On the growth of meromorphic functions with radially distributed zeros and poles

Theorem 1. Let the nonnegative integer p = p(q, X, Y) be associated with the class $\mathcal{M}_a(X, Y)$ in the following way. - (a) If q = 0, p = 0. - (b) Suppose $q \ge 1$ . For each integer $m_0$ , $0 \le m_0 \le q$ , consider the system of $q m_0 + 1$ equations (1.3) $$\sum_{j=1}^{M} a_{kj} e^{-ik\theta_j} - \sum_{j=M+1}^{L} a_{kj} e^{-ik\theta_j} = 0, \qquad m_0 \le k \le q,$$ subject to the following conditions: (1.4) (i) $$a_{kj} \ge 0$$ , $m_0 \le k \le q$ , $1 \le j \le L$ ; (ii) $$\sum_{j=1}^{L} a_{kj} = 1, \quad m_0 \le k \le q;$$ and (iii) for $$1 \le j \le L$$ , if $a_{kj} = 0$ then $a_{k'j} = 0$ for $k < k' \le q$ . If, for every $m_0$ , $0 \le m_0 \le q$ , system (1.3) has solutions satisfying conditions (1.4), let p = 0. Otherwise let p be the largest $m_0$ , $0 \le m_0 \le q$ , for which system (1.3) has no solutions satisfying (1.4). Then for all $f \in \mathcal{M}_q(X, Y)$ , we have (1.5) (i) $$\lim_{r \to \infty} \frac{T(r, f)}{r^p} = \infty \text{ if } p > 0,$$ and (ii) $$\lim_{r \to \infty} \frac{T(r, f)}{\log r} = \infty \text{ if } p = 0.$$ Furthermore, given $\psi(r) \to \infty$ as $r \to \infty$ , there exists $f \in \mathcal{M}_q(X,Y)$ such that (1.6) (i) $$\liminf_{r \to \infty} \frac{T(r,f)}{\psi(r)r^p} = 0 \quad if \, p > 0,$$ and (ii) $$\liminf_{r \to \infty} \frac{T(r, f)}{\psi(r) \log r} = 0 \quad if \, p = 0.$$ Clearly (1.5) asserts that $f \in \mathcal{M}_q(X, Y)$ has lower growth at least order p, maximal type, and (1.6) asserts that this result is best possible. It is trivial that p = q if $Y = \emptyset$ and M = 1, giving the result for entire functions with zeros on a single ray in [1] and [5]. If $Y = \emptyset$ and M = 2, an easy verification gives $p \ge \max(0, q - 1)$ , in agreement with the result in [5] and [11]. A geometric interpretation can be given to the integer p in most cases. Let us suppose that $p \ge 1$ and note that (1.3), (1.4i), and (1.4ii) express the fact that 0 is in the convex hull of $S_k$ (defined in (1.2)) for $m_0 \le k \le q$ . For $p \ge 1$ we thus have in the cases where we may ignore the rather technical condition (1.4iii) that p is the largest integer $m_0 \le q$ for which 0 does not lie in the convex hull of $S_{m_0}$ . It would perhaps be helpful to consider an example in which the above geometric interpretation of p fails, i.e. an example in which condition (1.4iii) plays an essential role. Suppose $X = \{0, \pi/4, \pi/3\}$ , $Y = \emptyset$ , and q = 4. It is elementary that the only solution of (1.3) with k = 4 subject to (1.4i) and (1.4ii) is (1.7) $$a_{41} = 1/2, \quad a_{42} = 1/2, \quad \text{and} \quad a_{43} = 0.$$ Similarly the only solution of (1.3) with k = 3 satisfying (1.4i) and (1.4ii) is (1.8) $$a_{31} = 1/2$$ , $a_{32} = 0$ , and $a_{33} = 1/2$ . There is no solution of (1.3) with k=2 subject to (1.4i) and (1.4ii). Thus from (1.4iii), (1.7), and (1.8), it is clear that p=3, even though 2 is the largest integer $m_0$ not exceeding 4 for which 0 is not in the convex hull of $S_{m_0}$ . Although Theorem 1 gives complete information concerning possible lower growth rates of $f \in \mathcal{M}_q(X,Y)$ in terms of q, X, and Y, it does not give information in terms of q and L alone concerning possible lower growth rates of a function of genus q with zeros and poles restricted to any L distinct rays. It would be of interest to determine $$\mu(q, L) \equiv \inf p(q, X, Y),$$ where X and Y vary over all disjoint sets in $[0, 2\pi)$ whose union has L members, and also to consider only entire functions and to determine $$\mu_{e}(q, M) \equiv \inf p(q, X, \emptyset),$$ where X varies over all sets of M members in $[0, 2\pi)$ . From [1], [5], and [11] we have (1.9) $$\mu_e(q, M) = \max(0, q - M + 1)$$ for M=1 or M=2. The possibility of extending (1.9) to other values of M is considered in [11]. In particular it is shown there that if M is a positive integer and $X \subset [0, 2\pi)$ consists of M members, then $$\inf_{f \in \mathscr{G}_q(X)} \mu(f) = \max(0, q - M + 1),$$ where, for general X, $\mathscr{G}_q(X)$ is the subclass of $\mathscr{E}_q(X)$ consisting of functions with zeros regularly distributed on each ray, and, for sets X whose members are themselves regularly distributed in $[0, 2\pi)$ , $\mathscr{G}_q(X) = \mathscr{E}_q(X)$ . Theorem 1 shows that (1.9) does not hold in general. Suppose, for example, that $X = \{0, \pi/180, \pi/90\}$ and q = 120. Using Theorem 1, we have $$\mu_e(120,3) \le p(120, X, \varnothing) = 90 < 120 - 3 + 1.$$ The quantity $\mu_e(q, M)$ has also been studied by E. V. Gleizer. It is my understanding that Gleizer, in a paper [4] submitted to the Ukrainian Journal of Mathematics simultaneously to the submission of this paper, showed $$\mu_e(q,3) \ge \max\left(0, \frac{q}{3} - 1\right).$$ Gleizer also obtained a result for entire functions very close to Theorem 1 applied to $\mathscr{E}_{a}(X)$ . The estimate $$\mu_e(q, M) \ge \left\lceil \frac{q}{5^M} \right\rceil$$ appears in [2, p. 25]. (The lower growth of entire functions of infinite order with radially distributed zeros is also dealt with in [2, p. 25].) An exact determination of $\mu(q, L)$ and $\mu_e(q, M)$ remains open in the general case, as does the probably easier question of whether or not $\mu(q, L) = \mu_e(q, L)$ . We also consider the ratio $T(\alpha r, f)/T(r, f)$ for $f \in \mathcal{M}_q(X, Y)$ .

