda_3, \cdots$$ of increasing integers such that $$(6.6) \hspace{1cm} |\ \lambda_s \omega_j - L_{sj} 2\pi \ | \le \frac{\pi}{120M} \hspace{1cm} (j=1,2,\cdots k;s=1,2,3,\cdots),$$ where the $L_{sj}$ are integers. Weyl's theorem also asserts that the sequence (6.5) has a positive density. The latter property is unnecessarily precise for our purposes; we only need the obvious implication $$(6.7) \lambda_{s+1} < 2\lambda_s (s \ge s_0).$$ We set $$K = \sigma \lambda_{s_0}$$ and observe that the integer K depends only on the $\omega$ 's. By the assumptions (9) and (10), there exists an integer q such that $$q \geqq K$$ , $\sum_{\mu} \frac{1}{|a_{\mu}|^q} = + \infty$ , $\sum_{\mu} \frac{1}{|a_{\mu}|^{q+1}} < + \infty$ . Define h by the inequalities $$\sigma \lambda_h \leq q < \sigma \lambda_{h+1}.$$ In view of the definition of K and (6.7) $$(6.9) q < 2\sigma \lambda_h.$$ We now obtain Theorem 2 by verifying that Lemma 6 may be applied with the value of q chosen above and $$(6.10) p = \sigma \lambda_h.$$ It is clear that we only have to ascertain that the zeros of f(z) lie in regions such as (3.4) with p defined by (6.10). Using (6.6) and (6.4) in (6.3), we obtain $$(6.11) \qquad |\sigma \lambda_n \omega_l - \varLambda_{nl} 2\pi| \leq \frac{\pi}{120} \qquad (l = k+1, k+2, \cdots m),$$ where the $\Lambda$ 's are integers. By (6.6) and (6.4), it is clear that (6.11) holds also for $l = 1, 2, \dots k$ , with $$A_{hl} = \sigma L_{hl} \qquad (l = 1, 2, \cdots k).$$ Hence, by (6.9), (6.10) and (6.11) $$\left|\omega_{l}-\frac{\varDelta_{hl}2\pi}{p}\right|\leq\frac{\pi}{60q} \qquad \qquad (l=1,2,\cdots m).$$ This shows that the location of zeros allows the application of Lemma 6. Theorem 2 is an immediate consequence.