A new property of meromorphic functions and its applications.

*(English)*Zbl 1070.30009
Begehr, Heinrich G. W. (ed.) et al., Analysis and applications–ISAAC 2001.

Proceedings of the 3rd international congress, Berlin, Germany, August 20–25, 2001. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1384-1/hbk). Int. Soc. Anal. Appl. Comput. 10, 109-120 (2003).

Proceedings of the 3rd international congress, Berlin, Germany, August 20–25, 2001. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1384-1/hbk). Int. Soc. Anal. Appl. Comput. 10, 109-120 (2003).

Let \(w\) be a nonconstant meromorphic function in the plane. In this paper the authors study a {proximity} property of some \(a\)-points of \(w\) and a {similarity} property of the \(w^{-1}\)-preimages of certain small polygons, then they apply those results to show an interesting theorem about \(15\) points.

First the authors introduce a lot of terms to express these properties, such as “Ahlfors’ sets of simple \(a\)-points or preimages of polygons” [Definitions 1 and 4], “right distributed Ahlfors’ sets” [Definitions 2 and 4], “closely located points or sets (for a given function \(\varphi\))” [Definitions 3 and 5], “close points” [Definition 3], “small sets” [Definition 5], “a curvilinear polygon \(\varepsilon\)-similar to a polygon” [Definition 6], and “\(\varepsilon\)-distant sets” [Definition 7]. They are all measured with the unintegrated Shimizu-Ahlfors characteristic function \[ A(r,w)=\frac{1}{\pi}\int \int_{| z| <r} \frac{| w'(z)| ^2}{\bigl(1+| w(z)| ^2\bigr)^2}rdrd\theta\,, \quad z=re^{i\theta}\,, \] as well as a given positive and monotone increasing function \(\varphi(r)\) tending to infinity as \(r\to\infty\).

For given \(q\) distinct complex numbers \(a_{\nu}\), \(1\leq \nu\leq q\), the authors call sets \(\Omega(r,w)\) of simple \(a_{\nu}\)-points of \(w\) in \(| z| <r\) { Ahlfors’ sets} of \(a\)-points [Definition 1], if \[ \sum_{\nu=1}^q n_0\bigl(\Omega(r,w),a_{\nu}\bigr)\geq \bigl(q-4-o(1)\bigr)A(r,w) \] holds as \(r\to\infty\), \(r\not\in E\), where \(E\) is a set of finite logarithmic measure. Here \(n_0\bigl(\Omega(r,w)\bigr),a_{\nu})\) denotes the number of simple \(a_{\nu}\)-points in \(\Omega(r,w)\) for each \(\nu\) and \(r>0\). (The number \(q-4\) is replaced by \(q-2\) when \(w\) is entire, each of which is actually attained by, for example, some elliptic and trigonometric functions, respectively.) The corresponding definition is given for sets \(\Omega^P(r,w)\) of \(w^{-1}\)-preimages of some \(q\) polygons \(P_{\nu}\) [Definition 4]. The second fundamental theorem of Nevanlinna and Ahlfors ensures the existence of those sets. As a proximity property of \(a\)-points or preimages of polygons, the authors show the existence of some rich (“right distributed” and “closely located”) Ahlfors’ sets of them by using results obtained in previous works of the authors. Here the authors say that i) points \(a\) and \(b\) in \(| z| <r\) are { closely located} (for \(\varphi(r)\)) or { close} when \(| a-b| \leq \varphi(r)rA(r,w)^{-1/2}\), and ii) a set \(S\) or a family of sets \(S_{\nu}\) in \(| z| <r\) is { small} or { closely located} when the diameter \(d(S)\) of the set \(S\) or the diameter \(d(\cup_{\nu} S_{\nu})\) of the union of the sets \(S_{\nu}\) is less than \(\varphi(r)rA(r,w)^{-1/2}\), respectively. But a collection of { \(\varepsilon\)-distant} of polygon, which are located relatively { far} from each other, is defined in a different way.

Concerning a similarity property on geometric shapes, the authors say that a curvilinear polygon \(\tilde{P}\) is { \(\varepsilon\)-similar} to a polygon \(P\), when these two figures have the same number of vertices \(t_k(\tilde{P})\) and \(t_k(P)\) and sides \(s_k(\tilde{P})\) and \(s_k(P)\), respectively, and their shapes are similar in the sense that \[ (1-\varepsilon)^4/(1+\varepsilon)^4\leq m_k(\tilde{P})/m_k(P)\leq (1+\varepsilon)^4/(1-\varepsilon)^4 \] and \[ | \alpha_k(\tilde{P})-\alpha_k(P)| \leq -4\log(1-2\varepsilon)\,, \] where \(m_k(\tilde{P})\) (resp. \(m_k(P)\)) is the ratio of the length of two sides of \(\tilde{P}\) (resp. \(P\)) emanating from a vertex \(t_k(\tilde{P})\) (resp. \(t_k(P)\)) and \(\alpha_k(\tilde{P})\) (resp. \(\alpha_k(P)\)) is the angle formed by three points \(t_j(\tilde{P})\) (resp. \(t_j(P)\)), \(j=k-1,\, k,\, k+1\).