Theorem 2. For $\alpha > 1$ and $f \in \mathscr{E}_q(X)$ of finite order $\lambda$ , there exists $K = K(\lambda, \alpha, X) > 0$ such that (1.10) $$T(\alpha r, f) < KT(r, f), \quad r > r_0(f).$$ Theorem 2 generalizes to meromorphic functions in many, but not all, cases. A discussion of the possibility of such a generalization appears in §4. It is elementary that (1.10) implies (1.11) $$\lim_{r \to \infty} \frac{T(r+1,f)}{T(r,f)} = 1.$$ (Compare to Corollary 2 of [5].) In [12] it is shown that (1.11) implies that the Nevanlinna deficiency is independent of the choice of the origin. From Theorem 2 we thus conclude that any entire function of finite order for which the Nevanlinna deficiency is origin dependent cannot have its zeros restricted to a finite number of rays through any one point. (See for example [8].) We conclude the Introduction by collecting certain elementary facts needed in the proofs of Theorem 1 and Theorem 2. Our arguments depend heavily on the Fourier series of $\log |f(re^{i\theta})|$ , where f has the form (1.12) $$f(z) = (\exp h(z)) \frac{\prod_{\nu} E(z/z_{\nu}, q)}{\prod_{\nu} E(z/w_{\nu}, q)} = (\exp h(z)) g(z),$$ E(z,q) is the Weierstrass factor of genus q, $$E(z,q) = (1-z)\exp(z+z^2/2+\cdots+z^q/q),$$ and $$h(z) = \sum_{m=1}^{\infty} d_m z^m, \quad |z| < \infty.$$ Letting $$c_m(r,f) = \frac{1}{2\pi} \int_0^{2\pi} e^{-\mathrm{i}m\theta} \log |f(\mathrm{re}^{i\theta})| d\theta,$$ we have (1.13) (i) $$c_0(r, f) = N(r, 0) - N(r, \infty);$$ (ii) $$c_m(r,f) = \overline{c_{-m}(r,f)}, m < 0;$$ (iii) $$c_{m}(r,f) = \frac{d_{m}}{2}r^{m} + c_{m}(r,g)$$ $$= \frac{d_{m}}{2}r^{m} + \frac{1}{2m} \left\{ \sum_{|z_{\nu}| \le r} \left( \left(\frac{r}{z_{\nu}}\right)^{m} - \left(\frac{\overline{z}_{\nu}}{r}\right)^{m} \right) \right\}$$ $$- \frac{1}{2m} \left\{ \sum_{|w_{\nu}| \le r} \left( \left(\frac{r}{w_{\nu}}\right)^{m} - \left(\frac{\overline{w}_{\nu}}{r}\right)^{m} \right) \right\},$$ $$1 \le m \le q;$$ and (iv) $$c_m(r,f) = \frac{d_m}{2}r^m + c_m(r,g)$$ $$= \frac{d_m}{2}r^m - \frac{1}{2m} \left\{ \sum_{|z_\nu| \le r} \left(\frac{\overline{z}_\nu}{r}\right)^m + \sum_{|z_\nu| > r} \left(\frac{r}{z_\nu}\right)^m \right\}$$ $$+ \frac{1}{2m} \left\{ \sum_{|w_\nu| \le r} \left(\frac{\overline{w}_\nu}{r}\right)^m + \sum_{|w_\nu| > r} \left(\frac{r}{w_\nu}\right)^m \right\},$$ $$m \ge q + 1.$$ A derivation of these formulas, originally due to F. Nevanlinna [10], can be found in many places, including [9]. Letting $m_1(r, f)$ and $m_2(r, f)$ denote the $L^1$ and $L^2$ norms of $\log |f(\mathrm{re}^{i\theta})|$ respectively, we observe trivially from Nevanlinna's first fundamental theorem that for each interger m $$(1.14) \quad \frac{|c_m(r,f)|}{2} \le \frac{m_1(r,f)}{2} \le T(r,f) \le m_2(r,f) + N(r,\infty).$$ We shall need the following elementary lemma. LEMMA A. Suppose $m_1 < m_2 < \cdots < m_k$ and $n_1 < n_2 < \cdots < n_k$ are nonnegative integers. If $\pi$ is any permutation of $\{1, 2, \dots, k\}$ other than $\pi(j) \equiv k - j + 1, 1 \leq j \leq k$ , then (1.15) $$\sum_{j=1}^{k} m_j n_{k-j+1} < \sum_{j=1}^{k} m_j n_{\pi(j)}.$$