After those definitions the authors state Theorem 1, which shows the existence of much nicer Ahlfors’ sets of preimages \(z_i(P_{\nu})\) of \(q(>4)\) { \(\varepsilon\)-distant} polygons \(P_{\nu}\), that is, there exist { right distributed} Ahlfors’ sets consisting of { closely located} and { small} curvilinear polygons \(z_i(P_{\nu})\) each of which is \(\varepsilon\)-similar to \(P_{\nu}\). The proof is done by applying Ahlfors’ covering theory to study some phenomena related to geometry of neighborhoods of the { simple} \(a\)-points and estimations for the inverse functions \(F(w)\) of \(w(z)\) around those points. Theorem 1 is a refinement of the results given in their previous paper [Complex Variables, Theory Appl. 44, No. 1, 13–27 (2001; Zbl 1022.30027)].

As an application of Theorem 1, they study the following problem which is motivated by M. Ozawa [J. Kodai Math. Sem. Rep. 20, 159–169 (1968; Zbl 0157.39501)]: Consider meromorphic functions \(w\) for which there is a set \(L^{(w)}\) of values \(a\) such that all the \(a\)-points of \(w\), \(a\in L^{(w)}\), lie on a finite collection of non-parallel straight lines. How many lines do we need to reduce \(w\) to rational functions?

Taking a function \(\varphi(r)=A(r,w)^{1/4}\), {five} ({three}) \(\varepsilon\)-distant triangles \(P_{\nu}\), \(\nu=1,2,\ldots , 5\) (resp. \(3\)), and \(\varepsilon <(1-e^{-\pi/8})/2\), the authors give an answer to the problem in Theorem 2 (resp. Theorem 3) that \(15\) (resp. \(9\)) values \(a_{\mu}\), \(\mu=1,2,\ldots , 15\) (resp. \(9\)) are enough to show that any meromorphic (resp. entire) function \(w\) in the plane with \(L^{(w)}= \{ a_{\mu} \}\) must be a rational function (resp. ploynomial).

(One might want to know whether \(15\) (resp. \(9\)) is the best possible number in this problem or not. But this seems quite difficult to answer. In fact, so is the number \(5\) (resp. \(3\)) in Theorem 1 in view of elliptic (resp. trigonometric) functions as above. On the other hand, these functions are periodic, and thus it is not likely that all the \(a_{\mu}\)-points are located on a finite collection of non-parallell lines.)

There are some typographical problems in the paper, but they are not at all serious.

For the entire collection see [Zbl 1031.35002].

First the authors introduce a lot of terms to express these properties, such as “Ahlfors’ sets of simple \(a\)-points or preimages of polygons” [Definitions 1 and 4], “right distributed Ahlfors’ sets” [Definitions 2 and 4], “closely located points or sets (for a given function \(\varphi\))” [Definitions 3 and 5], “close points” [Definition 3], “small sets” [Definition 5], “a curvilinear polygon \(\varepsilon\)-similar to a polygon” [Definition 6], and “\(\varepsilon\)-distant sets” [Definition 7]. They are all measured with the unintegrated Shimizu-Ahlfors characteristic function \[ A(r,w)=\frac{1}{\pi}\int \int_{| z| <r} \frac{| w'(z)| ^2}{\bigl(1+| w(z)| ^2\bigr)^2}rdrd\theta\,, \quad z=re^{i\theta}\,, \] as well as a given positive and monotone increasing function \(\varphi(r)\) tending to infinity as \(r\to\infty\).

For given \(q\) distinct complex numbers \(a_{\nu}\), \(1\leq \nu\leq q\), the authors call sets \(\Omega(r,w)\) of simple \(a_{\nu}\)-points of \(w\) in \(| z| <r\) { Ahlfors’ sets} of \(a\)-points [Definition 1], if \[ \sum_{\nu=1}^q n_0\bigl(\Omega(r,w),a_{\nu}\bigr)\geq \bigl(q-4-o(1)\bigr)A(r,w) \] holds as \(r\to\infty\), \(r\not\in E\), where \(E\) is a set of finite logarithmic measure. Here \(n_0\bigl(\Omega(r,w)\bigr),a_{\nu})\) denotes the number of simple \(a_{\nu}\)-points in \(\Omega(r,w)\) for each \(\nu\) and \(r>0\). (The number \(q-4\) is replaced by \(q-2\) when \(w\) is entire, each of which is actually attained by, for example, some elliptic and trigonometric functions, respectively.) The corresponding definition is given for sets \(\Omega^P(r,w)\) of \(w^{-1}\)-preimages of some \(q\) polygons \(P_{\nu}\) [Definition 4]. The second fundamental theorem of Nevanlinna and Ahlfors ensures the existence of those sets. As a proximity property of \(a\)-points or preimages of polygons, the authors show the existence of some rich (“right distributed” and “closely located”) Ahlfors’ sets of them by using results obtained in previous works of the authors. Here the authors say that i) points \(a\) and \(b\) in \(| z| <r\) are { closely located} (for \(\varphi(r)\)) or { close} when \(| a-b| \leq \varphi(r)rA(r,w)^{-1/2}\), and ii) a set \(S\) or a family of sets \(S_{\nu}\) in \(| z| <r\) is { small} or { closely located} when the diameter \(d(S)\) of the set \(S\) or the diameter \(d(\cup_{\nu} S_{\nu})\) of the union of the sets \(S_{\nu}\) is less than \(\varphi(r)rA(r,w)^{-1/2}\), respectively. But a collection of { \(\varepsilon\)-distant} of polygon, which are located relatively { far} from each other, is defined in a different way.

Concerning a similarity property on geometric shapes, the authors say that a curvilinear polygon \(\tilde{P}\) is { \(\varepsilon\)-similar} to a polygon \(P\), when these two figures have the same number of vertices \(t_k(\tilde{P})\) and \(t_k(P)\) and sides \(s_k(\tilde{P})\) and \(s_k(P)\), respectively, and their shapes are similar in the sense that \[ (1-\varepsilon)^4/(1+\varepsilon)^4\leq m_k(\tilde{P})/m_k(P)\leq (1+\varepsilon)^4/(1-\varepsilon)^4 \] and \[ | \alpha_k(\tilde{P})-\alpha_k(P)| \leq -4\log(1-2\varepsilon)\,, \] where \(m_k(\tilde{P})\) (resp. \(m_k(P)\)) is the ratio of the length of two sides of \(\tilde{P}\) (resp. \(P\)) emanating from a vertex \(t_k(\tilde{P})\) (resp. \(t_k(P)\)) and \(\alpha_k(\tilde{P})\) (resp. \(\alpha_k(P)\)) is the angle formed by three points \(t_j(\tilde{P})\) (resp. \(t_j(P)\)), \(j=k-1,\, k,\, k+1\).

After those definitions the authors state Theorem 1, which shows the existence of much nicer Ahlfors’ sets of preimages \(z_i(P_{\nu})\) of \(q(>4)\) { \(\varepsilon\)-distant} polygons \(P_{\nu}\), that is, there exist { right distributed} Ahlfors’ sets consisting of { closely located} and { small} curvilinear polygons \(z_i(P_{\nu})\) each of which is \(\varepsilon\)-similar to \(P_{\nu}\). The proof is done by applying Ahlfors’ covering theory to study some phenomena related to geometry of neighborhoods of the { simple} \(a\)-points and estimations for the inverse functions \(F(w)\) of \(w(z)\) around those points. Theorem 1 is a refinement of the results given in their previous paper [Complex Variables, Theory Appl. 44, No. 1, 13–27 (2001; Zbl 1022.30027)].

As an application of Theorem 1, they study the following problem which is motivated by M. Ozawa [J. Kodai Math. Sem. Rep. 20, 159–169 (1968; Zbl 0157.39501)]: Consider meromorphic functions \(w\) for which there is a set \(L^{(w)}\) of values \(a\) such that all the \(a\)-points of \(w\), \(a\in L^{(w)}\), lie on a finite collection of non-parallel straight lines. How many lines do we need to reduce \(w\) to rational functions?

Taking a function \(\varphi(r)=A(r,w)^{1/4}\), {five} ({three}) \(\varepsilon\)-distant triangles \(P_{\nu}\), \(\nu=1,2,\ldots , 5\) (resp. \(3\)), and \(\varepsilon <(1-e^{-\pi/8})/2\), the authors give an answer to the problem in Theorem 2 (resp. Theorem 3) that \(15\) (resp. \(9\)) values \(a_{\mu}\), \(\mu=1,2,\ldots , 15\) (resp. \(9\)) are enough to show that any meromorphic (resp. entire) function \(w\) in the plane with \(L^{(w)}= \{ a_{\mu} \}\) must be a rational function (resp. ploynomial).

(One might want to know whether \(15\) (resp. \(9\)) is the best possible number in this problem or not. But this seems quite difficult to answer. In fact, so is the number \(5\) (resp. \(3\)) in Theorem 1 in view of elliptic (resp. trigonometric) functions as above. On the other hand, these functions are periodic, and thus it is not likely that all the \(a_{\mu}\)-points are located on a finite collection of non-parallell lines.)

There are some typographical problems in the paper, but they are not at all serious.

For the entire collection see [Zbl 1031.35002].

Reviewer: Kazuya Tohge (Kanazawa)